Probability and potentials (A Blaisdell book in pure and applied mathematics)

(fE2 P).

We denote similarly by 2OO(Q,~) the space of bounded random variables with the norm of uniform convergence (which is independent ofP), and by Loo(Q,ff,P) the space of classes of elements of 2 00 (which depends on P) with the quotient norm. The norm of an element fE LOO is written Ilflloo (the essential supremum of Ifl). We use, without further reference, the following properties of LP spaces: the fact that LP is a Banach space [see e.g., Dunford and Schwartz, (67), p. 146]; Holder's inequality [Dunford and Schwartz, (69), p. 119]; and the fact that the dual of L1 is LOO [(67), p. 289]. This last result is a consequence of the Radon-Nikodym theorem [(67), p. 176] which we also have occasion to use. * 9

Here are two useful remarks: (a) Letfbe an integrable random variable, measurable relative to a sub-O"-field f§ of §". Then f is a.s. positive, if and only if:

{f(w)dP(w)

>0

for every

A

E

'§

(take for A the event {f < 0}). This implies, in particular, that two random variables f and g, both f§-measurable and integrable, are a.s. equal if and only if they have the same integral over every set in f§. (b) Letfandg be two integrable random variables. We say thatfandg are orthogonalif the product fg is integrable and has zero expectation. Denote by (j} a sub-O"-field of §", by U the closed subspace of L1 consisting of classes of f§-measurable random variables, and by V the subspace of LOO consisting of classes of bounded random variables orthogonal to every element of U. It follows from the Hahn-Banach theorem that a random variable f E 2 1 is a.s. equal to a f§-measurable function if and only iff is orthogonal to every element of V. Convergence of random variables

10 We now recall some definitions concerning convergence of random variables, limited to the case of sequences. Let (fn) be a sequence of real-valued random variables defined on (Q,§",P). The sequence (fn) is said to converge to a random variable f: almost surely ifP{w:fn(w) -+ few)} = 1 in probability ifP{w: Ifn(w) - f(w) I > e}~ n---+oo 0 for all e > 0, in the strong sense in LP if the fn and f belong to 2 P, and if E[lfn - flP] n--oo --+ 0, in the weak sense in L1 [or alternatively in the sense of the topology O"(L1,L 00)] if the functions fn and f belong to 2 1 , and if, for every random variable g E 2 00 ; lim E[fn . g]

= E[f· g],

n

in the weak sense in L2 [or in the sense of the topology 0"(L2,L2)] if the functionsfn andf belong to 2

2

and if, for every random variable g E 2

2;

lim EU"n . g] = E[f' g]. n

• It will also be proved in Chapter VIII, as an application of martingale theory.

15

11, 011, TI2, 013, TI4

A Resume of Integration Theory

We shall return to weak convergence in Ll in the study of uniform integrability. We limit ourselves here to pointing out that convergence a.s. and strong convergence in LP imply convergence in probability, and that a sequence that converges a.s. can be extracted from any sequence that converges in probability. More precisely, for every real-valued random variable f, let 7T[f] = E[lfl AI]; the function (f,g) JW+ 7T[f - g] is a pseudo-metric, which defines convergence in probability; if the sequence (fn) satisfies the property ~ 7Tlfn - fn+l] n

< 00

it converges in probability and almost surely. [See e.g., Ounford and Schwartz, (67), p. 150.]

Image probability laws Let (n,~,p) be a probability space, (E,lff) a measurable space, and fa random variable dejined on thejirst space with values in the second. The image law ofP underf, denoted by f(P), is the probability law Q on (E,lff) defined by

Dll

DEFINITION

Q(A) = P(f-l(A))

for all

A

E

lff.

This law is also called the law (or the distribution) of the random variable f.

Let g be a measurable mapping of (E,lff) into a measurable space (G/§). The following equality ("transitivity of image laws") then holds: g(f(P))

-+ T12

= (g of)(P).

Let h be a real-valued random variable defined on (E,lff); h is Q-integrable if and only if h of is P-integrable, and then: THEOREM

t

h(x) dQ(x)

=

L

h 0 f(w) dP(w).

Integration of probability laws-Fubini's theorem D13 DEFINITION Let (n,~ and (E,lff) be two measurable spaces. A family (P:Il):IlEE of probability laws on (n,~) is said to be lff-measurable if the function x ~ P:Il(A) is lff-measurable for every A E~. Let (P:Il):IlEE denote such a family in the following statement.

-+ T14

Let Q be a probability law on (E,lff). Denote by (U, 0/1) the measurable space (E X n, lff X ~. (1) Let f be a real-valued random variable dejined on (U,O/I). Each of the partial mappings x JW+ f(x,w), W JW+ f(x,w) is measurable on the corresponding factor space. (2) There exists a unique probability law on (U,O/I), which we denote by S, such thatfor every FUBINI'S THEOREM

AElff,BE~:

S(A

X

B)

=

L

P.(B) dQ(x).

(14.1)

11, D15, T16, D17 Probability Laws and Mathematical Expectations

(3) Let f be a positive* random variable on (U,Cllt). The function x is tt-measurable and

fu

!(x,w) dS(x,w)

.A./'It+

1

= IE dQ(x)

16

r f(x,w) dPa;(w)

In

!(X,w) tIP.<w).

(14.2)

This relation is still true if the random variable f, without being positive, is S-integrable,. one can then only affirm that the function w .A./'It+ f(x,w) is P a;-integrable for Q-almost every x E E. Remarks If the function f is neither positive, nor S-integrable, it may happen that the second member of (14.2) is defined although the first is not. When all the laws Pa; are equal to the same law P, the law S is called the product law of the laws Q and P, and written Q ® P. The probability space (U,Cllt,Q ® P) is not, in general, complete. The Fubini theorem is usually stated only in the case of product laws, and in the following form: with the suppositions that the factor spaces are complete, and that the functionfis measurable over the completed product space, assertion (1) is no longer true, but one can affirm that the partial maps x ~ f(x,w) [respectively, w ~ f(x,w)] are still tt-measurable (respectively, ~-measurable) for Q-almost all x E E (respectively, for Palmost all w E Q). The definition of the product of a finite number of probability laws is obvious. We do not study here infinite products of probability laws, which enter into the general theory of stochastic processes, treated in Chapter IV. D15

With the notation of No. 14, by the integral of the family Pa; with respect

DEFINITION

to Q, denoted by

L

Pa; dQ(x), we mean the image law ofS under the projection ofE

X

Q on Q.

The following theorem is obtained by combining Theorems 12 and 14. T16

THEOREM

Denote by P the law

(O,.iJO). The function x

~

In f(

a positive random variable on

w) dP.( w) is then C-measurable, and

1!(W) dP(w)

f

IE Pa; dQ(x), and by f =

L

dQ(x)L!(W) dP.(w).

This relation is still true for any P-integrable random variable f; one can then affirm that is Pa;-integrable for Q-almost all x E E.

2.

Uniformly Intesrable Random Variables

All the random variables in this section are real-valued, and are defined on the same probability space (Q,~,P). D17 DEFINITION Let.YE be a subset ofthe space ePl(Q,~,P). One says that .YE is a uniformly integrable collection of random variables, if the integrals

Jrqfl ~c} If( w) I dP( w) tend uniformly to 0 as the positive number c tends to

f

E

.YE

+ 00.

• Recall that the integral has been defined for all positive measurable functions (No. 6).

(17.1)

17

Uniformly Integrable Random Variables

Notation

11, 18, T19

Letfbe a random variable. We denote by fC the function defined by

f(W) for fC(w) =

(

If(W) I < c

0

for f(w)

o

for f(w)

>c < -c

We putfc = f - fC. Definition 17 then takes the following form: £7 is uniformly integrable if and only if there exists, for every 8 > 0, a number c such that Ilfc 111 < 8 for every function fE£7. Remarks (a) Definition 17 is clearly compatible with the a.s. equality of random variables. * Note that the random variables enter into the definition only through their absolute values. (b) Every finite family of random variables and, more generally, every family of random variables dominated in modulus by some fixed integrable function is uniformly integrable. 18

...

Let £7 be a subset of 'pI,. then £7 is uniformly integrable ifand only if the following conditions are realized: (a) The expectations E[lfll,fE £7, are uniformly bounded.t (b) Whatever 8 > 0 is, a number t5 > 0 can be found such that the conditions A E~, P(A) < t5, imply the inequality T19

THEOREM

L11(w)1

dP(w)

<•

for every

fE£7.

(19.1)

Proof To establish the necessity of conditions (a) and (b), we note that, for every integrable function f and every set A E ~ :

L11(w)1

dP(w)

< cP(A) + EII1,1]·

(19.2)

Supposing that £7 is uniformly integrable, choose c large enough so that

E[If:1l

< 812

for every

f E £7

First we obtain (a) by taking A = Q, then (b), by choosing t5 = 812c. Conversely, supposing that properties (a) and (b) are satisfied and given an 8 > 0, associate with it some t5 > 0 satisfying (b) and take c = SUP/EJf' E[lfll/t5, a finite quantity in view of (a). Apply formula (19.2), taking for A the set {If I ~ cl, whose probability is less than t5 in view of the inequality

P{lfl > c} < ! E[lfll. c

We obtain the inequality

r

J
If(w)1 dP(w)

<8

for every

fEJf,

and £7 is thus uniformly integrable. • We can thus speak of uniformly integrable subsets of £1.

t One can prove that Ca) is a consequence of Cb), if the measure P has no atomic part.

11, T20, T21

T20

Probability Laws and Mathematical Expectations

18

Let.Ye be a uniformly integrable subset of LI. Its closed convex hull is also uniformly integrable. THEOREM

Proof We begin by noting that the closure in LI of a uniformly integrable set still is

uniformly integrable. This is an immediate consequence of Theorem 19. It will thus suffice to show that the convex hull of .Ye is uniformly integrable. Denote by Us the sphere of radius e in LI, by Bc the sphere of radius c in Loo, considered as a subset of LI. Definition 17 then takes the following form: A subset .Ye of LI is uniformly integrable if and only if for every e > 0 a number c > 0 can be found such that .Ye c Bc + Us' Observe that such a sum set is convex: If it contains .Ye, it therefore contains its convex hull. Let Hand K be two uniformly integrable subsets of LI. Their union H U K is clearly uniformly integrable, and hence so is the convex hull of H U K. It then follows from the inclusion: 1 2(H + K) c the convex hull of H U K, Remark

that the sum H + K is uniformly integrable. This result can also be deduced from Theorem 19. The following is a generalization of Lebesgue's theorem. T21

Let (fn)neN be a sequence of integrable random variables that converges almost everywhere* to a random variable! Then f is integrable, and the convergence offn to f takes place in the LI norm, if and only if the fn are uniformly integrable. If the random variables fn are positive, they also are uniformly integrable if and only if THEOREM

lim E[fnl = E[fl n

< 00.

Proof Let us suppose first that fn converges to f in norm (which presupposes the integrability off) and show that conditions (a) and (b) of Theorem 19 are verified. Denote by A any

measurable set. We have

L

If.(w)1 dP(w)


If(w)1 dP(w)

+

IIf. -

fill'

(21.1)

Condition Ca) follows immediately. On the other hand, let us choose an integer N such that Ilfn - fill < e/2 for every n > N, and a number ~ such that the inequality peA) < ~ implies

L

Igl dP

< e/2,

when g ranges over the finite collection consisting of the functions h,

f2' ... ,fN,! The first member of (21.1) is then smaller than e for every n when peA) is

less than ~, and condition (b) is verified. Conversely, suppose that the functions fn are uniformly integrable. The expectations E[lfnll are then uniformly bounded, and Fatou's lemma implies the inequality E[lfll < 00. Let us show that fn converges to f in norm. We have E[lfn -

fll < E[lf~

- fCIl

+ E[lfncll + E[lhll.

(21.2)

Given e > 0, choose a number c large enough so that the two last expectations of the second member are majorized by e/3, independently of n. Choose n large enough so that the first expectation is majorized by e/3, which can be done in virtue of Lebesgue's theorem,

* Or only in probability.

Uniformly Integrable Random Variables

19

11, T22

since the functions If: - fCI are uniformly bounded and converge almost everywhere to O. The first member is then smaller than c, and the convergence in norm is established. It remains to show that the convergence of E[fn] to E[f] < 00 implies, when the fn are positive, the convergence of E[lfn - fl] to 0 (and consequently the uniform integrability of the fn)' To do this, we write

f

+ fn

= (f Afn)

+ (fv fn)'

E[f Afn] tends to E[f] from Lebesgue's theorem. On the other hand, E[f + fn] tends to 2E[f] from the hypothesis. It follows that E[f V fn] tends to E[f]. One then deduces from the formula that the expectation E[lf - fnl] tends to O. We shall prove the following theorem (due to La Vallee Poussin) completely, because it will clarify the meaning of uniform integrability. In practice, only the implication (2) => (1) (which is the easiest to establish) is used, G(t) being generally a function tP(p > I). In particular, every bounded subset of "p2 is uniformly integrable. T22 THEOREM Let YE be a subset of "pI. The following properties are equivalent: (I) YE is uniformly integrable. (2) There exists a function G(t) defined on R+ which is positive, increasing, and convex, *

such that lim G(t) =

and

+ 00,

t

t-++oo

sup E[G

leX

0

IflJ .

< 00.

(22.1)

Proof In order to establish the implication (2) => (1), given an c > 0, put a = M/ e, where Mis the value of (22. I). Choose a number c so large that G(t)/t > a for t > c. We then have If I < (G Ifl)/a on the set {If I > c} and consequently 0

f

IfI dP < .! a

{lf1:2:c}

f

(G

IfD dP

0

{l/1:2:C}

< !. M = a

e

for each functionf E YE. Definition 17 is thus verified. We establish the converse implication by constructing a function G(t) of the form

it

g(t) dt, where g denotes an increasing function, tending to

constant value gn on each interval

E[G

0

with t, which has a

rn, n + I[ (n EN). Put, for each functionfE YE, aif) = P{lfl

Let us take go

+ 00

= 0; we have

Ifll < glP{l < If I < 2} + (gl +

> n}. 00

g2)P{2

< If I < 3} + ... =.2 gnaif)· n=l

It remains to show that one may choose coefficients gm which tend to infinity with n, such that the sum .2n gnan(f) is uniformly bounded. Choose a sequence of integers Cm which increases to infinity, such that sup leX

• This property is not used for (2) => (1).

r J

If I dP {Ill :2:Cn}

< 2- n ,

11, T23

Probability Laws and Mathematical Expectations

20

which is possible by virtue of the uniform integrability. Then

fill

,,:,")Ifl dP

>

ml

mP{m

< If I < m + I}

00

00

m=Cn

m=Cn

> I P{lfl > m} = I

am(f)·

It follows that the sum 2.n2.~am(f)is uniformly bounded for f E £; but this sum is of the form 2.m gmam(f), gm denoting the number of integers n such that Cn < m. The following theorem plays a large role in what follows. We prove it only in part, since the implication (2) => (1), the more difficult to establish, is not very useful. The reader will find a complete proof in Dunford and Schwartz [(67), p. 294]. T23 THEOREM (Compactness criterion of Dunford-Pettis). Let £ be a subset of the space LI. The following three properties are equivalent: (1) £ is uniformly integrable. (2) £ is relatively compact in LI in the weak topology (J(Ll,L 00). (3) Every sequence of elements of £ contains a subsequence that converges in the sense of the topology (J(Ll,L 00).

Proof We first show that (1) => (2). Let U be an ultrafilter on £; for each function f each set E E!F, put liE)

E

£,

= Lf(W) dP(w).

From the relation IIlE)\ < E[lf/], and condition (a) ofT19, the numbers llE) are uniformly bounded in modulus. The limit

I(E)

= Hm IlE) u

thus exists for every E E!F. It is clear that the set function E ~ I(E) is additive and bounded. From condition (b) of T19, there exists for every e > 0 a number b > 0 such that P(E) < b implies Il(E) 1 < e; I is hence a measure and absolutely continuous with respect to P. There then exists, by the Radon-Nikodym theorem, a function ep ELl such that for every measurable subset E:

l(E)

=

L

ep(w) dP(w).

Assertion (2) will be established if we show that the ultrafilter U converges to topology. Now evidently lim E[f· g] = E[ep . g]

ep in the weak

u

for every function g E ,poo, which is a finite linear combination of indicator functions of sets. It then suffices to remark that every function g E ,poo is a uniform limit of such functions. The equivalence of assertions (2) and (3) is a consequence of difficult theorems from the theory of topological vector spaces [Eberlein-Shmulian theorem; see Dunford and Schwartz (67), p. 430]. We only need the implication (2) => (3), which we now establish directly. Let (fn)nEN be a sequence of elements of ,pI, whose classes belong to £; denote by:T the (J-field generated by the functions fn' and by:To the smallest collection of subsets of Q, closed under (uf, (), which contains the sets of the form {In < a} (n E N, a rational). It is easily verified that :To is a countable collection, which generates the (J-field :T. By means

21

11, T24

Construction of Measures. Radon Measures

of the diagonal procedure, extract from the sequence (fn) a subsequence the integrals Lf;'(W) dP(w) (E E,r.)

(f~t)mEN'

such that

have a limit when m --+ 00. We shall show that the sequence (f:n) is weakly convergent. It suffices for this, according to assertion (2), to show that this sequence has a single limit point in LI. Let cl> and cI>' be two limit points: these two functions are a.s. equal to ff-measurable functions [see 9(b)]; in order to establish their a.s. equality, it thus suffices to show [in virtue of 9(a)] that 4>(w) dP(w) = c/>'(w) dP(w)

L

L

for every set E E ff. Now this equality holds for E E ff o• Denote by...ll the collection of subsets E E ff for which this equality is true; it follows from Lebesgue's theorem that...ll is closed under passage to monotone limits, and from I.T19 that Jt =:T. Thus cl> = cI>' a.s., and the theorem is established.

3. Construction

cif Measures.

Radon Measures

We have as yet given no method for the construction of a measure on a measurable space. The beginning of this section indicates the two theorems which are most often used for that purpose, as well as a procedure which sometimes permits the enlargement of the a-field on which a measure is defined. We limit ourselves to the case of probability laws. Almost all the measures we shall find on our way will be Radon measures (in potential theory) or abstract measures arising from the modification of Radon measures (stochastic processes taking values in compact spaces). The obvious reference for Radon measures is Bourbaki's treatise (Integration, Chapter III and IV). The exposition we present here is very schematic, its goal being only the explanation of the roles played by the Baire and Borel a-fields. The reader will find in Chapter III the proofs of several theorems cited in this section. Extension theorems

The Daniell extension theorem is proved, for example, in Loomis [(90), p. 29]. See also III.T2S. Let Ye be a vector space ofreal-valuedfunctions defined on a set Q, which contains the constants and is closed under the operation V. Let I be a positive linear functional on Ye such that l(l) = 1. There then exists a probability law P on the a-field ff generated by Ye, such that every function f E Ye is P-integrable and T24

THEOREM (Daniell)

I(f) -'Inf(oo) dP(oo)

if and only if the following condition holds: for every decreasing sequence (In) of elements of Ye such that limfn = 0, lim l(fn) is equal to O. n

The law P then is unique.

n

(24.1)

11, T25, D26, T27, 28

Probability Laws and Mathematical Expectations

22

The following theorem can be deduced from T24, by taking for YE the set of all finite linear combinations of indicators of elements of §'o. T25 THEOREM (Caratheodory) Let §'O be a nonempty collection of subsets of 0, closed under (uf, (), and let 1 be a positive, additive set function defined on§'o such that 1(0) = 1. There exists a probability law P on the a-jield §' generated by §'0, such that P(A) = I(A) for every A E §'0' if and only if the following condition holds:

lim I(A n) is equal to 0 for every decreasing sequence n

n

(An)neN of elements of §'O such that An The law P then is unique. n

=

0.

(25.1)

Internally negligible sets

D26 DEFINITION Let (O,§',P) be a probability space. We shall say that a set A c 0 is internally P-negligible* if every §'-measurable subset of A has probability zero. T27 THEOREM Let JV be a collection ofsubsets of 0 that satisfies the following conditions: 1. JV is closed under (Uc). 2. Every element of JV is internally P-negligible. Let §" be the a-jield generated by §' and JV,. then the law P can be extended in a unique manner to a law p' on §", such that every element of JV is P'-negligible. Proof We only indicate the steps in the reasoning and leave the details to the reader. Let vii be the collection of all subsets of 0 contained in some element ofJV, and let ~ be the collection of subsets of the form F 6. M (F E §', M E vii). It is easily verified that ~ is a a-field. Since.A contains the empty set, we have §' c ~, and similarly vii c ~. Let A = F 6. M be an element of ~; put Q(A) = P(F). It can be checked that Q(A) depends only on A, and not on its representation as F 6. M. In order to show that Q is a probability law on ~, consider a sequence (An)neN of disjoint elements of~, and their union A; each is of the form F n 6. M n (Fn E §', M n Evil). Let F be the union of the F n • Since they are disjoint up to negligible sets, we have P(F) = ~n P(Fn); on the other hand, A and F differ only by an element of vii. Thus Q(A) = ~n Q(A n )· The law P' then is the restriction of Q to §". In order to establish the uniqueness of p', consider another law P" on §" satisfying the same assumptions. Every element of vii then is internally P"-negligible, and P" thus can be extended to a law on ~ such that every element of vii is negligible. This law must be identical to Q, and hence P' = P".

28 Remarks (a) This result is often applied to families JV consisting of one single internally negligible set. (b) This theorem implies the existence of the completion of a probability space (O,§',P) (No. 3), JV denoting then the collection of all subsets of negligible sets. Let §'P be the completed a-field. Every element of §'P can be written F 6. M, where F belongs to §' and M is contained in a P-negligible set NE§'. F 6. M then is included between F""'N and F 6. N, which belong to§' and differ by a negligible set. The usual approximation of real-valued measurable functions by step functions then gives the following result: A real-valuedfunction f is §'P-measurable if and only if there exist two functions g and h, §'-measurable, such that g < f < h, P{g ~ h} = O.

* Sets of probability 0 are often called P-negligible sets.

23

Construction of Measures. Radon Measures

II, T29, T30, 31

Let (n,~) be a measurable space; for each law P on (n,~), we consider the completed a-field ~P, and we denote by ~" the intersection (over P) of all these a-fields. The measurable space " (n,~) is called the universal completion of (n,~). The reader can easily check the following properties: " (1) Every law P on ~ can be uniquely extended to a law P" on" ~, and P J\,/I/'+ P is a one-to-one mapping onto the set of all laws on~." (2) Let (E,tff) be a measurable space, andfbe a measurable function from (n,~) to (E,tff);fthen " to (E,tff). " also is measurable from (n,~) (c)

We start now the study of Radon measures with some preliminaries on the relations between a-fields and topology. T29 THEOREM Let E be a compact space. A compact subset K of E belongs to the Baire a-field if and only if K is the intersection of a sequence of open sets in E.

Proof Let K be a compact Baire set of E, and let (fn) be a sequence of continuous functions, such that K belongs to the a-field generated by the fn(1.9). Let fbe the continuous mapping (fn)neN of E into RN: There exists a measurable subset A of RN such that K = f-l(A) (1.12), and we thus have K = f-l(f(K». Now RN is a compact metrizable space, and the compact setf(K) therefore is the intersection of a sequence of open sets G n • We then have K = nnf-1(G n). Conversely, assume that the compact set K c E is the intersection of a sequence (G n ) of open sets. Let gn be a continuous function with values in [0,1], equal to 1 on K and to off Gn. Then K is equal to {gn = I}, and thus belongs to the Baire a-field.

nn

T30

°

Let (Ei)ieI be a family of topological spaces, and let E be their product. (a) Assume that each E i is metrizable and separable, and that I is countable. Then the Borel a-field !!I(E) is equal to the product a-field I1ieI !!I(Ei). (b) Assume that each E i is compact (I being arbitrary). The Baire a-field !!Io(E) then is equal to IIieI !!IO(Ei). THEOREM

Proof Since the Baire and Borel a-fields are equal for metrizable spaces, and since the product E in statement (a) is metrizable, we may consider only Baire a-fields. Let us denote by (Xi)iel the coordinate functions on E; each Xi' being continuous, is measurable from (E, !!Io(E» into (Ei,!!Io(Ei» [1.10,d)]. The identity mapping of (E,!!Io(E» onto (E, IIiel !!IO(Ei» thus is measurable (I.TI2), which means that the product a-field is contained into !!Io(E). We now establish the reverse inclusion. In case (a), it follows from the existence of a countable base for the topology of E, the elements of which belong to the product a-field. In case (b), let us prove that every real-valued continuous function on E is measurable with

respect to the product a-field. This follows indeed from the Stone-Weierstrass theorem, * according to which continuous functions can be approximated uniformly on E by polynomials: where each he (k = 1, ... , n) is a continuous function on E, which depends on one coordinate only; f is obviously measurable with respect to the product a-field.t 31 Let E be a topological space, A be a subspace of E, and i be the inclusion mapping from A into E. We show that the Borel sets in A are exactly the intersections with A of Borel sets in E. Let or be the collection of these intersections. Since the mapping i is continuous, we have or c 8l(A)

* Dunford and Schwartz (67), p. 272, Loomis (90) p. 9, Bourbaki (15), Section 4, No. 2.

t Statement (b) is false for Borel a-fields of nonmetrizable compact spaces, even for finite products.

11, 32, 33, T34, T35 Probability Laws and Mathematical Expectations

24

[1.10(d)]. Conversely, or is a a-field and contains the open sets of A (intersections with A of open sets of E); or thus contains 8i(A).

32 Let X and Ybe two random variables on a probability space (n,.F,p) with values in a separable metric space E. Since the diagonal of E x E belongs to 8i(E x E) = 8i(E) x 8i(E), the set {X = Y} is an event, which has probability one if and only if the following equality holds for every measurable bounded function f, or only (I. T20) for continuous [: (32.1)

E[[(X, Y)] = E[[(X,X)]

According to I.T20, it suffices to verify (32.1) for functions [(x,y) bounded and continuous respectively on X and Y.

= g(x)h(y),

where g and hare

Radon measures We could have omitted the subject of Radon measures, and referred the reader to the treatise of Bourbaki. We have not done this because Bourbaki associates directly, with every Radon measure ft, a set function defined on the a-field of ft-measurable sets. For probability theory we need to be a little more delicate in the matter of the a-fields, and to examine the different extension procedures with more care. The case of compact spaces suffices to illustrate this point. 33 Let E be a compact space. Recall that a positive Radon measure on E is a positive linear functional ft on the space ~(E), and that a Radon measure is a linear functional on ~(E) equal to the difference of two positive Radon measures (or, alternatively, continuous in the topology of uniform convergence). Let fl be a positive Radon measure. According to the classical lemma of Dini, * every sequence of continuous functions that decreases to 0 converges uniformly to 0 on E. The linear functional fl thus satisfies condition (24.1), and the Daniell extension theorem gives us the "Riesz representation theorem": ...

T34

THEOREM

Let 4> be the mapping that associates with each bounded positive measure m

on (E,PAo(E)) the linearfunctional f

-¥I'+

fE/(X) dm(x) on ~(E). Then 4> is a bijection ofthe set

of bounded positive measures onto the set ofpositive Radon measures. We have, on the other hand, the following result: ...

T35 THEOREM Let m be a positive bounded measure on PAo(E); then m can be extended, in a unique way, to a measure mon PA(E), which possesses the following property: Let (Ki)iEI be a family of compact sets in E, which is filtering to the left. Then we have m(ni K i ) = infi m(Ki).t (35.1). The measure mthen is regular (see 111.28): For every Borel set A: m(A)

=

sup

m(K)

(35.2)

KcA Kcompact

Let us denote by 1p the inverse of the bijection 4> of No. 34; we simplify our language by calling "Radon measures" the original Radon measure ft, the measure 1p(ft) on 860 (E), and • See Bourbaki (15), §4, No. 1 (p. 53). See also X.6 below.

t This holds in particular when the compact sets and their intersection belong to 8io(E}-a result not obvious from the definition of ft.

25

Construction of Measures. Radon Measures ~

11, T36, 37, D38 /"-..

its extension "P(fl) to f1l(E), and further by writing fl instead of "P(p), "P(p,). If f denotes a Borel function, the integral of f with respect to fl can be written as p(f), (p,f), or Sf(x)dfl(X), Note that a l.s.c. (lower semicontinuous) function is Borel. We then have the following statement, * which appears in some way as a generalization of T35, since the indicator of a compact set is upper semicontinuous.

-+

T36 THEOREM We then have

Let (h)ieI be a family of l.s.c. functions, which is filtering to the right. (36.1)

The first member of (36.1) makes sense, since the function sUPih is l.s.c. Note that this result only concerns positive functions, but doesn't require their integrability. Theorems 35 and 36 can also be considered from an "abstract" point of view (see III.T33 and T34). 37 Bounded positive measures on the Borel field ~(E) of a nonmetrizable compact space E are not necessarily Radon measures. A positive measure A on ~(E) is a Radon measure if and only if it is bounded (on locally compact spaces this condition must be replaced by finiteness on compact sets), and if it verifies either one of the following (equivalent) properties: 10 Let f be a positive bounded, l.s.c. function; then we have A(f) = sup A(g)

(37.1)

+ ge"C:r g:5: I

20

A is a regular measure [Le., verifies (35.2)]

(37.2)

This implies in particular that the sum of a series of Radon measures is itself a Radon measure if and only if it is bounded. Completion of a Radon measure The following definition given for compact spaces extends in fact to the case of a locally compact and a-compact space, but more care is required for general locally compact spaces. D38 DEFINITION Let fl be a Radon measure on (E,f1l(E». We denote by f1l Jl the completion of f1l(E) with respect io fl; the elements of f1l Jl are called fl-measurable sets. The elements of the intersection p-field

where p ranges over the collection ofall positive Radon measures on E, are called universally measurable sets. If Eis metrizable, f1liE) is just the universal completion of f1l(E) (28,c).

t Bourbaki (18), Chap. 4, Section 1, Theorem 1, p. 105.

11, D39, D40, T41, 42 Probability Laws and Mathematical Expectations

26

4. Independence. Conditionina The notion of independence is seldom used in this book. The definitions given below are principally intended to prepare the reader to understand the notion of conditional independence, essential to the theory of Markov processes. Definition of independence D39 DEFINITION Let (Xi)iEI be afinitefamily ofrandom variables defined on the probability space (Q,~,P), with values in the measurable spaces (Ei,tffi)iEI' Let X be the random variable (Xi)iEI with values in the space (IIiEI E i , IIiEI tffi)' The random variables Xi are said to be independent if the law of X is the product of the laws of the Xi' Let (Xi)iEI be any family of random variables. The Xi are said to be independent random variables of every finite subfamily are independent.

if the

This definition can take the following form (No. 14): The random variables (Xi)iEI are independent if and only if for every finite set J c I, and every family (Ai)iEJ such that Ai E tff i for i E J, P{Xi E Ai for every

i E J} =

IT iEJ

P{Xi E Ai}'

The definition of independence can take another equally interesting form: D40 DEFINITION Let (Q,~,P) be a probability space and let (~i)iEI be a family of sub-(]fields of ~. These (]-.fields are said to be independent if, for every finite subset J c I and every family of sets (Ai)iEJ such that Ai E ~ifor i E J,

Definitions 39 and 40 are easily shown to be equivalent. The random variables (Xi)iEI are, in fact, independent (in the sense of D39) if and only if the (]-fields 5""( Xi) are independent (in the sense of D40). In the same way, the (]-fields (~i)iEI are independent if and only if the random variables Xi are independent, where Xi denotes the identity mapping of (Q,ff) onto (Q'~i)' T41 THEOREM Let ~1' ~2' ••• , ~ n be independent (]-.fields, and let h,f2' ... ,fn be real-valued random variables, measurable with respect to the corresponding (]-.fields ~b ••• , ~ n' The product flf2 ... fn then is integrable if and ehdf fb ... ,fn are integrable and, moreover,

42 Let X and Y be two independent real-valued random variables, and let A and ft be their respective laws. The law of the pair (X, Y) in R2 is the product law A (2) ft. The law of the random variable X + Y is the image of this product law under the mapping (x,y) ~ x + y from R2 into R. In other words, the law of X + Y is the convolution A * ft.

27

11, T43, D44, 45

Independence. Conditioning

Conditioning

The notion of conditional expectation often appears a little disconcerting at first sight. Its use in this book will present no difficulties, however, since we shall use only formal properties of conditional expectations, all summarized in this section. Let (Q,~,P) be a probability space, and f a random variable defined on (Q,~) with values in a measurable space (E,C). Let Q be the image law ofP by f Let X be a P-integrable random variable on (Q,~). There exists a Q-integrable random variable Y on (E,C) such that for every set A E C: T43

THEOREM

f

Y(x) dQ(x)

A

=J.

X(w) dP(w).

(43.1)

r1Ul

Let Y' be another random variable satisfying (43.1); then Y = Y' a.s. Proof The assertion regarding the uniqueness of Y is an immediate consequence of

remark 9(a). In order to establish the existence of Y, we begin by considering the case where X is square-integrable relative to P. Associate with every element Z of "p2(E,C,Q) the number

In (Z

0

f)X dP, which depends only on the equivalence class of Z. We thus construct a

linear functional on L2(E,C,Q), whose norm is at most equal to a function YE "p2(E,C,Q) such that

L

(Z 0 f)X dP =

IE ZY dQ

11

X112. Hence there exists

for every such Z.

The function Y is as desired. Suppose that the random variable X is positive: the random variable Y then has a positive integral over every set A E C. It is thus a.s. positive, from 9(a). Consider now the case where the random variable X is only supposed integrable. The same then holds true for its positive part X+, and for its negative part X-. The random variables X~ = X+ A n (n EN) thus belong to L2(Q,~,P), so we can associate with them random variables Y n + as above. According to the preceding remark, these random variables are a.s. positive, increase a.s., and their integrals are bounded by E[X+]. We can thus choose an integrable random variable Y+, equal a.s. to the limit of the Y n +. In the" same way construct a random variable Y_, starting with X-. The integrable random variable Y = Y+ - Y_ satisfies relation (43.1), and the theorem is established. D44 DEFINITION Let Y be an integrable real-valued random variable defined on (E,C,Q), which satisfies relation (43.1). JlPe:say that Y is (a version of) the conditional mathematical expectation of X, given f

45 Remarks (a) When X is the indicator function of an event B, Y is called the conditional probability of B, given f It is important to keep in mind that this "probability" is a random variable defined only up to an a.s. equality, and not a number. (b) Consider a partition of the set Q into a sequence of measurable sets Am and denote by fthe mapping of Q into N equal to n on An. The image measure Q on N is then defined by

11, D46, 47

28

Probability Laws and Mathematical Expectations

Let X be an integrable random variable on Q; it is very easy to compute Y:

Yen) =

f

XdP

An

P(A n )

for every n such that P(A n ) ~ O.

IfP(A n ) is zero, Yen) can be chosen arbitrarily. Suppose in particular that Xis the indicator function of an event B; then Y(n) = P(B n An)fP(A n) if P(A n) is not zero. One recognizes here the number called, in elementary probability theory, the conditional probability of B given that An has occurred. It would be tempting to say the same in the general case, and to call the value Y(x) (x E E) "the conditional expectation of X given thatf(m) = x", but this

terminology would be improper, because the random variable Y is defined only up to a.s. equality, and one can specify its value at a point x only if Q({x}) ~ O. (c) Let X be a positive nonintegrable random variable. The passage to a monotone limit used in the proof of T43 still applies, and yields a positive random variable, not necessarily finite, defined up to a.s. equality, that satisfies formula (43.1). We speak in this case of the generalized conditional expectation.

We have begun with Definition 44 of conditional expectation because we believe it is the most intuitive one. The following is a variation much more important in practice, which we use constantly. It is obtained by taking in statements 43-44: for E the set Q, for tff a sub-a-field of /F, and for fthe identity mapping of Q onto itself. The image measure Q is then the restriction of P to tff and we have the following definition: D46

Let (Q,/F,P) be a probability space, let tff be a sub-a-.field of /F, and let X be an integrable real-valued random variable. A (version of the) conditional expectation of X relative to tff is an integrable tff-measurable random variable Y such that DEFINITION

L

X(w) dP(w)

=

L

for every

Y(w) dP(w)

A

E

C.

(46.1)

I

In the following we omit the word "version." We generally use the notation E[X tff]* for Y. When tff is the a-field ff(ft" i E I) generated by a family of random variables, we speak of the conditional expectation of X relative to theft" and we write simply E[X 1ft" i El]. If X is the indicator function of an event A, we speak of the conditional probability of A relative to C (or to the ft,) and we write P(A I tff) [or peA 1ft" i E I)]. It often happens that conditional expectations are superimposed in the form E[E[fl /Ft] I /F2], where /Ft and /F2 are sub-a-algebras of /F. We then employ the notation E[XI/Ft 1/F2], which is more intelligible. Remark Returning to the notation of statements 43-44, and denoting by f/ the a-field ff(f), we have a.s. the equality E[X f/] = Y f Theorem 1.18 permits us to recover Definition 44 from Definition 46.

I

0

Fundamental properties of conditional expectations ...

47 Under this title we group all the properties of conditional expectations that we use in what follows. In particular, we state anew Definition 46, in another form. The random variables considered are defined on the space (Q,/F,P).

* The notation E"'[X] is also widely used.

29

11,48

Independence. Conditioning

1 Let X and Y be two integrable random variables, a, b, and c be constants. For any l1-field C c !F we have

PROPERTY

E[aX + bY + cl C]

= aE[XI C] + bE[Y/ C] +

c a.s.

(47.1)

<

Ya.s. One then

2 Let X and Y be integrable random variables, such that X has E[XI C] < E[yl Cl. PROPERTY

3 Let X n (n EN) be integrable random variables that increase to an integrable

PROPERTY

random variable X. The following relation then holds

I

E[X C] = lim E[X n I Cl

a.s.

(47.2)

n

4 (Jensen's inequality) Let c be a convex mapping ofR into R, and let X be an integrable random variable such that coX is integrable. The following inequality then holds PROPERTY

(47.3)

Proof The function c is the upper envelope of a countable family of affine functions L n such that Ln(x) = anx + bn. The random variables L n 0 X are then integrable, and thus L n 0 E[XI C] = E[L n 0 Xl C]

< E[c

0

Xl Cl.

It then suffices to pass to the upper envelope in the first member. If the random variable X takes its values in an interval I of the line, it evidently suffices that c be defined and convex on I.

5 Let X be an integrable random variable; the random variable E[X I C] is Cmeasurable. If X is C-measurable we have X = E[X I C] a.s. PROPERTY

(Repetition of a part of the definition of conditional expectations, and an immediate consequence of the uniqueness.) 6 Let!Z, C be two sub-l1-fields of!F such that !Z c C. Then for every integrable random variable X (47.4) E[X ICI !Z] = E[X I!Z] a.s., and in particular (47.5) E[E[X Cl] = E[X]. PROPERTY

I

(The first formula is an immediate consequence of the uniqueness. The second is deduced by taking !Z = {0,Q}.)

7 Let X be an integrable random variable and Y an C-measurable random variable such that the product XY is integrable. We then have

PROPERTY

I =

E[XY C]

YE[X / C]

a.s.

(47.6)

Proof In the case where Y is elementary, (47.6) is an immediate consequence of the definition of conditional expectations. The general case then follows by means of passage to a monotone limit. 48

By application of Jensen's inequality, taking for c the function (0), one obtains the inequality

CONTINUITY PROPERTIES

x ~ Ixl

P

(1

IIE[XI

C]ll p

< II X ll

p'

(48.1)

11, D49, 50, T51

Probability Laws and Mathematical Expectations

30

This inequality is clear for p = 00. The mapping X ~ E[X Ilt] is thus an operator of norm < 1 in LP (1 < P < (0). Now every continuous linear operator on a Banach space B is still continuous when B is given its weak topology a(B,B') [see, for instance, Bourbaki (17) p. 103; Dunford-Schwartz (67) p. 422]. The conditional expectation operator, in particular, is thus continuous with respect to the topologies a(Ll,L 00) and a(L2,L2). Let (Xn)neN be a sequence of integrable random variables that converges a.s. to an integrable random variable X. One may wonder whether this implies the a.s. convergence of the conditional expectations E[Xn Ilt] to E[X Ilt], for any sub-a-field It. Doob has shown that the answer is positive when the sequence is dominated in modulus by an integrable random variable, and Blackwell and Dubins proved in (11) that this condition could not be weakened. Conditional independence

This notion will not be used until the chapter devoted to Markov processes. The proof of Theorem 51, nevertheless, constitutes a good exercise on the application of properties 1-7, and we recommend its immediate study. D49 DEFINITION Let (a,~,p) be a probability space, and let ~1' ~2' ~a be sub-a-fields of §'. ~1 and §'a are said to be conditionally independent relative to ~2 if the following relation holds (49.1) whenever Y 1 , Ya denote positive random variables measurable, respectively, relative to the a-fields ~b ~2' 50 Remarks (a) By taking for ~2 the a-field {0,a}, the definition of independence (Nos. 39,41) is recovered. One can, moreover, define in the same way the conditional independence of several a-fields relative to a given a-field. (b) The usual procedure of approximation by step functions easily shows that, if (49.1) holds for indicator functions of sets, then it holds as stated above, and also when Y 1 , Y a, Y1 Ya are integrable. (c) The terminology is in fact rather flexible. If ~2 is the a-field .r(f) generated by a random variable f, one can say that ~1 and ~a are conditionally independent "givenf"; if ~1 and ~a are the a-fields generated by random variables fl and fa, one can say that ''/1 andfa are conditionally independent relative to §'2'" etc.

T51 THEOREM Let ~12 be the a-field generated by ~1 and ~2' The a-fields ~1 and ~I are conditionally independent relative to §'2, if and only if the following relation holds:

!

E[Yal §'12] = E[Ya ~2]

a.s.

(51.1)

for every integrable, §'a-measurable random variable Y a. Proof (a) (49.1) => (51.1). We have to verify that the two members of(51.1) have the same integral over any element of ~12' Now the collection of elements of ~12 for which this occurs is closed under the operations (umc, nmc). On the other hand, the family ri' of finite disjoint unions of sets of the form Al n A 2 (AI E ~1' A 2 E §'2) generates the a-field §'12' It thus suffices, according to I.TI9, to verify the relation: E[a 1a2E[Ya l ~12]] = E[a 1a2E[Ya l §'2]]

31

II, T51

Independence. Conditioning

where Q 1 and Q2 denote, respectively, the indicator functions of Al and A 2. Now we have (the parenthetical numbers indicate the properties used): E[a I Q2E [ Yal ff12 ]] = E[E[Q1a2Yal ffI2 ]] = E[a I Q 2 Ya]

(7) (5)

= E[E[aI Q2Yal ff 2]] = E[a 2E[a l Ya / ff 2]] = E[a 2E[Q 1 1 ff 2 ]E[Ya / ff 2 ]] = E[a2E[(Q1E[Ya / ff 2]) ff 2]]

l

= E[E[(Q 2a IE[Ya ff 2 ]) = E[Q2QIE [ Ya / ff 2 ]]·

I Iff

2 )]

(5) (7) (49.1) (7) (7) (5)

(b) (51.1) => (49.1). We have

E[Y1Yal ff 2 ] = E[Y1 Ya lff12 1ff 2 ] = E[(Y1E[Yal ff12 ]) ff 2] = E[( Y1E[ Yal ff 2]) ff 2]

I I

I

= E[ Y1 ff 2 ]E[ Yal ff 2]·

(6) (7) (51.1) (7)

The properties of conditional mathematical expectations will hereafter be used without special reference.

CHAPTER

III

Complements to Measure Theory

The larger part of this chapter is devoted to the capacitability theorem of Choquet (in its "abstract" form) and to results connected with it. The remainder of this book furnishes several important applications of Choquet's theorem, mostly to potential theory, but also to the general theory of stochastic processes. In contrast to the first two sections, which refer to results worthy of being considered as classical, the last section contains some theorems of lesser importance, of interest mainly to the professional in probability theory. It can be omitted without inconvenience, at least by readers possessing a good knowledge of Radon measures.

1.

Compact Pavings. Ana!ytic Sets

1 Let E be a set. A paving on E is a collection of subsets of E that contains the empty set; the pair (E,tff) consisting of a set E and a paving tff on E is called a paved set. This terminology is used only in this chapter and in the applications depending on it. Let (Ei,tffi)iEI be a family of paved sets. The product paving of the tff i (respectively, sum paving. of the tffi ) is the paving on the set IIiEI E i (respectively, on ~iEI E i ) consisting of the subsets of the form IIiEI Ai (respectively, ~iEI Ai), where Ai C E i differs from E i (respectively, from 0) only for a finite number of indices, for which Ai belongs to tff i . It is important to note, when the tff i are a-jields, that the product paving of the tff i is not identical with the product a-field of the tff i (the latter is generated by the product paving). Hence, there is ambiguity in using notations such as IIiEI tff i or tff x IF to denote a product paving. We shall use them nevertheless, in this chapter only. * Compact and semicompact pavings 2 Let (E,tff) be a paved set, and let (Ki)iEI be a family of elements of tff. We say that this family has the finite intersection property if niEl o K i ;;c 0 for every finite subset 10 C I. This amounts to saying that the sets K i belong to a filter or alternatively, from the ultrafilter theorem,t that they belong to some ultrafilter U on E. • A solution consists in adopting the sign @ for product a-fields, as in Neveu (105). We didn't think it useful to do so here. t Bourbaki (13), 3rd edition, Section 6, No. 4, Theorem 1.

32

33

Compact Pavings. Analytic Sets

Ill, D3, T4-6

D3 DEFINITION Let (E,C) be a paved set. The paving C is said to be compact (respectively, semicompact) if every family (respectively, every countable family) of elements of C, which has the finite intersection property, has a nonempty intersection.

For example, if E is a Hausdorff topological space, the paving consisting of the compact sets of E is a compact paving. Let C be a compact [semicompact] paving on E; then the paving C U {E} is compact [semicompact] . Properties of compact pavings T4 THEOREM Let E be a set given a compact (respectively, semicompact) paving C, and let C' be the paving obtained by closing C under the operations (Uf, fla) [respectively, (U/, flC)]. The paving C' is then compact (respectively, semicompact).

Proof Let §" be the paving obtained by closing C under (Uf). The paving C' is obtained by closing §" under (fla) [respectively under (flc)]. Since this last closure evidently preserves compactness, it will suffice to show that §" is a compact paving (respectively, semicompact). Consider thus a family (Ki)iEI (respectively, a countable family) of elements of §", which has the finite intersection property; let U be an ultrafilter such that K i E U for every i E I. Each set K i is a union U jEJi K ii of elements of C, where J i is a finite set. Hence there exists an index ji E J i such that K iii E U. * The family (Kii)iEI thus has the finite intersection property, its intersection therefore is nonempty, and that of the family (Ki)iEI is nonempty a fortiori. T5 THEOREM Let (Ei,Ci)iEI be afamily ofpaved sets. If each of the pavings C i is compact (respectively, semicompact), then so are the product paving IIiEI Ci and the sum paving LiEI C i •

Proof The proof is immediate concerning the product paving. Let Je be the paving on the sum set LiEI E i consisting of the subsets of the form !iEI Ai' where Ai = 0 for all the indices except at most one, for which Ai belongs to C i . This paving is evidently compact (semicompact). It then suffices to note that the sum paving is obtained by closing Je under (uf)· The following theorem will be used only for semicompact pavings, so we neglect the version for the compact case. T6 THEOREM Let (E,C) be a paved set, and let f be a mapping of E into a set F. Suppose that, for every x E F, the paving consisting of the sets f-1({x}) fI A, A E C, is semicompact. Then,for every decreasing sequence (An)nEN of elements of C,

En

yEn

Proof It suffices to show that with every x n f(A n) an element n An can be associated such that f(y) = x. But the family of sets of the form f-l({x}) fI An has the finite intersection property, and hence it has nonempty intersection, so it suffices to choose y in this intersection. • Bourbaki, (13), 3rd edition, Chapter 1, Section 6, No. 4, prop. 5. This proof was communicated to us by G. Mokobodzki.

Ill, D7, T8-IO

Complements to Measure Theory

34

§"-analytic sets D7 DEFINITION Let (F,!F) be a paved set. A subset A of F is said to be §"-analytic if there exists an auxiliary set E with a semicompact paving tt, and a subset BeE X F belonging to (tt X §")alJ such that A is the projection of B on F. The paving on F consisting of the §"-analytic sets is denoted by de§). * T8

THEOREM

~

is contained in de§). The paving d(§) is closed under (uc,nc).

Proof The first assertion is evident. To establish the second, consider a sequence (An)neN of §"-analytic sets. There exists by definition, for each integer n: A set En with a semicompact paving tt m A subset B n of En X F, belonging to (ttn X ff)alJ [and hence equal to the intersection of a sequence (Bnm)meN of elements of (ttn X §")a] whose projection on F is An" Let E be the product set ITn Em with the semicompact paving ITn ttn; let TT be the projection of E X F on F. Denote by Cn the cylinder based on B n in E X F, i.e., the set (ITm#nEm) x B n ; nnAn is equal to TT(nn Cn). The closure under (nc) will thus be established if we show that the set n Cn belongs to (tt x §)alJ. It suffices for this to note that each Cn belongs to (tt x §)alJ. Now let E be the sum In Em with the semicompact paving In tt m and let TT be the projection of E x F on F. We have TT(In B n) = Un An [identifying (In En) X F with In (En X F)]. It then suffices to show that In B n is an element of (tt x §")(1lJ. But this set is equal to nm In B nm , and In B nm evidently belongs to (tt x §)a. Thus the closure under (uc) is established.

n

T9

THEOREM

(a) Let (E,tt) and (F,§) be two paved sets; we then have d(tt) x d(§) c d(tt x §).

(b) Assume that the paving tt is semicompact; let A' belong to d(tt A of A' on F then belongs to de§).

X

§). The projection

Proof Let A x B belong to d(tt) x de§); D7 implies that A is contained in some Al E tta' B in some B I E §" We obviously have d(tt) x ~ c d(tt x ~);therefore, A x BI E d(tt x §") from T8. The same is true for Al X B, and thus for A x B = (A x BI ) n (AI X B). Let us now prove (b): Since A' belongs to d(tt x §), one may find a semicompact paved set (G/~) and a set A" c G x (E x F), belonging to (~ x (tt x ff»alJ' such that A' is the projection of A" on E X F. Now observe that the paving ~ x tt is semicompact, and that A" can be considered as a «~ X tff) x §)(1lJ subset of (G x E) x F whose projection on F is A. (1.

TIO

THEOREM

We have d(d(§» = de§).

Proof Let A be an d(ff)-analytic set. There exists a set E, with a semicompact paving tt, and a set A' E (tt x d(§»(1lJ such that A is the projection of A' on F. Now we have tt x d(§) c d(tff) x d(§) c d(tt x §) (T9(a» , and therefore A' belongs to d(tt x §) (T8). The conclusion A E d(§) now follows from T9(b). • These sets in fact are the same as the .iF-Suslin sets, i.e., as the sets obtained by applying to elements of .iF the operation (A) of Suslin. This result can be easily proved by the method of Choquet (26); see also Sion (110, 111). Our definition is easier to use in the setup of capacity theory and stochastic processes.

35

Ill, TIl-13

Compact Pavings. Analytic Sets

Tll THEOREM Let (F,~) and (G,~) be two paved sets, and f a mapping of F into G such that f-l(~) c d(~). Then we also have f-l(d(~» c d(~. Proof Let A be an element of d(~), and let (E,C) be a semicompact paved set, such that there exists B E (C X ~)u~ whose projection on G is A. Denote by h the mapping (x,y) ~ (x,f(Y» of E x F into E X G. The set C = h-l(B) obviously belongs to (C x d(~»u~ c (d(C x ~)u~ c d(C X ~) (T9 and T8); f-l(A) is equal to the projection of C on F, and therefore is ~-analytic (T9). T12 THEOREM d(~) contains the a-field .r(~) generated by plement of every element of ~ is ~-analytic.

~

if and only if the

com-

Proof The condition is clearly necessary. To show that it also is sufficient, consider the collection.r of all sets B c F such that B and CB belong to d(~;!T is a a-field contained in d(~, and the condition implies ~ c !T. We thus also have .r(~ c !T c d(~. We have established the necessary theorems for proving and using Choquet's theorem on capacities. The reader can therefore omit the end of this section without inconvenience. We begin with a result concerning direct images of analytic sets. T13 THEOREM Let F be a separable metric space, and ~ = fJj(F) be its Borel a-field. (a) Let E be a compact metric space, C = fJj(E) be its Borel a-field, and f be a measurable function from (E,C) into (F,~). For every C -analytic set A in E, the image f(A) then is ~-analytic in F. (b) The statement remains true if the hypothesis on (E,C) is replaced by the following: (E,C) is a measurable space; there exists a compact metric space E' and a (13.1) measurable function ep from (E',fJj(E'» to (E,C) , which maps E' onto E. (c) Let E be a Polish space *; the measurable space (E,fJj(E» then possesses the property (13.1). Proof (a) Let $' be the paving of all compact subsets of E; $' generates the Borel a-field C, and the complement of every element of $' belongs to $' u. We then have $' c C c de$') (TI2) and therefore de$') = d(C) (TI0). Let G be the graph off, and g be the mapping (x,y) ~ (f(x),y) of E x F into F x F; G is the inverse image by g of the diagonal of F x F, which belongs to fJj(F x F), i.e., to the product a-field !T(~ X ~ (11.31), and finally to d(~ X ~ (TI2). On the other hand, we have g-l(~ X ~ C C x ~ c de$') x ~ c de$' x ~), and therefore G E de$' x ~ according to Tll. Let A belong to d(tC) = d($'); A x F belongs to de$' x ~), and the same holds for (A X F) n G. We now observe that the projection of this set on F is f(A), and apply T9. (b) Let ep be a measurable mapping of E' onto E, and let A be C-analytic in E; ep-l(A) = A' then is fJj(E')-analytic in E' (Tll); we have f(A) = (f 0 ep)(A'), and (a) applied to f 0 ep shows that f(A) is ~-analytic. (c) Let E be a Polish space, N be the one-point compactification of the discrete space N, and E' be the compact metrizable space NN. We construct a Borel mapping ep from E' onto E. It will be sufficient to find a Borel subset V of E', and a continuous mapping f from V onto E, since we may then set ep(x) = f(x) if x belongs to V ep(x) = Xo if x belongs to CV,

* According to Bourbaki (14), a topological space E is a Polish space if it is separable, and can be metrized in such a way that it becomes complete.

Ill, TI4

Complements to Measure Theory

36

denoting some point in E. We now provide E with a distance compatible with its topology, under which E is complete, and choose for each nE N a countable covering (A~mEN of E by closed sets, the diameter of which does not exceed 2- n • For any finite sequence of integers s = (s(O), s(l) ... s(n)), we set Xo

and for every infinite sequence (J E NN A(F =

n As

S-<(F

where the symbol s -< (J means that (J begins with s, i.e., s = «J(O), (J(I) ... (J(n)) for some n. Let us denote by U (respectively, V) the set of all (J E NN such that A(F is empty (respectively nonempty). Since E is complete, the condition on the diameters implies that A(F = 0 if and only if As = 0 for some finite sequence s -< (J. The set of all infinite sequences that begin with a finite sequence s being open (and closed) in NN, U is open; V therefore is a closed set in NN, and a Borel set in E' = NN. Assume now that (J belongs to V; A(F is not empty, its diameter is zero, and so it must be reduced to one single pointf«J). The function f obviously maps V onto E, and we must only show that f is continuous on V. Let, indeed, «Jp)pEN be a sequence of elements of V, which converges to an element (J of V. If (Jp is different from (J, let i p be the smallest integer m such that (Jp(m) #: (J(m), and let jp be (Jp(ip - I) = (J(ip - I). The points f«J) and f«Jp) belong to A~p-I, and their distance cannot exceed 2i p-I. It is now sufficient to observe that i p tends to in"finity with p, according to the definition of convergence in NN. The separation theorem

We now give a version of the separation theorem for analytic sets. Let us denote by (F,!F) a paved set, by A and A' two subsets of F, by ~(~) the closure of ~ U {F} under (Uc, nc). We say that A and A' can be separated by elements of~(!F), if one can find two elements Band B ' of ~(!F), such that B n B ' = 0, A c B, A' CB'. T14 THEOREM Assume that ~ is a semicompact paving. Then any two disjoint ~-analytic sets A and A' can be separated by elements of~(!F).

Proof We shall begin with a lemma. Let (C n) and (D m) be two sequences of subsets of F, such that C n and D m can be separated, for each pair (n,m), by elements of~(/F). Then Un Cn and U m D m also can be separated by elements of ~(~). Let us, indeed, choose, for each pair (n,m), elements E nm , Fnm of~(~) such that Cn c E nm , D m c Fnm , E nm n Fnm = 0; if we set E' = Un E nm , F ' = U n Fnm , we obtain disjoint elements of ~(!F) that contain, respectively, Un C n and UmD m • Let now A and A' be disjoint ~-analytic sets. At the cost of a preliminary construction (that of a product space), we may assume that there exists one single semicompact paved set (E,8), such that A and A' are the projections on F of the following subsets of E x F:

nm

mn

J

=

n U J nm' n

m

l =

n U J: m n

m

where J nm = E nm x Fnm , J~m = E~m x F~m are elements of 8 x ~. We shall say that two subsets of E x F can be separated if their projections on F can be separated by elements

Ill, DI5

Compact Pavings. Analytic Sets

37

ofCC(§). The proof will consist in assuming that J and J' cannot be separated, and deducing from it that A and A' have a nonempty intersection. Let mh m2' ... , m i be integers. We set

L'm 1m2 ... mi will have a similar meaning. Since we have J = U ml L m1 , J' = U m1' L m1" and J and J' cannot be separated, the lemma implies the existence of integers mh m~ such m ,Lm ' L 'm 'm" ' , = Um2 that L m1 and L m1 ' cannot be separated. But we have L m1 = U m 2 L m1 21 12 and the lemma implies the existence of m2' m~ such that L m1m2 and L'm 1'm2' cannot be separated. By proceeding inductively, we obtain two infinite sequences m lm2 ... , m~m~ ..., such that L m1m2 ... mi and L'm 1'm2' ... m/ cannot be separated for any integer i. These sets cannot be empty, since every subset of E X F can be separated from 0. We have, therefore, E lm1

n

E2m2

n ... Eimi

:F- 0,

In the same way (Flm1

n F2m2 n ... Fim ) n (F{ml' n

F~m2'

n ... F!m) :F-

0

because these two sets belong to CC(§), and contain, respectively, A and A'. Since the pavings tff and §" are semicompact, there exist x E U i Eimi , x' E Ui E!m/' yE Ui (Fimi n F!m/); we have then (x,y) EJ, (x',y) El', and therefore yEA n A', which is the result we were looking for. Blackwell spaces

The spaces we now describe have been introduced by Blackwell (10), under the name of Lusin spaces. We have not kept this name, since it leads to confusion with the spaces called "espaces lusiniens" by Bourbaki (14). We limit ourselves to proving Blackwell's main result, and refer the reader to (10) for other interesting properties of these spaces. We first recall some measure theory. A measurable space (O,§) is said to be separable if the a-field §" is generated by a countable family (An) of its elements. The atoms of (O,§) are the equivalence classes in for the relation

°

for every

A

E§".

Every real-valued measurable function on (O,§") can be approximated by elementary functions, and therefore is constant on each atom (this extends to LCC-valued random variables). If §" is separable, generated by the sequence (An), then the atom of §" containing x is the intersection . of those An that contain x; the atoms thus are measurable sets. We denote by .Yt" the paving on R consisting of all compact sets; we have seen that d(.Yt") = d(8I(R)) (No. 13). Elements of d(.Yt") are thus called analytic sets ofR, without mentioning the paving.

D15 DEFINITION A measurable space (O,§) is said to be a Blackwell space if the following properties hold: (1) Each point of is an atom, and §" is separable. (2) For every measurable function f from (O,§) to R, and every A E §", the image f(A) is analytic in R.

°

Ill, T16, T17

Complements to Measure Theory

38

The assumption on the atoms of 0 is not at all essential: Blackwell, in fact, omits it in (10). On the other hand, it does not really restrict the generality, and it simplifies the exposition. The reader can easily verify that a statement equivalent to (2) is obtained, if one only assumes that f(O) is analytic in R. The following is another characterization of Blackwell spaces. A measurable space (O,~) is a Blackwell space if and only if it is isomorphic to some space (A,&I(A», where A is an analytic subset of R [or, more generally, a &I(E)analytic subset of a Polish space E].

T16

THEOREM

Proof Let E be a Polish space, and A be a &I(E)-analytic subspace of E; we denote by i the inclusion mapping of A into E. The characterization of &I(A) given in 11.31 immediately shows that assertion (1) of D15 is verified. On the other hand, letfbe a real-valued Borel measurable function on A, B a Borel subset of A, and let us choose some x E f(B). If we set g = f· I B + X • I A"B' we obtain a Borel function on A such that g(A) = f(B). Since &I(A) is equal to .r(i), Theorem 1.18 implies the existence of a Borel function h on E such that g = h 0 i; the image h(A) = g(A) then is analytic in R, according to T13. Conversely, let (O,~ be a Blackwell space, and let (An) be a sequence that generates~. Consider the following Borel mapping of 0 into the interval [0,1]:

and set A = f(O); A is analytic in R according to 15(2), and f is one-to-one according to 15(1). In order to show thatfis an isomorphism of(O,~ to (A,&I(A», we must only prove that the direct imagef(H) of a set H E ~ belongs to &I(A). Let G be O""H;f(G) andf(H) are analytic in R, and their intersection is empty. They can therefore be separated by disjoint Borel sets G' and H' of R (TI4). We now have but to observe that G' n A and H' n A are Borel subsets of A, equal, respectively, to f(G) andf(H). Let (O,~) be a Blackwell space, &I a sub-a-field of~, and ~ a separable sub-a-field of~. The inclusion &I c ~ then holds if and only if every atom of &I is the union of a set of atoms of~.

T17

THEOREM

Proof The condition is obviously necessary, let us prove it is also sufficient. Let B be an element of &I, B' be its complement, and (C n ) be a sequence that generates ~. Let us set few) =

~ Io~.(w) . ~ 3n

Then f is a measurable function, and the classes for the equivalence relation RI associated withf are precisely the atoms of~. The condition thus implies that Band B' are saturated for RI' and therefore f(B) and f(B') are disjoint analytic sets in R. Hence one can separate them by disjoint Borel sets D and D'. The relations B c f-l(D), B' c l-l(D') imply that B is equal to l-l(D), and therefore belongs to ~. Remarks (a) The theorem no longer is true if~ is not separable. For instance, take 0 = R, ~ = &I = &I(R), and take for ~ the a-field generated by all the singletons {x}, x E R; f!jJ and re have the same atoms, and are not equal.

39

Capacities

Ill, D18, T19

(b) Letfand g be two real-valued random variables on (Q,$0; assume some functional relation f = hog exists between f and g, h being an arbitrary function from R to R. Then f is a measurable function of g, i.e., there exists a measurable mapping h' from R to R such

thatf = h' 0 g. Indeed, the a-fieldY(g) is separable, and every atom of Y(f) is the union of a family of atoms of Y(g); we therefore have Y(f) c Y(g), f is Y(g)-measurable, and I.T18 shows thatfis a measurable function of g.

2. Capacities It isn't necessary to have read the preceding paragraph in order to take up this one; we use only the definition of analytic sets and some elementary properties of compact pavings (T6).

D18 DEFINITION Let F be a set with a paving §' closed under the operations (Uf, nf). A Choquet capacity on F (~-capacity if there is danger ofambiguity) is an extended-real-valued set function I, defined for all subsets of F, and having the following properties: (a) I is increasing (A::> B => I(A) > I(B)). (b) For every increasing sequence (An)nEN of subsets of F:

(18.1) (c) For every decreasing sequence (An)nEN of elements of §':

l( QAn) = i~f I(A n). A subset A of F is said to be capacitable

(18.2)

if

leA) = sup I(B). BE~"

(18.3)

BeA

In classical potential theory the "Newtonian outer capacity" is a Choquet capacity relative to the paving §' consisting of the compact subsets of Rn (n > 3). One then has §'" = §', and equality (18.3) expresses the fact that the outer capacity of A can be evaluated "from inside." Newtonian capacity (and similar ones) are discussed in Brelot (21) and (22). ...

T19 THEOREM (Choquet) capacitable for I.

Let I be a Choquet §'-capacity. Then every §'-analytic set is

We follow Bourbaki* for the proof of this theorem, which will comprise two lemmas. LEMMA

1 Every element of ~(1" is capacitable for I.

Proof Let A be an element of §'(1" such that I(A) > - rot; A is the intersection of a decreasing sequence (A n)n;;'l of elements of §'(1' and each An is the union of an increasing sequence (A nm )m;;'l of elements of §'. We show that there exists, for each number a < I(A), an element B of~" such that B c A and I(B) > a. We prove first the existence of a sequence • (14) Section 6, Prop. 14 (p. 138). t The result is obvious if I(A) = -

00

«18.3) holds with B = 0).

Ill, D20

Complements to Measure Theory

(Bn)n~l

of elements of §" such that B n C An and l(Cn) B 2 n ... n B n • Let us construct B 1 • We have from (18.1) that

> a,

40 where Cn = A n B1 n

leA) = I(A n AI) = sup I(A n AIm)' m

It suffices to take B1 = AIm' where m is chosen large enough so that I(A n AIm) > a. Suppose then the construction is done up to the (n - I)st step. We have by hypothesis C n- l C A, l( Cn-l) > a. Consequently,

I(Cn_l )

= I(Cn- 1

n An)

=

sup I(Cn- 1 n A nm). m

One then takes for B n a set A nm , where m is large enough so thatl(Cn_1 n A nm) = I(Cn) > a. The sequence (B n) having been constructed, put B~ = B 1 n B 2 n ... n B n and B= nnBn= nnB~. The sets B~ belong to §" and decrease, we have Cn C B~; hence I(B~) > a, and I(B) > a from (18.2). We have B n C Am and hence B C A. The set B therefore satisfies the given conditions, and the lemma is established. Now let A be an §"-analytic set. There exists an auxiliary set E, with a semicompact paving c!, and an element B of(c! x ~a;; such that the projection of B on Fis equal to A. Denote by 7T the projection of E x F on F, and by ':§ the paving consisting of finite unions of elements of c! x §". We have LEMMA

2

The set function J defined, for every H

C

E x F, by

J(H) = 1(7T(H» is a ':§-capacity on E

X

F.

Proof The function J is evidently increasing, and satisfies (18.1). Property (18.2) follows immediately from the relation

~ 7T(B

n)

=

7T( ~ B n )'

which holds, by virtue of T6 and T4, for every decreasing sequence (Bn)neN of elements of ':§. We can then conclude the proof. The set B being capacitable for J, there exists an element D of ':§;; such that D C B, and J(D) > J(B) - e (e > 0). Let C be the set 7T(D); the above equality shows that C is an element of §";;, and that C C A and l( C) > leA) - e. Construction of capacities The hypotheses of Choquet's theorem are very general but sometimes difficult to verify. One rarely encounters set functions defined at once for all the subsets of a set F; it is more natural to consider functions defined on a paving, and to try extending them to the whole of ~(F) as a Choquet capacity. We are going to describe, still following Choquet, such an extension process for "strongly subadditive" set functions. We limit ourselves to the case where these functions are positive, but this restriction is not at all essential.

Let §" be a paving on a set F, closed under the operations (Uf, nf). Let I be a set function defined on §", positive and increasing. We say that I is strongly subadditive if, for every pair (A,B) of elements of §", D20

DEFINITION

I(A U B)

+ I(A

n B)

< leA) + I(B).

(20.1)

41

Ill, T21, T22

Capacities If the sign

"<" is replaced by "=," the idea of an additive function is obtained.

T21 THEOREM Let:F be a paving on F, closed under the operations (Uf, nf), and let I be a positive, increasing setfunction defined on:F. The following three properties are then equivalent: (a) I is strongly subadditive. (b) I(P U Q U R) + I(R) < I(P U R) + I(Q U R)for every P, Q, RE:F. (c) I( Y U Y') + I( X) + I( X') < I(X U X') + I( Y) + I( Y') for every system ofelements X, X', Y, Y' of:F such that Xc Y, X' c Y'. Proof In order to show that (a) => (b), put, in (20.1), A = PUR, B = Q U R. It follows that I(P U Q U R) + I«P n Q) U R) < I(P U R) + I(Q U R).

Since the function I is increasing, this inequality implies (b). To show that (b) => (c), put P = Y, Q = y', R = X in (b). It follows that I(Y U Y' U X)

+ I(X) < I(Y U

X)

+ I(Y'

U X).

Adding I(X') to both sides, and taking into account the equalities Y U Y' U X = Y U y', Y U X = Y, Y' U X = Y' U X U X', it follows that

+ I(X) + I(X') < I( Y) + [/( Y' U X U X') + I(X')]. Apply (b) again, putting P = y', Q = X, R = X'. We thus can dominate the bracket in the I( Y U Y')

second member of the preceding inequality and obtain I(Y U Y')

+ I(X) + I(X') < I(Y) + I(Y'

U X')

+ I(X U

= I(Y) + I(Y') + I(X U

X')

X')

i.e., (c). To show finally that (c) => (a), it suffices to put, in (c), X = A nB, Y = B, X' This yields I(A U B) + I(A n B) + I(A) < I(A) + I(B) + I(A).

=

Y'

=

A.

There are then two possibilities: either I(A) = + 00 and inequality (20.1) is trivially satisfied, or I(A) < + 00, and the inequality above implies (20.1). 22 Remarks Formula (c) extends immediately by induction in the following manner: Let Xl' X 2 , ••• , X m Y I , Y 2 , ••• , Y n be elements of:F such that Xi c Y i for i = 1,2, ... , n. We then have

l( Yy.) + .p<x,) < l( YX,) + F<¥;)·

(22.1)

This formula has a more pleasing aspect when all the quantities I(Xi ) are finite. It can then be written (22.2) Inequality (b) is less useful. When all the quantities used are finite, it can be written I(P U Q U R) - I(P U R) - I(Q U R)

+ I(R) < o.

The first member is the "second difference" of the set function I.

Ill, T23

Complements to Measure Theory

42

We are going to associate an "outer capacity" with every increasing and strongly subadditive set function, and to seek conditions under which this procedure gives us a true Choquet capacity.

-+

Let F be a set with a paving :F closed under the operations (Uf, nf). Let I be a set function defined on :F that is ~ '., positive, increasing, strongly subadditive, and satisfies the following property: For every increasing sequence (An) of elements of:F, whose union A belongs to :F, T23

THEOREM

(23.1)

I(A) = sup I(An)' n

Define, for every set A

E

:Fa' I*(A) = sup I(B)

(23.2)

Re91' ReA

Then put, for every subset C of F, I*( C)

=

(23.3)

inf I*(A)t

Ae91' a A:JO

The function 1* is then increasing, and has the following properties: (a) For every increasing sequence (Xn)n2::1 of subsets of F, I.(

~

x.) = s~p I·(X.).

(23.4)

(b) Let (Xn), (Yn) be two sequences of subsets of F such that X n C Ynfor all n. Then,

I.( ~ Y.) + p·(X.) < I.( ~ x.) + p.(y.).

(23.5)

The function 1* is a Choquet capacity if and only if

I.( QA.) = i~f I(A.)

(23.6)

for every decreasing sequence (A n)n2::1 of elements of :F. Proof We begin by remarking that definition (23.2) gives us an extension of I to :Fa' and definition (23.3) an extension of 1* to all of '.l3(F). In other words, the definition of 1* is consistent. It is clear that 1* is increasing on '.l3(F). (1) Let (A n)n2::1 be an increasing sequence of elements of :Fa' and let A = Un An; then I*(A) = sUPn I*(A n)· It evidently suffices to show that I(B) < sUPn I*(A n) for every B E:F such that B c A. Let (A nm)m2::1 be an increasing sequence of elements of :F whose union is An. By replacing if necessary each A nm by the set A~m = AIm U A 2m U .•. U A nm , it can be supposed that A nm is an increasing function of n for each m. One then has sUPn I*(A n) = sUPn (suPm I(A nm)) = sUPn I(A nn ). Let B be an element of :F contained in A; then B = Un (B n Ann) and, from (23.1), I(B)

= sup I(B n n

Ann) ::;;; sup I(A nn ) n

t 1* is called the outer capacity associated with 1.

= sup I*(A n). n

43

III

Capacities

(2) The function I* is strongly subadditive on ~(F). First let A and B be two elements of :Fa and let (An), (B n) be two increasing sequences of elements of :F whose unions are equal, respectively, to A and to B. The sets An ( l Bm An U B n then belong to :F and A

(l

U (An

B =

(l

n

B n);

We thus have from (1) that I*(A U B)

+

I*(A

(l

B) = lim I(A n U B n)

+

n

Hm I(A n

(l

n

B n)

< lim [l(A n) + n

I(B n)] = I*(A)

+

I*(B).

Consider then any two subsets of F, X and Y. Denote by A and B elements of :Fa containing, respectively, X and Y. We have I*(X U Y)

+ I*(X ( l

Y)

< I*(A

U B)

+

I*(A

(l

B)

< I*(A) + I*(B).

We then obtain the sought-for inequality, I*(X U Y)

+

I*(X

(l

Y)

< I*(X) + I*(Y),

by passing to the lower envelope with respect to A and B. (3) Let (Xn)n~l be an increasing sequence of subsets of F, and let X = Un X n; then I*(X) = sUPn I*(Xn). It evidently suffices to consider the case where the second member is finite. Let h be a

number> 0; we are going to show that one can find an increasing sequence (Yn)n~l of elements of :Fa such that Y n ::J X m and I*( Y n) < I*(Xn) + h. If Y then denotes the union of the Ym which belongs to :Fa and contains X, we have from (1) that I*(X)

< I*(Y) =

sup I*(Yn) n

< sup I*(X n) + h n

and the theorem will be established, h being arbitrary. Begin by choosing for each n a set Zn E:Fa such that X n C Zn and I*(Xn) < I*(Zn) < I*(Xn) + hj2 n ; then put Y n = Zl U Z2 U ... U Zn. We show by induction that I*(Xn) < I*( Y n) < I*(Xn) + h(1 - Ij2 n), which implies the desired result. Since these inequalities are evidently satisfied for n = 1, suppose them verified up to step n. We have Yn+l = Y n U Zn+l; strong subadditivity implies the inequality I*( Yn+t)

< I*(Zn+t) +

[I*( Y n) - I*( Y n

(l

Zn+l)).

Now the value of the bracket is less than h(1 - Ij2 n), since Y n ( l Zn+t is an element of :Fa included between X n and Yn and we have, from the induction hypothesis, I*(X.)

< I*(Y.) < I*(X.) + h( 1 -

;.).

Ill, 24, 25

44

Complements to Measure Theory

where the last inequality follows from the definition of Zn+l. The formula is thus true for the (n + l)st step, and property (3) is established. It only remains to prove (23.5). This inequality is obtained immediately by passage to the limit in the relation

1{9, y,) + .!1*(

X.)

<

1{9,

Xi )

+ ~/*(Y,),

which is a consequence of property (2) [see (22.1)], the passage to the limit being justified by property (3). Finally, it is immediate that condition (23.6) is necessary and sufficient for 1* to be a Choquet ~-capacity. An application to measure theory

24 Let (Q,~,P) be a complete probability space, and let ~ be a collection of subsets of Q, contained in ~ and closed under (uf, nf). Let I be the restriction of P to ~. Evidently, I*(A) = P(A) for every element A of ~a' and it follows from (23.3) that I*(A) = P(A) for every element A of ~lJ. Conditions (23.1) and (23.6) are clearly satisfied. Let A be a ~-analytic subset of Q. Choquet's theorem implies the equality sup P(B) = inf P(C) Re ~lJ

Oe l§(f

ReA

O;)A

Consequently there exist an element B' of ~lia and an element C' of ~ali' such that B' c A c C' and P(B') = P(C'). This impI~es in particular that A is ~-measurable. This very important result was known long before Choquet's theorem. [See Saks (107), p. 50.] 25 Let us return to the general situation of Theorem 23, and suppose that the function I is additive on~. Theorem 23 then implies all the fundamental theorems of measure theory. We shall illustrate this method by proving the Daniell extension theorem (II.T24). Keep for a moment the hypotheses of T23, suppose that I is additive on ~, and that (23.6) is satisfied. Let (An)n e Nand (Bn)n eN be two decreasing sequences of elements of ~. A passage to the limit [legitimate in view of (23.6)] in the formula

shows that 1* is additive on ~lJ. Then let A and B be two elements of d(~), or more generally, two capacitable sets for 1. Let 8 be a number> 0; choose two sets A' and B' belonging to ~li' and contained, respectively, in A and B, such that I*(A')

> I*(A) -

8;

I*(B')

> I*(B) -

8.

We then have I*(A u B)

+ I*(A n

B)

> I*(A' U B') + I*(A' n B') = > I*(A) + I*(B) - 28.

Since the function 1* is strongly subadditive, and additive on d(~.

8

I*(A')

+ I*(B')

> 0 is arbitrary, we see that 1* is

45

Capacities

Ill, D26

We now prove the Daniell theorem: Let Q be a set, and .Yt' a vector space of real functions defined on Q, closed under the operations V and A. Let I be a positive linear form on Je, satisfying Daniell's condition: For every decreasing sequence of elements hn of Je+ such that lim n hn = 0, lim n I(h n) is equal to 0. Let F be the set R+ x Q; associate with each non-negative function g defined on Q the set Wg c F consisting of the points of F situated below the graph of g. The correspondence g.JVV'+ Wg is one to one. In particular, we denote by:F the paving formed from sets Wh , hE .1r+. This paving is closed under (uf, nf) in virtue of the relations

Wh 1 U Wh 2 = Wh 1Vh2 and Wh 1 1'1 Wh 2 = Wh 1Ah2 Let J be the set function defined on :F by J(Wh ) = I(h) (h EJe+). The function J is additive in view of the equality hI V h2 + hI A h 2 = hI + h2 ; it is clear, on the other hand, that J satisfies property (23.1), in virtue of Daniell's condition. We can then consider the set function J* on ~(F) and put, for every non-negative function g defined on Q, I*(g) = J*(Wg ). Let us show that J also satisfies property (23.6). The verification is reduced to that of the following statement: Let (fn)neN be a decreasing sequence of elements of .11'+, and let (gn)neN be an increasing sequence of elements of Je+ such that sUPn gn > infn fn; then I*(suPn gn) > infn I(fn)· Put h n = fa - fn. These functions belong to .1r+ and increase with n. The relation sUPn (g n + h n ) > fa implies, using (23.4), the relation

I*{ s~p (gn

+

») = s~p [/(gn) + I(h n)] > I(fo),

hn

and the desired property follows. Let Je' be the collection of functions g, measurable relative to the d-fieldff(£) generated by Je, such that Wg + E .9I(:F) and Wg _ E.9I(:F): Je' contains .1r, is closed under the operations V and A, and under passage to monotone limits; reasoning analogous to that of I.T20 shows that .1r' contains all the ff(£)-measurable functions. In order to obtain the Daniell integral, it now suffices to put

l(g) = l*(g+) - I*(g-) for every function g E.1r' such that this expression makes sense and is finite. Lebesgue's theorem on monotone convergence then reduces to formula (23.4). Right-continuous capacities

Most of the capacities encountered in potential theory are right continuous in the followmg sense. D26 DEFINITION Let E be a Hausdorff topological space, and let tff be a paving on E, closed under (uf, nf). Let I be a positive and increasing set function defined on tff. We say that I is right continuous (on tff) if it satisfies the following property:

For every set K E tff, for every number e > 0, there exists an open set U containing K such that I(K') < I(K) + e for every element K' of tff contained in U. (26.1)

Ill, T27

Complements to Measure Theory

46

We consider a paving 8 of a Hausdorff topological space E, consisting of compact subsets of 8, closed under (vI, (1/), and having the following "denseness" property: For every pair consisting of a compact set K and ofan open set U containing K, there exists a set K' c 8 such that K c K' c U. (26.2)

The collection of compact Baire sets of a locally compact space E satisfies this property. Let in fact K be compact, U an open set containing K, and f a function with compact support with values in [0, I], zero off U and equal to I on K. It suffices to take for K' the set {x:f(x) ..

> 1/2}.

T27 THEOREM Let 8 be a paving on E that satisfies property (26.2), and let I be a set function defined on 8, which is finite, positive, increasing, strongly subadditive, and right continuous. The function 1* then is a Choquet capacity, right continuous on d(8).

Proof We start by establishing an auxiliary result of a topological nature: Let Ub U2 be two open sets, K 1, K 2 compact sets contained, respectively, in Ub U2 , and L an element of 8 such that K 1 V K 2 C L C U1 V U2 • There exist two elements L 1, L 2 of 8 such that L = L 1 V L 2 , K 1 C L 1 C Ub and K 2 c L 2 C U2 •

Consider in fact the sets K 1 (I ( U2 , K2 (I ( U1 ; they are compact and disjoint in the compact space L, and are contained, respectively, in U1 and U2 • They thus admit in L two disjoint open neighborhoods, of the form VI (I L, V2 (I L, where it can be supposed that the two open (in E) sets VI and V 2 are contained, respectively, in U1 and U2 • Put M 1 = L (I ( V2 , M 2 = L (I ( VI; M 1 and M 2 are compact and their union is L. The relation K 1 c VI implies K 1 c L"" V 2 = M 1. The relation L"" V2 c L (I VI implies M 1 c VI CUI' M 2 satisfies analogous relations. The desired construction is achieved, then, by choosing two elements Nb N 2 of 8 such that M i c Ni C U i (i = I, 2) and putting L i = L (I Ni' This lemma immediately extends by induction to a finite system of pairs (Ki , Ui ) such that K i CUi' Let (Kn)n2:1 be an increasing sequence of elements of 8, and let E be a number> O. Associate with each set K n an open set

u.n containing K n such that I(K') < I(Kn) + 2En for

every K' E 8 such that K' c Un' Put V n = U1 V U2 V ... V Un' and let L be an element of tff included between K n (= K 1 V K 2 V ... V K n) and V n : L is of the form Uf=1 L i , where L i is an element of 8 such that K i c L i CUi' It follows then from formula (22.1) that I(L) - I(Kn ) = I(Uf=1 L i ) - I(Uf=1 K i ) < E. One can then deduce the inequality I(K') < I(Kn ) + E for every element K' of 8 contained in Vn • Put V = Un V n. Every set LE 8 contained in V is contained in one of the V n from the 'Borel-Lebesgue property, hence, I(L) < sUPn I(Kn) + E. This implies the following results: Suppose that the union K = Un K n belongs to 8; then I(K) = sUPn I(Kn); in other words, property (23.1) is satisfied. Let A be an element of 8(1; there exists an open set V containing A such that I(L) < I*(A) + E for every element L of 8 contained in V. Let us verify then property (23.6), which will imply that 1* is a Choquet capacity. Let (Kn ) be a decreasing sequence of elements of 8, and let K be its intersection. We show that infn I(Kn ) < I*(A) + E for every E > 0 and every A E e(1 containing K. To do this, associate with A an open set Vas above; we have K n c V, hence K n C V for n large enough, and finally I(Kn ) < I*(A) + E for n large enough. The first assertion of the statement is thus established.

nn

Ill, D28, T29

Regular Measures

47

Let X be any subset of E, let A be an element of C containing X such that I*(A) < 1* + e/2, and let V be an open set containing A such that I(K) < I*(A) + e/2 for every set K E C contained in V. The same property holds for every set L E C (j contained in V, since L is the intersection of a decreasing sequence (Kn ) of elements of C, and K n is contained in V for large enough n. We then have (I

I*(B)

< I*(X) + e

for every capacitable subset B contained in V, and in particular for every B E d(tff). This proves the theorem. Remarks When the function I is right continuous, the set function 1* * obtained by replacing the elements of Ca by open sets in formulas (23.2) and (23.3) is a Choquet capacity. See Brelot (21). Suppose that E is a locally compact space, and that C is the family of compact Baire sets of E. For every pair K, U consisting of a compact set K and an open set U::::> K, there exists an open set V E C such that K eVe U. All of the reasoning we have done above could thus be done in this case with open Baire sets, belonging to Ca. The continuity from the right of 1* then shows that in the definition of 1* the elements of Ca can be replaced by open sets belonging to C (I

(I.

3. Regular Measures This section is independent of the preceding one and of the theory of analytic sets. We restrict ourselves to the study of probability laws. Definition of regular measures D28 DEFINITION Let (Q,~,P) be a probability space, and let % be a semicompact paving of Q whose elements are ~-measurable. The law P is said to be regular (relative to f ) if the following equality is satisfied: P(B) = sup peA) for every

B

E~.

(28.1)

.Ae% .AcE

In general, the expression in parentheses is omitted. Clearly every law regular with respect to % is also regular with respect to the closure of % under (uf, nc) (T4). We can hence always suppose that % is closed under these operations. The following theorem is due to Alexandrov (l). T29

~o

be a collection of subsets of Q, containing 0 and closed under (Uf, nf, (). Let $'0 be a semicompact paving consisting ofelements of~o. Denote by ~ the a-field generated by ~o, and by f the closure of f o under (uf, nc). Let P be a set function defined on ~0, which is additive, positive, and such that P(Q) = 1. Suppose that P(B) = sup peA) for every B E ~o. (29.1) THEOREM

Let

Ae%o AcB

The function P can then be extended in a unique manner to a probability law on law is regular with respect to %.

~,

and this

Ill, 30

Complements to Measure Theory

48

Proof The existence (and the uniqueness) of the extension ofP to a probability law on:F follows from the Caratheodory extension theorem (1I.T2S) if we can show that lim n P(B n) = o for every decreasing sequence (Bn)n?l of elements of :Fo such that B n = 0. Let e be a number> 0, and choose for each n a set K n E:Ko such that K n C B n and P(Kn) ~ P(B n) - e/2 n. Put L n = K l (\ K 2 (\ ••• (\ K n ; we have L n C B n and

nn

n

P(B n) - P(L n)

< L [P(B

i) -

P(Ki )]

< e.

i=l

Since the intersection of the L n is empty, we have L n = 0 for n large enough; P(B n) is hence less than e for large enough n, and the result is established. It remains to show that the extension obtained (which we still denote by P) is regular relative to :K. We apply theorem 1.19 to show this, taking for ~ the collection :Fo, and for vii the collection consisting of all the subsets B E :F such that P(B) = sup P(A). .Aef .AcB

The reader can verify without difficulty that.A is closed under the operations (Vmc, (\mc), and hence .A = :F. Projective limits of regular measures

We will not consider here the more general projective systems, but only those which will be of use in the theory of stochastic processes in Chapter IV. The notation is that of Chapter IV. We denote by (E,C) a measurable space, by T any index set, and by V the collection of finite subsets of T. The elements of V will be denoted by small boldface letters: u, v, .... Let u be an element of V (respectively, u and v two elements of V such that u C v). We denote by 7Tu (respectively by 7TUY ) the canonical projection of ET on E U (respectively of El' on E U ). The mappings 7Tu' 7TUy are measurable over the natural product O"-fields, and we have the relations: (u, V E V,. U C v) 30

(u, v,

WE V, U

eve w).

Suppose we are given a probability law P on every measurable space (E U , r) (u E U). The family (PU)UEU constitutes a projective system of probability laws if, for every pair of elements u, v of V such that u c v, the following relation holds 7TUY (Py) = P u (30.1) (see II.Dll). We say that the projective system (Pu) admits a projective limit if there exists a probability law P on (ET, fffT) such that 7Tu(P) = P u for every

u E V.

(30.2)

As we shall see in a moment, the law P, ifit exists, is unique; it can thus be called the projective limit of the laws Pu. The collection of subsets of ET of the form 7T;;-l(A u)(u E V, A u Er) constitutes a paving fff:, which is closed under (vI, D. If we put for A = 7T;l(A u) E fff:, P(A) = P u(AJ,

49

Regular Measures

Ill, T3I, T32

we obtain a function of A, independent of the representation chosen for A. Suppose there are, in fact, two representations for A, of the form 7T;;1(AJ and 7T-;l(A y ) (u, v E U). Let w be an element of U, which contains u and v; we have 7T;;;(AJ = 7T;;(A,,) = 7TiA). It then follows from formula (30. I) that Pu(A u) = P w(7T;;;(AJ) = P w(7T;;(AJ) = Py(Ay).

Given that the paving tff~ generates the a-field tffT, the uniqueness of the projective limit Jaw follows from Theorem I. T21; there exists at most one probability law on tffT, which induces the function P on tff~. The problem of the existence of the projective limit law is reduced to the verification of Caratheodory's condition (1I.T2S) for P. The following theorem (which was communicated to us by J. Neveu) gives a simple sufficient condition for the projective system (Pu) to admit a projective limit. We shall simplify notation a little by writing Pt instead of p{t} for the finite subset of T consisting of t alone. T31 THEOREM Suppose that there exists, for each t ET, a semicompact paving ~ c tff such that the law Pt is regular relative to .x;. The projective system (PJueu then admits a projective limit. Proof We can suppose that E belongs to each of the pavings Kt. Denote by ~ (respectively by f T) the closure under (uf, (lc) of the product paving rrteu ~ (respectively rrteT ~); from T4 we see that each of these pavings is semicompact. Denote on the other hand by f~ the paving on ET consisting of the subsets of the form 7T;;l(AJ (u E U, A u E .Jt'J; f~ is contained in ~, and hence is semicompact.

Let us first show that the law P u is regular relative to~. To do this we apply Theorem 29, taking for ~ the collection of finite unions of sets of the form TIteu At (At E tff), and for ~ the collection of finite unions of sets of the form TIteu Kt (Kt E ~). We are left with verifying the relation pu(rr At) = teu

sup p(rrKt ).

Kte)("t Kt cAt

teu

To this end suppose we are given a number E > 0, and take numbers Et > 0 (t E u) such that !teu Et = E. Choose for each t E U a set Kt E~, such that Kt c At and P(At""Kt) < Et. We then have, denoting by B t the inverse image in E U of the set At""Kt under the projection of E U onto the factor space of index t, pu[(rr At)""'(rr Kt)] teu "'" teu

~ Pu( teu U Bt) < ! P,(At""Kt ) < E. 'eu

The law P u is thus regular relative to ~ for each U E U. The set function P on tff~ satisfies the hypotheses of T29 relative to the paving f~; it can thus be extended to a regular probability law relative to ~ on tffT. The following theorem implies that every probability law on the Baire a-field of a compact space is regular. T32 THEOREM Let f be a semicompact paving on a set Q, closed under (uI, (lc), such that the complement of every element of f belongs to f". Every probability law P on the a-field F generated by f is then regular relative to f.

Ill, T33, T34

Complements to Measure Theory

50

Proof Let
KcA

We have
K running over the collection of all compact subsets of A. This amounts to the regularity ofP. It is well known that every separable and metrizable space E can be imbedded in a compact metrizable space F. If E is Polish, E is analytic in F (T13), and therefore universally measurable (No. 24). According to the above reasoning, every law on fJ6(E) is regular. Another proof of this result (due to Prokhorov) is given in Neveu (105), p. 61.

Regular measures relative to a compact paving The following theorems illustrate the extension process which leads to Theorem 11.35. The reader can remark that the point lies in the replacement of semicompact pavings by compact ones. T33 THEOREM Let ('o,S;,P) be a probability space such that the law P is regular relative to a compact paving :Yt c S;. Let (Ki)ieI be a family of elements of :Yt, which is filtering to the left, and whose intersection K belongs to S;. We then have P( K) = inf P( K i ). i

Proof We may and do assume that :Yt is closed under (uI, nc). Let l' be a countable subset of I such that inf P(Ki) = inf P(Ki), and let K' = K i . We leave it to the reader ieI'

ieI

n

ieI'

to verify that P(K'",,-Ki) = 0 for every i E I. We prove that K'''''-K is negligible. According to the regularity of P, it suffices to show that P(L) = 0 for every LE:Yt contained in K'",,-K. Since L n K i) = 0, the compactness of :Yt implies the existence of some

(n

ieI

j

E

I such that L n K j

= 0. Hence L c

K'''''-Kj , which implies P(L)

= O.

T34 THEOREM Under the same hypotheses as above, denote by :Yt' the compact paving obtained by closing :Yt under (uf, na), and by S;' the a-field generated by S; and by yt". There then exists on S;' a law p', which extends P and which is regular relative to yt",. this law is unique. Proof Consider the following ordered set f/: an element of f/ is a triple (2, ri, Q), where 2 is a paving closed under (uf, nf) such that :Yt c 2 c :Yt', ri is the a-field generated by S; and 2, and Q is an extension of P to ri, regular with respect to 2. On the other hand, the order relation -< on f/ is defined by (2, ri, Q) -< (2', ri', Q') if and only

51

Ill, 35

Regular Measures

if 2 c 2' (which implies C§ c C§'), and Q' induces the law Q on C§. The set Y is nonempty, since it contains (%,:F, P), and the reader can easily prove, using T29, that Y is inductive. According to Zorn's lemma, Y contains a maximal element (2, C§, Q). The existence part of the statement will be proved if we show that assuming 2 =F: %' leads to a contradiction. Let K belong to %'''''''2; K is the intersection of a family (Ki)iel of elements of %, which is filtering to the left. Let l' and K' have the same meaning as in the proof of T33 ; the reasoning of T33 shows that K'''''''K is internally Q-negligible. The law Q can therefore (II.T27) be extended to a law Q' on the a-field C§' generated by K and C§, such that Q'(K'''''''K) = O. Let 2' be the compact paving consisting of all sets (H (I K) U L, where Hand L belong to 2. Since C§' is the collection of all sets (A (I K) U (B (I (K), where A and B belong to C§, the verification of the regularity of Q' relative to 2' easily reduces to the fact that Q'«(K) = sup Q'(L), with LE 2', L c (K. This follows from the relation Q'«(K) = Q«(K')

=

L

sup Q(L), with LE 2, L c (K'. Therefore (2', C§', Q')

E

Y, a

L

contradiction to the maximality of (2, C§, Q). The uniqueness of the law P' is obvious: T33 determines the value of P'(K) for every K E %', and the regularity condition then determines uniquely P' (A) for every A E:F'. Hence, when P is a Radon law on the Baire a-field of a compact space, the law P' is the canonical extension of P to the Borel a-field. 35 The following statements are left to the reader as exercises. We denote by (O,:F, P) a probability space, such that P is regular relative to a compact paving % c :F, closed under (uf, (la). (1) Assume that P is complete. Let A be a set such that A (I K is measurable for every K E %; prove that A is measurable. (2) We say that a set U c 0 is open, if K""" U E % for every K E %. These sets are the open sets for a topology on 0 (generally not Hausdorff). Let (Ui)i el be a family of open sets, which is filtering to the right, and let U = U Ui ; then P(U) = sup P(Ui ) (prove iel

iel

that every K E % contained in U is contained in some Ui , and use regularity). (3) Let I be a non-negative lower semicontinuous (l.s.c.) function, with respect to this topology, and let e be >0. Denote by Uk' for k EN, the open set {f> ke}, and set '" kelu I I = k7N

k

\u k+l

Prove that III is l.s.c. ({Ill> ke} = UH1 is open). Let (h)iel be a family of non-negative l.s.c. functions, which is filtering to the right: Then E[I] = sup E[.t:] (first reduce the iel

problem to the case of bounded functions, and then, by considering the functions I:, to the case of l.s.c. functions taking only the values 0, e, 2e, ... , Ne. If g is such a function, N-1

E[g] =

L P{g > ke}. Conclude by applying (2».

k=O

CHAPTER

IV

Stochastic Processes

1.

General Properties

of Processes

1 Notation We denote by T an index set, which we call the time set, although the interpretation of the parameter t ETas a time (we often speak of "the instant t") requires that T at least be given the structure of an ordered set. Most of the time, T will be an interval either of the extended line ~ ("continuous case"), or of the set of integers Z, possibly compactified by the addition of the points + 00, - 00 ("discrete case"). We denote by (E, C) a measurable space, which we call the state space. Generally, E will be a compact metrizable space, or be easily imbedded in such a space. Definition of stochastic processes

D2

A stochastic process [with time set T and state space (E,tf)] is a system (n,~,p,(Xt)teT) consisting of A probability space (n,~,p) and Afamily (Xt)teT of random variables defined on (n,~ with values in (E,C). The random variable (Xt ) is called the state of the process at time t; the function t ~ X t ( co) from T into E is called the path of co; the measurable space (n,~ is called the base space of the process. * We usually omit the word "stochastic," and simplify our language further by speaking, when it causes no ambiguity, of the "process (Xt)teT" or even of the "process (Xt )". DEFINITION

Equivalent processes

D3 DEFINITION Consider two stochastic processes hat1ing the same time set T and the same state space (E,G): (n,~,p,(Xt)teT) and (n',~',p',(X;)teT)· We say that the processes (Xt ) and (X;) are equivalent if P{Xt1 E AI' X tll E A 2 , ••• , Xt"E An} = P'{X:1 E AI' X:lI E A 2 , • •• , X:n E An} (3.1) for every finite system of times t 1 , t 2 , ••• , t n and elements Ab A 2 , ••• , An of G. • Added in prOOf There is a tendency now to distinguish between a stochastic process (D2) and a random function, the difference being that in the latter case no probability law is given on the base space (O,oF). This terminology certainly is very convenient.

S2

53

General Properties of Processes

IV, 4, D5, 6, D7, 8

4 The importance of the notion of equivalence follows from the following considerations: A stochastic process is a mathematical representation of a natural phenomenon whose evolution is governed by chance. Suppose that we have observed a very large number of independent realizations of this phenomenon. We know then with arbitrary precision (thanks to the laws of large numbers) the expression that figures in formula (3.1) for the arbitrarily large number of times t I , t 2 , ••• ,tn, but observation can help us no further. In other words, "Nature can give us stochastic processes only up to an equivalence." The probabilist is thus free to choose, from a class of equivalent processes, those he desires to work with. We shall see later how useful this freedom of choice can be. Here is a notion similar to that of equivalence, but more restrictive, which we use often at the end of this chapter: D5 DEFINITION Let (Xt)teT and (Yt)teT be two stochastic processes defined on the same probability space (O,j'=",P), with values in the same state space (E,tff). The process (Yt)teT is a modification of the process (Xt)teT if Y t = X t a.s. for every t ET.

First canonical process

°

6 Consider a stochastic process (O,j'=",P,(Xt)teT)' Denote by fthe mapping of into ET, which associates with each WE the point (Xt(w))teT of ET, i.e., the path of w. The mappingf is measurable when ET is given the product a-field tffT (see No. 1.12). One can thus consider the image law f(P) on the space (ET,tffT). Denote by Yt the coordinate mapping of index t on ET. The processes (O,j'=",P,(Xt)teT) and (ET,tffT,f(P),( Yt)teT) are then equivalent (by the very definition of image laws), and we can make the definition:

°

D7

DEFINITION

With the same notation as in No. 6, the process (ET,tffT,f(P),( Yt)teT)

is called the first canonical process associated with (or equivalent to) the process (Xt).

Two processes (Xt ) and (X;) are equivalent if and only if they are associated with the same canonical process. This canonical process is almost never used directly when the time set T is uncountable: The a-field tffT contains, in fact, only those events which depend on at most a countable infinity of the variables Yt, while the more interesting properties of processes (continuity of paths, for example) depend on all the random variables. The first canonical process is mainly used as a stage in the construction of more elaborate processes, as we shall see later.

Construction of processes 8 Recall the situation envisaged in No. 4: We have observed a certain "random phenomenon" in the course of its development, and we desire to represent it by means of a process. It is natural to begin by constructing the simplest of the processes in the equivalence class, Le., the first canonical process of this class. We thus use the measurable space (ET,tff~, and the coordinate mappings (Yt)teT' It remains to construct a probability law P on this measurable space, such that

P{ Y t1 E AI' ... , Ytn E A n}= ep(tb

••• ,

t n ; AI' ... , An)

54

Stochastic Processes

IV, T9, 10

for every finite subset u = {t I , ... , t n} of T, and every finite collection AI' ... , An of measurable subsets of E, where the functions 4> are given by observation of the phenomenon. The construction is possible if and only if the set function Al

x A2 X

•..

x

An ~ 4>(tb t 2, ... , t n; AI, A 2, ... , An)

can be extended to a probability law P u on (EU,CU), a probability law uniquely determined by the function 4> (from Theorem I.T21, applied to the collection of finite unions of subsets of E U of the form Al X A 2 X ••• X An). It is necessary on the other hand that (8.1) for every pair of finite subsets u, v of T such that u c v, 7Tuv denoting the projection of E V on E U • We recognize here the definition of a projective system of probability laws (111.30), and see that the possibility of the construction of the law P is equivalent to the existence of a projective limit for the projective system (Pu). Such a projective limit does not necessarily exist; Theorem II1.T31 gives us a simple sufficient condition for its existence. In the most important case for applications, where E is a compact space, a very simple proof of the existence of the law P can be given independently of the results of Chapter Ill.

-+-

T9 THEOREM Let E be a compact space, and let C be the Baire a-field of E. For each finite subset u ofT, let P u be a probability law on (EU,CU). If the laws P u satisfy condition (8.1), there exists a probability law P on (ET,CT) such that 7Tu (P) = P u for every finite subset u of T, U 7Tu denoting the projection of ET onto E • Such a law is unique. Proof Let CCf(ET) be the vector space consisting of the continuous functions defined on the compact space ET, which depend only on a finite number of variables. We know from the Stone-Weierstrass theorem that CCf(ET) is dense in CC(E T). Let g be an element of U CC/E~; there exists a finite subset u of T and a continuous function gu defined on E such that g = gu ° 7Tu. It follows from relation (8.1) that the integrals

r

JEU

gu dP u depend

only on g, and not on the representation chosen for g. If we associate this integral with g E CC/ET), we define a positive linear form of norm I on CCf(ET), which extends by continuity to a positive linear form of norm 1 on CC(ET), i.e., to a Radon law P on ET. We have, for every finite subset u of T and every continuous function gu on E U

r

JET

gu0'7TudP=r gudPu

JE

U

by the very definition of P. Relation (9.1) follows then from I.T20, and the same theorem yields the uniqueness of the law P. Second canonical process 10 Let E be a compact space, with the Baire a-field C = fJlo(E) , and let (Xt)teT be a process with values in (E,C). Let (ET,CT,p,( Yt)teT) be the first canonical process associated with the process (Xt ). Since the law P is a Radon law on the compact space ET, it is possible to extend it to a law P on the Borel a-field fJl(E T), which satisfies the condition of theorem II.T35. We thus define a new process equivalent to the process (Xt ): T" (E T ,fJl(E ),P,( Yt)teT)

IV, 11, 12, D13

Separable Processes

55

which we call the second canonical process associated with (Xt ). It should be remarked that the random variables (Yt ) are now measurable when E is given the Borel a-field. Although the a-field fJl(E T) is incomparably richer than the a-field eT = fJlO(E T), the second canonical process is not really satisfactory. It will rather serve us as a tool for the construction of "nice" processes, i.e., processes whose paths are as regular as possible.

2.

Separable Processes

The notion of separability appears extremely important to us. We point out, however, that it will be possible to avoid it in the rest of this book. The only indispensable numbers of this section are 20-22. All the processes envisaged in this section, with the exception of those in the appendix, have as time set an interval T of R, and take their values in a compact metrizable space E. We begin with an example (due to Doob) showing that the same event can have different probabilities for equivalent processes. 11 Consider the two processes (Xt ) and (Yt ) defined as follows with state space R, time set the interval [0,1], and base space (Q,~,P) the interval [0,1] with Lebesgue measure. We set for every co E [0,1] Xt(co) =

°

Y,(w)

=

{~

for t #: co for t

=

co

These two processes are equivalent (we have X t = Yt a.s. for each t). The set of co with continuous paths is measurable for the two processes. It has probability 1 for the first, probability for the second.

°

Definition of separable processes 12 Let (Xt)teT be a stochastic process with values in E, defined on the space (n,~,p). Let I be a subset of T, K a compact set in E. We denote by V(I,K) the set of co whose paths "remain in K for the times in I," namely, V(I,K) = {co: for every t

E

I, X t ( co)

E

K}.

Suppose that I is a countable set. It is then clear that V(I,K) E~, and that P[V(I,K)] = inf P[V(u,K)].

(12.1)

u finite

ueI

Let (Q,~,P,(Xt)teT) be a process with values in E. We say that the process is separable (relative to the collection of compact subsets of E) if for every open interval leT and every compact set K c E: D13

DEFINITION

E ~

(13.1)

(a)

V(I,K)

(b)

P[V(I,K)] = infP[V(u,K)] u

where

U

runs over the collection offinite subsets of I .

(13.2)

IV, 14, 15, D16

Stochastic Processes

56

14 Remarks (a) Properties (13.1) and (13.2) are then verified in the obvious way for any interval leT, open or not. (b) Let f be a continuous function defined on E, with values in a compact space F. If the process (Xt ) is separable, then so is the process (f 0 X t ) with values in F. (c) A process (X t ) whose paths are right (or left) continuous is separable. Indeed, under this hypothesis we have V(I,K) = V(I n Q,K)

for every open interval I,

where Q denotes the set of rational numbers. Property (13.1) is thus verified, and we can easily verify the equality (13.2). Example of a separable process

15 Let (Xt ) be a process with values in the compact metrizable space E. Consider the second canonical process associated with (Xt ) (see No. 10), T

T\

(E ,~(E j,P,( Yt)teT)' A

Denote by I an open interval of T, by UI the collection of finite subsets of I, and by K a compact subset of E. The sets V(u,K), V(I,K) are compact in ET, and hence measurable Since V(I,K) is the intersection o~ the family (V(u'K)ue UI) of compact subsets, which is filtering to the left, and the law P is a Radon law, it follows from II.T35 ~_hat A

A

P[V(I,K)] = inf P[V(u,K)]. ueUI

The second canonical process is thus separable. It should be noted that the metrizability of E plays no role, and that the hypothesis according to which I is an interval has not been used (see Nos. 26-29). This example shows that every process with values in E is equivalent to a separable process. This is a weak form of a result due to Doob, which we shall prove later, according to which every process admits a separable modification (TI9)-this last result actually uses the metrizability of E. Universal separating sets

D16 DEFINITION With the same notation as in D13, a countable subset SeT is said to be a separating set for the pair (I,K) if P[V(S nI, K)] = inf P[V(u,K)]. u finite uel

(16.1)

A countable set SeT, dense in T, is said to be a universal separating set ifS is a separating set for every pair (I,K). It is easy to construct a set S separating for a given pair (I,K). It suffices to choose fin'ite subsets Un of I such that for each n E N

P[V(umK)]

and to put S

=

< u inf finite uel

Un Un'

P[V(u,K)]

+ 1, n

'-----

IV, T17-T19

Separable Processes

57

The existence of a universal separating set S for a given process (Xt ) by no means implies the separability of the process: S is still a universal separating set for every version of (Xt ), separable or not. Example Let (Xt) be a process with right-continuous paths. Every countable dense set in T is then a universal separating set.

T17 THEOREM There exists a universal separating set for every stochastic process with values in E (compact metrizable). Proof We keep the notation of the preceding numbers. Let f be a countable collection of compact subsets of E such that the complements of the elements of f constitute a base for the topology of E. Let J be the collection of open intervals in 1 with rational endpoints. For each pair (I,K) (I E J, KEf), denote by SI,K a separating set for 1 and K. Let S be a countable dense set in T, containing the union of the sets SI,K (I E J, KEf). We will show that S is a universal separating set. Indeed, let J be an open interval of T, and let L be a compact subset of E. J is the union of an increasing sequence (Im)mEN of elements of J, L the intersection of a decreasing sequence (Kn)nEN of elements of f. It suffices to show that, for every finite subset U of J P[V(S

n J, L)]

< P[V(u,L)].

But we have u c I m for large enough m and it follows that P[V(S

n J,L)]

= p[n V(S n

n J,K n)]

= inf P[V(S n

< inf P[V(S n n

n J,K n)]

Im,K n )]

< inf P[V(u,Kn )] =

P[V(u,L)].

n

The following theorem allows us to study the regularity of the paths of a separable process. We denote by Xw(w) (where W is any subset of T) the closure of the image of W under the mapping t ~ X t(w); X w( w) is also the intersection of the sets KEf such that wE V(W,K). ..

T18 THEOREM Let (n,~,p,(Xt)tET) be a process with values in E, and let S be a universal separating set for this process. (a) Suppose that the process (Xt) is separable. There then exists a P-negligible set A E~, such that for every wE n""-A (18.1) Xi w ) = XSf"'Iiw) whatever be the open interval JeT. (b) Conversely, if the law P is complete, condition (18.1) implies that the process (Xt) is separable.

We shall prove simultaneously Theorem 18 and the following theorem: ..

T19

THEOREM

If the law P

is complete, the process (Xt) admits a separable modification.

Proof The symbols J and f have the same significance as in the proof of T16. For each pair (I,K) (I E J, KEf), put A(I,K) = V(S

n I, K)""- V(I,K)

and let A be the union of the sets A(I,K). Suppose that the process is separable and admits S as a universal separating set. The set A is then P-negligible. Let J be an open interval,

IV, T19

58

Stochastic Processes

(Im ) an increasing sequence of elements of f such that J = U m I m , and W an element of n which does not belong to A. We verify relation (18.1): The set x:;(W) [respectively, X s (')iw)] is the closure of the union of the sets X1m(w) [respectively, XS(,)lm(W)]; it thus suffices to verify that XI m(w) = XS(')1 m (w) for w f/= A. But the first (respectively, the second) member is the intersection of the elements K of :ft, which contain it, i.e., which are such that WE V(lm,K) [respectively, WE V(8 n I m , K)]. It then suffices to remark that the relations W E V(lm,K) and W E V(8 n I m , K) are equivalent for W f/= A and K E:ft. Conversely, suppose condition (18.1) is satisfied. We have the equivalences (where K is a compact set c E): WE V(J,K) <=> x:;(W) c K

X,;((0

c K <=> X s (,)J(W) c K

X s (')iw) c K <=> WE V(8

for

W f/= A

n J, K).

The sets V(J,K) and V(8 n J, K) thus differ only by a subset of A, i.e., by a negligible set. The process (Xt ) is thus separable, and admits 8 as a universal separating set. We now proceed with the proof of Theorem 19. For each t ET, define a function Yt(w) in the following manner. We put if

Xt(W)

n XI (')s(w),

E

(19.1)

IeJ lel

and in the opposite case, we take for Ylw) an arbitrary point of the set that appears in (19.1) (which as the intersection of a family of nonempty compact sets, filtering to the left, is nonempty). We show that the functions Y t are measurable, and that the process (Yt ) is a separable modification of (Xt ). Since the set 8 is a universal separating set, we have for each pair (I,K) (I E f, K E :ft) that V(I n (8 U {t}), K) c V(I n 8, K) and P[V(I n (8 U {t}), K)] = P[V(I n 8, K)]. The set B(I,K) = V(I n 8, K)"""V(I n (8 U {t}), K) is thus negligible, and so is the union B of the sets B(I,K) (I E f, K E :ft). Now the relation (19.1) is verified for W f/= B. It then follows that X t = Y t a.s., that Y t is a random variable, and that the process (Yt) is a modification of (Xt ). It is clear that Xt(w) = Ylw) for every W when t belongs to 8. We thus have from the construction of Y t that Yt(W)

E

n Y (')s(w) I

for every

t

and every w.

leJ lel

Let then J be an open interval of T, and L a compact subset of E. The properties Ylw) EL

for every

t

EJ

Yl w) E L

for every

t

E8

and

nJ

are equivalent from (19.2). The process (Yt ) is hence separable (cf. DI3).

(19.2)

59

Separable Processes

IV, 20, 21

Remarks (1) For every subset W of T, denote by Gw( w) the set of points of T X E of the form (t,Xt(w)) (for t E W). Condition (18.1) is equivalent to the relation

for every

(19.3)

whatever be the open interval JeT, the closures being taken in the compact space T x E. We leave the verification of this equivalence to the reader. Here is one of its consequences: Let F be another compact metrizable space, and let f be a continuous mapping of T X E into T X F. Suppose that the process (Xt ) satisfies condition (18.1). We show that the process (Yt) defined by Yt(w) = f(t,Xt(w)) satisfies it also. It suffices to note that f(GJ(w)) = f(GJ(w)) = f(GsnJ(w)) = f(Gsni w )).

The equality (18.1) for (Yt ) is obtained by projecting on F. (2) It should not be thought that separability implies any regularity for the paths of a process. Let f be an arbitrary mapping of T into E. Define a "deterministic" process by choosing a space Q consisting of a single point w, for which we put Xlw) = f(t). This process has only a single path, which can be very irregular. It is nevertheless separable. The set V(I,K) is, in fact, identical either to the whole space Q (and one then has P[V(u,K)] = 1 for every finite subset u of I), or to the empty set (and there then exists a finite subset u or I such that V(u,K) = 0). The conditions of Definition 13 are thus satisfied. Oscillatory discontinuities of paths

20 Letfbe a mapping of an interval T of R into a Hausdorff topological space E. We say that f is free of oscillatory discontinuities* if the limit from the right

f(t+) = limf(s) s-+t s>t

exists at every point t of T (except for the right endpoint of T) and if the limit from the left

f(t-)

= limf(s) s-+t s
also exists at every point t of T (except for the left endpoint of T). We begin by dealing with real-valued functions and by giving (following Doob) a simple criterion for the absence of oscillatory discontinuities. 21 Let fbe a mapping of the interval T into R. Denote by a, b two finite real numbers such that a < b and by u a finite subset ofT, of which the elements are SI' S2' ••• , Sn (arranged in increasing order). Define by induction the times t b t 2, ... , t n E U in the following manner: Let t 1 be the first of the elements Si of u such that f(Si) < a, or Sn if there exists no such element. Let t k be, for every even [respectively, odd] integer k, 1 < k < n, the first of the elements Si of u such that Si > t k - 1 and f(Si) > b [respectively, f(Si) < a]. If no such element exists, we set t k = Sn. Consider the last even integer 2k such that one actually has f(t 2k- 1 ) < a and f(t 2k ) > b. If no such integer exists we put k = O. The intervals (tbt2), (t3,t 4), ••• ,(t2k-bt2k) of u • Such a function is called a "regulated" function by Bourbaki ("fonction reglee").

IV, T22

Stochastic Processes

60

represent the periods of time when the function f is rising from a to b, whereas the intermediate intervals represent the periods when f is descending from b to a. The number k is called the number of upcrossings byf(considered on u) of the interval [a,b] and denoted by U(f;u; [a,b D.

(21.1)

We similarly define the number of downcrossings by f(considered onu) of [a,b] to be D(f;u;[a,bD

=

U( - f;u;[ -b,-aD.

(21.2)

Upcrossings and downcrossings are also defined on intervals of the form ]a,b[, by replacing strict inequalities by weak inequalities in the definition of the times t i • * Now let S be an arbitrary subset of T. We set U(f;S;[a,bD =

sup U(f;u;[a,bD.

(21.3)

u finite ueS

Definition (21.2) is extended in an analogous manner. The principal interest in these numbers comes from the following theorem. -.

T22 THEOREM Let f be a real-valuedfunction defined on a compact interval T. The function f is free of oscillatory discontinuities if and only if U(f;T;[a,bD < 00 for every pair of rational numbers a, b such that a < b.

Proof Suppose that there exists a point t E T where the function f has an oscillatory discontinuity-for example, suppose it has no left limit. A sequence of points (tn) can then be found that increases to t in such a way that lim infj(tn) = c n-+ 00 nodd

>d=

lim sup f(t n ). n-+oo

neven

Choose then two rational numbers a and b such that d < a < b < c. It is immediately verified, by extracting finite subsets of the set of points t n , that U(f;T;[a,bD = + 00. The converse follows from a property which the reader can easily prove: If r, s, tare three times such that r < s < t, we have U(f;[r,t];[a,bD

< U(f;[r,s];[a,bD + U(f;[s,t];[a,bD + 1.

Let oc and {J be the endpoints of T. Suppose that the function f has no oscillatory discontinuities; one can then associate with each point t ET an open interval It containing t, such that the oscillation offin each of the intervals It n ]t,{J]. [oc,t[ n It is strictly less than b - a. We can cover the interval T with a finite number of intervals It 1 , It 2 , ... , It k . Arrange in increasing order of magnitude the points oc and {J, the points tl' t2 , ••• , tk and the endpoints of the intervals It 1 ,It, ... ,It. We obtain a finite set of points: oc = So < SI < ... < Sn = 2 k (J, such that the oscillation off on each of the intervals ]Si,Si+I[ is less than b - a. We thus have U(f;]Si,si+I[;[a,bD = 0, and consequently also U(f;[Si,si+I];[a,bD < 1. The inequality cited above then gives us U(f;T ;[a,bD < 2n - 1 and the converse is established. t

* The numbers U([;u;[a,b)), D([;u;[a,b)) have the advantage of defining l.s.c. functions of/for the topology of simple convergence. This property extends to the number of uperossings or of downcrossings on an arbitrary set S. t This proof was communicated to us by P. Gabriel.

61

IV, T23, 24

Separable Processes

Remark Let S be a countable dense set in T. Suppose that the function f is defined only on S, and that U(f;S; [a,b]) < 00

for every pair of rational numbers a, b such that a < b. It can then be proved, exactly as above, that the functionf admits a left and right limit through S at every point of T. We apply Theorem 22 to the theory of stochastic processes. Let (n,~,p,(Xt)tET) be a separable stochastic process with values in a compact metrizable space E. The set of WEn whose paths arefree ofoscillatory discontinuities is then measurable.

T23

THEOREM

Proof Let:Ye be a countable collection of continuous real-valued functions which separates

the points of E [for each pair of distinct points x, y of E, there exists a function h E :Ye such that hex) ~ hey)]. The function t ~ Xt(w) is free of oscillatory discontinuities, if and only if each of the functions t ~ h Xt(w) (h E £) is free of them. Denote by Uh(w;S;[a,b]) the number of upcrossings by the function t ~ h Xt(w) (considered on SeT) of [a,b], and, suppose, to simplify the argument, that T is a compact interval. The set of WEn whose paths are free of oscillatory discontinuities is the intersection of the sets 0

0

< oo} (23.1) where a and b are two rational numbers such that a < b, and where h belongs to :Ye. Since {w: Uh(w;T;[a,b])

there are countably many of these sets, it suffices for us to verify that each of them is measurable. In order that Uh(w;T;[a,b]) = + 00, it is necessary and sufficient that one can find, for every integer k, disjoint open intervals with rational endpoints I b 12 , ••• , Ik , contained in T and such that Uh(w;lj;[a,b]) > 0 for j = 1,2, ... , k. Since the collection of systems of intervals of this type is countable, it suffices to verify that the set {w: Uh(w;I;[a,b]) = O}

(23.2)

is measurable, which it is, since it is identical to the union: V(I, h- 1([ -

00,

b])

U ('

l.) V(I, h-1([p, p rational

+ ooD)

p>a

As the set of WEn whose paths are free of oscillatory discontinuities is measurable, its probability will not be affected, if the measure P is completed. Let S be a universal separating set for the process (Xt ); if w does not belong to the exceptional set A considered in the statement of Theorem 18, the reader can easily verify that, for every finite subset u of T, and every h E :Ye, 24

Remarks

Uh(w;u;[a,b])

< Uh(w;S;[a,b]).

i A, Uh(w;T;[a,b]) < Uh(w;S;[a,b]).

It then follows that we also have, for w

Since the reverse inequality is clear, we can replace the inequality by an equality. Thus, in order that the paths of the process (Xt ) be a.s. free of oscillatory discontinuities on the compact interval T, it is necessary and sufficient that Uh(w;S;[a,b]) < + 00 a.s. for every functionh E :Ye, and every pair of rational numbers a, b such that a < b. The following theorem will be useful in the theory of Markov processes.

IV, T25, 26, D27

62

Stochastic Processes

T25 THEOREM Let (12,§",P,(Xt)teT) be a stochastic process with values in the compact metrizable space E. Suppose that the set 120 of cv E 12 whose paths are free of oscillatory discontinuities has probability I, and that for each t E T Xl cv) = Xt+(cv)

a.s. on

120 ,

where Xt+(cv) denotes the right-hand limit of the path of cv at the time t. There then exists a process equivalent to (Xt) whose paths are free of oscillatory discontinuities and are right continuous. Proof Consider the set §"o of elements of F contained in 120 • §"o is clearly a a-field, and the restriction Po of the law P to §"o is a probability law on (12o,§"o). On this new probability space, consider the process Ylcv) = Xt+(cv) (t ET) It is clear that the processes (Xt ) and (Yt ) are equivalent.

The following construction is often useful. Denote by E(T) the collection of mappings from T into E that are right continuous and free of oscillatory discontinuities. E(T) is a subset of ET:~Let (Zt)teT be the restrictions to E(T) of the coordinate mappings, and let r§ be the a-field on E(T) generated by the functions Zt, t E T. If we associate with each cv E 120 the function t ~ Yt(cv), we define a measurable mappingf from (12o,~o) into (E(T),r§). The process (E(T)/§,f(Po),(Zt)teT) is equivalent to the process (Xt), has paths as regular as the process ( Yt), and presents a canonical character that renders it very helpful on occasion.

Appendix to Section 2 An extension of the notion of separability *

26 Let (12,§",P,(Xt)teT) be a stochastic process with values in a measurable space (E,C). We make no particular hypothesis here, on T, on the measurable space, or on the law P. In particular, the latter satisfies no hypothesis of "regularity" in the sense of Chapter Ill, Section 3. Let Kbe a semicompact paving on E, consisting of elements of C. We call a condition any family (Kt)teT of elements of K. Let I be any subset of T. We say that the element cv E 12 satisfies the condition (Kt)teT for t in I if we have Xlcv) E Kt for every tEl. The set of all cv E 12 that have this property is denoted by V(Kt , t E I); it evidently belongs to §" when I is countable. D27 DEFINITION We say that the process (Xt) is separable relative to the condition (Kt)teT if the set V(Kt, t E T) belongs to §" and if P[V(Kt, t ET)] = inf P[V(Kt, t E u)].

(27.1)

ueT u finite

We say that a countable subset S of T is a separating set for the condition (Kt)teT P[V(Kt , t

E

S)] = inf P[V(Kt, t

E

u)].

if we have (27.2)

ueT u finite

* See Meyer (94). This paper contains no detailed proofs, so we have included one here. The results of this appendix will not be used in the sequel.

IV, 28, T29

Appendix to Section 2

63

Example Suppose that T is an interval of R and that E is a compact metrizable space. Separability such as defined in No. 13 is equivalent to the separability of the process, in the sense of Definition 27, relative to all the conditions of the form (Kt)teT where

Kt

K

for

t E

J

= (E

for

t E

T""J

K ranging over the collection of compact sets of E, J over the open intervals of T.

Remarks (a) There actually exist separating sets. Consider a finite subset Un of T, such that P[V(Kt , t E un)] is within at most Iln of the second member of (27.2), and put S = un Un' It is clear that S is separating for the condition (Kt)teT' On the other hand, one cannot establish the existence of a universal separating set. (b) Let S be a separating set for the condition (Kt)teT' and let Z be a countable subset of T which contains S. It is evident that V(Kt, t E Z) C V(Kt, t E S), and these sets have the same probability, as they differ only by a negligible set. It can immediately be deduced that the set V(Kt, t E S) ~ V(Kt, t E S'), where Sand S' are two separating sets, is negligible. (c) The process (Xt) is separable with respect to a condition (Kt)teT' if and only if the set 28

(28.1) is P-negligible, where S denotes a separating set for the condition. We have just seen that this property does not depend on the separating set chosen. The following theorem generalizes the result of Doob which we established in No. 17. T29 THEOREM Let C be any collection of conditions. There exists a process (Yt ), equivalent to the process (Xt ), which is separable relative to the elements of C. Proof Consider the first canonical version of the given process:

We render the process (Yt ) separable by extending the law P in such a manner that all sets of the form (28.1), relative to the conditions which belong to C, become negligible for this extension. It is known (II.T27) that such an extension is possible if and only if every countable union ofsets of the form (28.1) is internally P-negligible. We establish this point and the theorem follows. First fix some notation: With I denoting a countable subset of T, we denote by YI the projection map of ET on El. We say that a subset A of ET is cylindrical with base in El if we have A = y I l( YI(A». For example, every set of the form V(Kt , t E I) is cylindrical, based in El. We reason in the following manner: We suppose that there exist conditions (K~)teT (n EN) such that the set U n[V(K~, t E sn)"" V(K~, t ET)], where each sn is a separating set relative to the condition (K7)teT, contains a non-negligible measurable set a. We then modify the set a without changing its probability, and show that a contradiction is obtained. The set a is of the form Yi 1( a J ), where J is a countable subset of T and a J is measurable in E J • Let I be the countable set J U (U n sn), and let a' be the set

U n

[V(K~,

t

E

sn)""V(K;, t

E

I)].

Stochastic Processes

IV, T29

64

The set fJ = (J.""- (J.' has a strictly positive probability, since it differs from (J. only by a negligible set. It is cylindrical, has a measurable base in El, and is contained in the set

U

[V(K:, t E 1)""- V(K~, t ET)].

n

We are going to shrink that

fJ a little more.

Let H be the collection of finite subsets heN such

p[nV(K:,tEI)] =0. neh

Since the set H is countable, the set

fJ'

=

U

heH

[n

neh

V(K:, t El)]

is negligible. Set y = fJ""-fJ'. This set is again a cylinder based in El, and its probability is strictly positive. Here then is the contradiction we meet: Since y is nonempty, choose a Z E y, and set x = YI(z) = (Xt)tel' We are going to construct a point y = (Yt)teT""-1 of ET""-I such that Yt E K: for every t E T""-I, and every n such that X t E K: for every t E I. In other words, if we put z' = (x,y) E El X ET".I = ET, we have

z'i U n

[V(Kf, t E I)""-V(K~, t ET)].

(28.2)

But this is impossible, since z and z' have the same projection on El, and hence both belong to y, which is contained in the above set (28.2). Let N x be the collection of integers n such that X t E K~ for every t E I [or z E V(K:, t El)]. As the paving .Y{ is semicompact, a family (Yt)teT"' 1 satisfying the above properties can be found if and only if for every finite subset h of Nx we have

n K: '# 0

neh

for every s E T""-I.

But this condition is actually satisfied. We have in fact z E y, hence z i fJ', hi H, and consequently p[nneh V(K~\ t El)] > O. Since the set I is separating for these conditions, this probability is equal, for every SE T""-I, to p[nneh V(K:, t El u {s})]. We thus have, a fortiori, that P{ Y s E K: for every n Eh} > 0, which excludes the possibility that the set nneh is empty.

K:

3. Measurable Processes. Stopping Times This section serves principally as an introduction to the terminology we use in what follows. It will perhaps be to the reader's advantage to study it in a summary fashion, and return to it later according to his needs. In order to avoid overburdening the text, we have limited ourselves to processes whose time set is the half-line R+. The reader can easily adapt the results to the other usual time sets.

65

Measurable Processes. Stopping Times

IV, 30, D31, 32, D33, 34

Processes adapted to an increasing family of a-fields

Let (n,~ be a measurable space, and let (~t)tER+ be a family of sub-a-fields of ~ such that the relation s < t implies ~s c !Ft. We say that (~t) is an increasing family of sub-a-fields of~, and we call ~t, for each t E R+, the a-field of events prior to t. We denote by ~ <Xl the a-field generated"by the union of the a-fields ~t, and we set 30

~t+

= s>t n ~s

(t

E

R+).

The family (!Ft) is said to be right continuous if !Ft =

~t+

for every t

E

R+.

Remarks (a) The family of a-fields (~t+)tER+ is right continuous. (b) When the index set is N instead of R+ (or more generally when the index set is dis-

crete) the notion of a right-continuous family and the definition of the a-fields ~t+ evidently have no meaning. D3! DEFINITION Let (Xt)tER+ be a stochastic process defined or a probability space (n,~,p) and let (~t)tER be an increasing family of sub-a-fields of ~. The process (Xt ) is said to be adapted to the family (~t) if X t is ~t-measurable for every t E R+.

Let (Xt) be a stochastic process. It is natural to consider (Xt) as a process adapted to the family of a-fields ~t = .9""(Xs' s < t). Example

32 The preceding definitions have this intuitive significance: If we interpret the parameter t as time, and each event A E ~ as a physical phenomenon, the sub-a-field !Ft consists of the events that occur prior to the instant t. The !Ft-measurable random variables are hence those which depend only on the evolution of the universe prior to t. In particular, imagine that an observer watches the appearance of a certain phenomenon in the universe, and notes the time T(w) when this phenomenon is produced for the first time. The event {T < t}, which occurs if and only if the phenomenon considered is produced at least once before the instant t, or at that instant, is evidently prior to t. From this comes the interest in the following definition. D33 DEFINITION Let (n,~ be a measurable space, and let (~t)tER+ be an increasing family ofsub-a-fields of ~. A positive random variable T defined on n is said to be a stopping time [relative to the family (~t)*] if T satisfies the following property: The event {T

< t} belongs to ~t

for every t E R+.

(33.1)

34 Remarks (a) Every random variable equal to a positive constant is a stopping time. (b) Let T be a positive random variable, which satisfies the condition

{T < t} E ~t for every t E R+. It is clear that we then have {T

(34.1)

< t} E ~t+e for every t E R+ and every e > o. In other

words, Tis a stopping time relative to the family of a-fields (~t+)tER+. In particular, if the family (~t) is right continuous, relation (34.1) implies that T is a stopping time relative to the family (~t). (c) We often allow stopping times to take the value + 00. • Or (better): A stopping time of the family ("t) (added in proof).

Stochastic Processes

IV, D35, T36-T40

66

D35 DEFINITION Let T be a stopping time relative to the family of (J-fields We denote by ~ T the collection of events A E ~ such that

(~t)tER+'

00

A n {T

< t} E ~ t

for every t

E

(35.1)

R+.

We call ~ T the (J-field of events prior to T. The reader will immediately verify that these events do constitute a (J-field, and that if the stopping time is equal to the constant t, the (J-field ~t is recovered. Properties of stopping times The stopping times figuring in the following statements are all relative to a single family of (J-fields (~t)tER + • T36 THEOREM Let Sand T be two stopping times. Then the random variables SAT and S V T are again stopping times. * The proof is immediate. T37

THEOREM

Let T be a stopping time. Then T is ~ T-measurable.

The proof is immediate. T38 THEOREM Let T be a stopping time and S an that S > T; S is then a stopping time.

~ T-measurable

random variable such

Proof We verify the relation {S < t} E ~t for every t E R+. The event {S ~ t} belonging to ~ T' we have {S < t} n {T < t} E ~t. It then suffices to note that this intersection is equal to the event {S < t}. Here is a generalization of formula (35.1). T39 THEOREM then have

Let Sand T be two stopping times, and let A be an element of ~s. We (39.1)

Proof In order to verify that

A n {S

< T} n

{T < t} E ~t for every t

E

R+

it suffices to write the left-hand side in the form

[A n {S

< t}]

n {T

< t}

n {S A t

< TAt}.

Each of these three events belongs to ~t. The first, by reason of the relation A E ~s; the second, from the fact that T is a stopping time; the third, finally, follows from the fact, which the reader can easily verify, that the functions SAt and TAt are ~t-measurable. T40

THEOREM

Let Sand T be two stopping times such that S

< T.

~s c ~T'

Proof Let A be an element of ~s' From T39 we have A=An{S
• It can also be proved that S

+ T is a stopping time, but we shall not need this result.

We then have

67 T41

{S >

Measurable Processes. Stopping Times

IV, T41, T42, D43, 44

Let Sand T be two stopping times. The events {S T} then belong to the a-jields .?Fsand .?F T' THEOREM

< T},

{S

=

T}, and

< T} E.?FT from T39, and consequently {S> T} E.?FT by taking complements. Denote by R the stopping time SAT. Then R is .?FR-measurable (T37), and hence.?FT-measurable (T40). It then follows that the events {R = T} = {S = T}, {R < T} = {S < T} belong to .?F T' These events also belong to .?Fs, since Sand T play symmetric roles. Let (Tn)neN be a sequence of stopping times. It can be verified immediately that sUPn T n is again a stopping time, which implies in particular that the limit of an increasing sequence of stopping times is a stopping time. We have more complete results in the following case. Proof We have {S

Suppose that the family (.?Ft) is right continuous. Let (rn) be a sequence of stopping times. The random variables lim infn~oo T m lim supn~oo T n are then stopping times. Suppose, moreover, that the sequence (Tn) is decreasing, and denote its limit by T. We then have .?FT = n .?FT' n

T42

THEOREM

n

Proof We prove only the second assertion. We have.?F T c nn.?FT n from T40. Conversely, let A be an element of this latter a-field. We have A n {Tn < t} E.?F t for every t E R+, and hence also Un [A n {T n < t}l = A n {T < t} E.?Ft for every t. This implies that A n {T < t} E.?Ft+ for every t, and we conclude by using the equality .?Ft+ = .?Ft.

The following notation is often used in what follows. D43

Let t be a positive number, and n a positive integer. We denote by t(n) the k single number of the form 2n (k E N) such that DEFINITION

k-l
2n

-

In addition we set + oo(n) = + 00. Let H be a positive random variable. We denote by H(n) the random variable m ~ (H(m»(n). It is evident that t

< t(n) and limn~oo t(n)

= t.

Let Tbe a stopping time. It is clear that the random variables T(n) are then stopping times. Elementary examples of stopping times 44 Let (~t) be an increasing and right-continuous family of sub-a-fields of ~, and let (Xt) be a stochastic process adapted to the family (~t), with real values, and with rightcontinuous paths. It is easy to extend the results to processes with values in an LCC space. (a) Let B be an open subset of R. Put DB(m) = (inf {s: Xs(m) E B}

+ 00

if this set is nonempty otherwise

We have

{m: DB(m)

< t} =

U {m:

r rational

Xr(m) E B},

r
from the right-continuity of the paths. The event on the left thus belongs to ~t, and this implies that DB is a stopping time, from remark 34(b). This stopping time is called the first passage time (or hitting time) of B.

IV, D45, T46

68

Stochastic Processes

(b) Suppose moreover that the paths of the process (Xt) are free of oscillatory discontinuities. Define the following functions by induction (where X s- denotes a left-hand limit, and where e is a number > 0): To(ro) = 0 T

() _ (inf {s: s > Tn(ro), IXi ro ) - Xs_(ro) 1 > e}. n+l ro + 00 if the above set is empty.

These functions are then stopping times. We show this only for the case n = 1, the general case being treated by induction in an obvious manner. Let t be a number> O. Given a rational number h < t, a number e' > e, and an integer m > 0, consider the set D h •m of pairs of rational numbers (r,s) such that O
and

1 s-r<-.

m

Denote by A h , e', m the event

{ro:

for some

(r,s) E

Dh,m

\Xiro) - Xr(ro) 1 > e'}

which evidently belongs to §'t. The relation T1(ro) < t implies the existence of an h, an e', and an integer m o such that ro E Ah,e',m for every m > m o [the rational numbers rand s, taken from opposite sides of T1(ro)]. Conversely, the reader can easily verify that the existence of h, e', mo having these properties implies the relation T1(ro) < t. The set {T1 < t} is thus §'t-measurable, and T 1 is thus a stopping time. It is often necessary to give proofs similar to the above-very easy, but extremely boring. Fortunately, there exists a general theorem that permits us to dispense with them (see 52). Measurable processes D45 DEFINITION Let (Q,§',P) be a probability space, and let (§'t)tER+ be an increasing family of sub-a-jields of §'. Let (Xt)tER+ be a stochastic process defined on this space, with values in a measurable space (E,tff). We say that (Xt) is progressively measurable with respect to the family (§'t) if, for each t E R+, the mapping (t,ro) ~ Xt(ro)from [O,t] X Q into (E,tff) is measurable with respect to the a-jield &I([O,t]) X §'t.

The process (Xt) is said to be measurable (without reference to a family of a-fields) if the mapping (t,ro).A./'.I'+ Xlro) is measurable over the product a-field &I(R+) X §'. This definition, in fact, can be obtained from the preceding by taking all of the §'t identical to §'. A process measurable relative to the family (§'t) is evidently measurable, and adapted to the family (§'t). The following theorem* shows to what degree the converse is true. We state it only for real-valued processes, although it can easily be extended to processes with values in an LCC space. T46 THEOREM Let (Xt) be a measurable real-valued process, adapted to the family of a-jields (§'t)tER+' There then exists a modification of the process (Xt) progressively measurable with respect to the family (§'t).

* Borrowed from the article of Chung and Doob (47), where it is completed by considerations relative to the separability of the modification. The adjective progressively measurable also comes from this article.

69

Measurable Processes. Stoppirw Times

IV, T46

Proof Denote by M the collection of classes of real-valued random variables defined on (O,~,P), and consider on M the metric TT of convergence in probability (11, No. 10). Every stochastic process (Yt ) determines a mapping of R+ into M, which associates with each t E R+ the class Yt of Y t. Consider the collection of processes (Yt) such that: (1) The map t ~ Yt takes its values in a separable sUbspace of M; (2) The inverse image of every open sphere of M under this map is a Borel subset of R+; (3) This map is the uniform limit of a sequence of measurable elementary functions with values in M. These properties are not independent: (3) evidently implies (1) and (2), and (1) and (2) imply (3) from 1.17. We can consider every process (Yt) as a function on R+ X 0, namely (t,w) ~ Yt(w). Under these conditions, the collection of processes envisaged appears as a vector space [use (3)] closed under sequential pointwise convergence [use (1) and (2)]. It contains, on the other hand, all processes of the form Yt(w) = I(t) Y(w), where I is the indicator function of an interval of R+, and Y is a bounded random variable. It therefore contains all bounded measurable processes (1.T20), and a last passage to the limit shows that all measurable processes have properties (1)-(2)-(3). Consider now a measurable process (Xt) adapted to the family (~t). For each nE N, there is a process (X~) having the following properties: (1) The function t ~ X~ is elementary and measurable: there exists a partition ofR+ into a sequence of Borel sets A; (k EN), and a sequence of random variables Hr: (k E N) such that X~ = Hr: for tEA;. (2) TT[X t -

X;]

< 2n1+1 for every t.

Modify the random variables Hr: in the following manner: Let If sr: belongs to Af:, put

If s; is not in Af:, denote by such that

s; be the inf of the set A;.

G; any random variable, measurable with respect

to ~B:+'

(46.1) Such random variables do exist. We might for instance take, (t1') denoting a sequence of elements of which decreases to sf:,

A;

G: = Hm inf X

tfJ

l'

Inequality (46.1) is satisfied for every kEN. It is easily verified that the process (Y;) defined by for

IV, T47, D48, T49

70

Stochastic Processes

is progressively measurable with respect to the family that

(~t).

On the other hand, it follows

Then set

li~ Y:(w)

l't(w) =

{°

for every point (t,w) where this limit exists. when the above limit fails to exist.

The process (Yt) is progressively measurable with respect to the family (~t), and it follows from 11.10 that it constitutes a modification of the process (Xt ). We resume the study of modifications of processes when we treat the theory of martingales (Chapter VIII). We limit ourselves for now to an example of a progressively measurable process. T47 THEOREM Let E be an LCC space, and let (Xt ) be a stochastic process with values in E, adapted to a family of a-fields (~t), and having right-continuous paths. The process (Xt) is then progressively measurable with respect to the family (~t). The same conclusion is true for a process with left-continuous paths. Proof Let t be a number

> 0, and n a positive integer. For every integer k < 2 n and every

set

and complete this definition by putting X: = Xt. It is clear that the mapping (S,W) ~ X:( w) of [O,t] x Q into E is measurable when the first space is given the a-field &l([O,t]) X ~t. The same then holds for (s,w) .I\,f\/'+ Xi w), since this mapping is the limit of the preceding one as n~ 00. The case where the paths are left-continuous is treated similarly. The theorem extends, moreover, to any Hausdorft'space E, given its Borel a-field (1.15). The following notation is used for the rest of the book, with the random variable H being, in general, a stopping time. D48 DEFINITION Let (Xt)teR+ be a measurable stochastic process defined on (Q,~,P) with values in a space (E,C), and let H be a random variable defined on Q, with values in R+. We denote by X H the random variable w .I\,f\/'+ X H(w)( w). This function is indeed a random variable, since it is equal to the composition of the two measurable functions w ~ (H(w),w) and (t,w) ~ Xt(w). The case where the time set of the process is R+ U {+ oo}, and where H is allowed to take the value + 00, is also frequently encountered. The following theorem, for example, is true in this situation without the finiteness restriction. T49 THEOREM Let (Xt ) be a progressively measurable process with respect to the family of a-fields (~t), and let T be a (finite) stopping time. The random variable X T is then ~ T-measurable.

71

Measurable Processes. Stopping Times

IV, 50, D51, T52

Proof We have to show that, for every measurable subset A of the state space E and every t E

R+, we have

Put S = TAt. This set is equal to [{Xs EA}

n {S < t}]

U

[{Xt EA}

n

{T = t}].

It thus suffices to show that X s is ~t-measurable. This follows immediately from the fact that X s is the composition of the measurable functions: w -AN'~ (S( W ),w) from (o,~t) into ([O,t] X 0, ~([O,t]) X ~t), and (s,w) ~ Xlw) from ([O,t] X 0, ~([O,t]) X ~t) into E.

Measurability of hitting times

50 We have noted that a process can be considered as a function defined on the product set R+ X O. We now use this remark in a more systematic manner (and we use it much more in Chapter VIII). Thus, given a function X defined on R X 0, with values in a measurable space (E,C), we denote by X t the partial mapping w ~ X(t,w), and we identify the function X with the family (Xt)teR+ of these partial mappings. This permits us to say, for example, that X is a process if the family (Xt) is. We say also that a subset A of R+ X 0 is progressively measurable with respect to the family (~t) if its indicator function is identified with a progressively measurable process with respect to this family. The reader can verify that the progressively measurable processes are identified with random variables on R+ x 0, given the a-field of progressively measurable sets. D51

Let A be a subset ofR X O. We denote by D A (the "debut of A") the function defined on 0 by DEFINITION

DA(w)

=(

inf {t: (t, w) EA}

+ 00

if the above set is empty.*

The idea of using Choquet's theorem to establish the measurability of functions of this type is due to Hunt, but the following theorem has been established by Blackwell and Freedman.

-+-

Suppose that the family (~t)teR+ is right continuous, and that each a-jield ~t is complete for the law P. Let A be a progressively measurable subset of R+ X O. The function DAis then a stopping time.

T52

THEOREM

Proof Since the family (~t) is right continuous, it suffices to verify that {DA

< t} E ~t

R+. Since the a-field ~t is complete for the law P, we show that the set {DA < t} is ~t-analytic (see 111.24). Now this set is the projection on 0 of A n ([O,t[ X 0), which belongs to the a-field ~([O,t]) X ~t due to the fact that A is progressively measurable. Let f be the paving consisting of the compact subsets of [O,t]. Every element of the product a-jield ~([O,t]) X ~t is analytic with respect to the product paving f X ~t from III.TI2. It then suffices 10 use Ill.T9 to establish the theorem.

for every

t E

• It should be observed that every stopping time T can be considered as the debut of a progressively measurable set, namely {(t,w): t > T(w)}.

IV, 53, D54, 55

72

Stochastic Processes

Remark Let!!7 be the a-field of all progressively measurable subsets of R+ x preceding theorem extends to the case where A is only !!7-analytic.

n.

The

53 We suppose that the family (~t) satisfies the hypotheses of the preceding statement. Let (Xt ) be a stochastic process, with values in a state space (E,tff), which is progressively measurable relative to the family (~t). (a) Let B be an element of tff. We set DB{w) = inf {t: Xt{w)

E

B}.

This function is called the hitting time [for the process (Xt)] of B. It is equal to the "debut" of the set {{t,W): Xt{w) E B}, which is progressively measurable, so that DB is a stopping time. This remains true, from n.T11 and the above remark, when B is only supposed tff-analytic. (b) Letfbe a real-valued measurable function defined on E. The process «f °Xo,f oXt))tER+' with values in R x R, is evidently progressively measurable relative to the family (§"t). The following function of W is hence a stopping time: inf {t: IfoXo{w) - foXt{W) I > e}

(e

> 0).

It is called "the first time for which f differs from its initial value by more than e (on the path of w)." (c) Let us take up again the situation of No. 44{b), and show that the functions T n are stopping times, by the method of No. 52. Since the processes (Xs) and (Xs-) are progressively measurable from T47, the same holds for the process (Xs - X s-). Reason by induction: By supposing that T n is a stopping time, the set

A

= {{t,w): t > Tiw)}

is progressively measurable. The same holds for the set A' = {{t,w): IXt{w) - Xt_{w) I > e}. It then suffices to note that Tn+l is the debut of A n A'.

Systems and chains of stopping times All stopping times envisaged below are relative to the same right continuous family of a-fields (~t). We also suppose themjinite, but this restriction can be removed if we consider processes with time set R+ U {+ 00 }. D54 DEFINITION Let I be a totally ordered set. A system of stopping times is a family of stopping times {Ti)iEI such that T i < T; for every pair (i,j) of elements of I such that i < j. Let (Xt) be a stochastic process progressively measurable with respect to the family (~t). The stochastic process (XT)i E I is said to be "transformedfrom (Xt) by the system ofstopping times {Ti )." 55 Examples Let T be a finite stopping time. Two particularly interesting systems of stopping times can be obtained from T in the following manner: (1) The set I is equal to R+, and the stopping time T i is equal to T + i. The transformed process is obtained by "translating the origin of times to the instant T."

73

Measurable Processes. Stopping Times

IV, T56, T57

(2) The set I is equal to R+, and the stopping time Tt is equal to T A i. The transformed process is obtained by "stopping the process (Xt ) at the instant T." This transformation is the origin of the name "stopping time." Other very important examples of systems of stopping times, the "changes of time," will be studied in Chapter VII. We now establish an important property of transformed processes, communicated to us by K. L. Chung, as was the following lemma (which is of interest itself). T56

Let (Xt ) be a process with values in a measurable space (E,8), progressively measurable with respect to the family (~t), and let T be a stopping time. The mapping THEOREM

(s,w) ~ XsAT(w)(W), from R+ X

(56.1)

n into (E,8) is measurable when the first space is given the a-jield &I(R+) x ~T'

Proof We begin with a remark. Let r be an element of R+; then A fl (R+

x {w: T(w)

~ r}) E

&I(R+) x ~ T

(56.2)

for every set A E &I(R+) x ~r' Since the collection of sets A having this property is a a-field, it indeed suffices to verify that the sets of the form I x J (I E &I(R+), J E ~r) have it, and this follows from the definition of the a-field ~T' The theorem i.s established if we show that {(s,w): XsAT(w) E

U} E &I(R+) x

~T

for every set U E 8. It suffices for this to show that each of the following sets belongs to &I(R+) x ~ T: {(s,w): s ~ T(w), XT(w) {(s,w): s

E

U}

< T(w), Xlw) E U}.

This is obvious for the first set, since the three mappings (s,w) ~ s, (s,w) ~ T(w), and (s,w) ~ XT(w) are measurable with respect to the a-field &I(R+) x ~ T' Write the second set in the form

< r, T(w) > r, Xs(w) E U}. Now each of these sets is of the form (56.2), with A = {(s,w): s < r, Xs(w) E U} E &I(~) U

rrational

{(s,w): s

x

~r'

from the fact that the process (Xt ) is progressively measurable. The conclusion then follows from relation (56.2). Consider next a system of stopping times (Tt)teR+ having the following property: The mappings t ~ Tt(w) are increasing and right continuous.

(57.1)

Denote by (~t)teR+ the "family of transformed a-fields" defined by ~t = ~Tt; this family is increasing and right continuous from T40 and T42. Set finally Y t = X Tt . We have the following theorem. T57

THEOREM

The process (Yt) is progressively measurable with respect to thefamity (~t).

Proof The process (Tt) is adapted to the family (~t) from T37, and its paths are right continuous by hypothesis. It is thus progressively measurable with respect to the family (~t) from T47. Let t be a positive number; since the mapping (s,w) ~ Ts(w) is measurable

IV, T58, D59, T60

74

Stochastic Processes

with respect to the a-field 86'([O,tD x r,gt, the mapping (s,w) -'\1'1'+ (Ts(w),w) from ([O,t] x 0, 86'([O,tD X r,gt) into (R+ X O,86'(R+) X r,gt) is measurable. Since the mapping (u,w) -'\1'1'+ XUATt(W) from (R+ X O,86'(R+) X r,gt) into (E,C) is measurable from the preceding theorem, the composed mapping (s,w) -A.N+ XTsATt(W) = Ys(w) is measurable with respect to 86'([O,tD X r,gt. The following proposition is often useful in effecting a change in the time origin (No. 55, 1). Here T denotes a stopping time, and we set r,gt = §"T+t. T58 (r,gt)

A positive random variable S is a stopping time with respect to the family if and only if T + S is a stopping time with respect to the family (§"t). THEOREM

Proof Suppose that T have from T41 that

+ {S

S is a stopping time with respect to the family (§"t). We then

< t} =

{T + S

< T + t} E §"T+t =

r,gt.

Conversely, suppose that S is a stopping time with respect to the family (r,gt). We have

{T

+ S < t}

=

U

a+b 0

{T

< a}

n {S

< b}.

It thus suffices to show that {T < a} n {S < b} = {T < a} n [{S < b} n {T + b < t}] belongs to §"t. But we have {T <-a} E §"a C §"t, {S < b} E r,gb = §" TH' and thus {S < b} n {T + b < t} E §"t.

°

D59 DEFINITION Let sand t be two numbers such that < s < t < + 00. A chain of stopping times on the interval [s,t] is a sequence of stopping times (Tn)neN' which has the following properties: (1) To = s, and T n < Tn+l < tfor every n EN; (2) The probability P{Tn < t} tends to as n tends to 00.

°

T60 THEOREM Let (Sn) and (Tn) be two chains of stopping times on an interval [s,t]. For each w E 0, denote by R p(w) (p E N) the p + 1st point of the sequence obtained by arranging in increasing order of magnitude the values Sn(w), Tn(w) (n EN). The random variables Rp then constitute a chain of stopping times called the superposition of the chains (Sn) and (Tn). Proof We show by induction that each Rp is a stopping time or, what amounts to the same thing, that the event {Rp < a} belongs to §"a for every a E R+. This is clear for R o = s; the event {Rp < a}, on the other hand, is the union of the events

{R p _ l

< a} n

{R p _ l

< Tm < a}

(m EN)

{R p _ l

< a} n

{R p -

< Sn < a}

(n EN).

and l

All of these events belong to §"a' from T41 and the inductive hypothesis. It is then evident that (Rp) constitutes a chain of stopping times. Remark This result can also easily be deduced from T52. Let A be the progressively measurable subset of R+ X 0, union of the graphs of the stopping times Sn' T n ; Rp-I being a stopping time by the induction hypothesis, let B be the progressively measurable set {(t,w): t > Rp-I(w)}; then Rp is the "debut" of A nB, and hence a stopping time.

~

Part B

MARTINGALE THEORY

CHAPTER

V

Generalities and the Discrete Case

The theory of martingales is one of the basic tools of probability theory. This is particularly true of the results contained in this and the following chapter, for the most part due to Doob. The main theorems on Markov processes can easily be deduced from them, and they also have a number of important applications to potential theory.

1.

Difinitions and General Properties

It is interesting to dispose of the notion of a martingale in all its generality. We therefore

denote by T, in the following definition, an arbitrary set ordered by a relation <. But we shall soon return to the case we are most interested in, that of an interval of :R. or Z. The definitions of an increasing family of O'-fields and of a process adapted to such a family (IV.DJO and D3l) clearly extend to general ordered sets. Dl DEFINITION Let (n,~,p) be a probability space, (~t)tET an increasing family of sub-O'-jields of ~, and (Xt)tET a real-valued stochastic process, adapted to the family (~t). The process (Xt ) is said to be a martingale (respectively, a supermartingale, a submartingale) with respect to the family (~t) if (1) Each random variable X t is integrable, and (2) For every pair of elements s, t of T such that s < t, we have

I

E[Xt ~s] = X s a.s.

(respectively, <Xs'

> X s).

(1.1)

2 Remarks (a) The processes we call submartingales (respectively, supermartingales) would be called "semimartingales" (respectively, "lower semimartingales") in Doob's book. This terminology is now abandoned. (b) Let (Xt ) be a supermartingale. The process (-Xt ) is then a submartingale, and conversely. We thus state the theorems for only one of the two species of processes, in general, that of supermartingales. (c) Suppose that the random variables (Xt ) are non-negative. Relation (1.1) then makes sense, even if the X t are not integrable. One can then speak of a generalized martingale (respectively, supermartingale, submartingale).

77

V, 3, T4, T5

78

Generalities and the Discrete Case

(d) A process (Xt ) given with no reference to a family of a-fields is called a martingale (supermartingale) if it is a martingale (supermartingale) with respect to the family ff t = S""(Xs , s < t). It can then be said that a process equivalent to a martingale (supermartingale) is again a martingale (supermartingale). (e) Definition 1 can be generalized in the following manner. Suppose we are given for each t ETa measurable space (Ot,ff t), and for each pair (s,t) of elements of T such that s < t a measurable mapping 7Tst of 0t into Os, such that 7Trs 0 7Tst = 7Trt for r < s < t. Measures flt (each of which is defined on the space of the same index) then constitute a martingale if for s < t we have fls = 7Tst Cflt). Supermartingales and submartingales are similarly defined. Definition 1 is then obtained by taking all the 0t equal to 0, all the mappings 7Tst equal to the identity mapping of onto itself, * and by setting flt = X t P. Unfortunately, one cannot translate into this language the most important results of martingale theory, since these concern the behavior of the paths of the process (Xt).

°

Examples of martingales 3 We limit ourselves to two examples; the rest of the book will furnish many others. (a) Let (O,ff,P) be a probability space, and let (fft)teT be an increasing family of sub-afields of ff. For each integrable random variable Y, set Y t = E[ Y fftl. The process (Yt) is then a martingale [with respect to the family (fft)l. (b) Let (O,ff,P) be a probability space. Denote by T the collection of all finite sub-afields of ff, ordered by inclusion, and by Q a positive additive set function defined on ff. Each element t of T is generated (as a a-field) by a partition of into a finite number of measurable sets AI' A 2 , ••• , An. Denote by X t the following function (where an arbitrary value is assigned to ratios of the form 0/0):

I

°

f

Q(A i ) lA .. • i=l P(A i )

This function is evidently t-measurable. The process (Xt)teT is then a supermartingale. It is a martingale if Q(A i) = 0 for every set Ai such that P(A i ) = O. This process will be used in Chapter VIII. Additivity and convexity theorems T4 THEOREM Let (Xt) and (Yt ) be two martingales (respectively, supermartingales) defined on (O,§",P), relative to the samefamily (§"t) ofsub-a-jields of§"o Let a and b be two constants (respectively, two non-negative constants). The process (aXt + b Y t) is then a martingale (respectively, a supermartingale). The process (Xt A Y t) is a supermartingale. Proof Immediate.

T5

Let (Xt) be a supermartingale relative to a family of a-jields (ff t ). In order that (Xt) be a martingale, it is necessary and sufficient that the function t""'vV'+ E[Xtl be constant. THEOREM

I

Proof Let s, t be two elements of T such that s < t. We have E[Xt ffsl two sides are then a.s. equal, if and only if they have the same expectation. • More precisely, the identity mapping of (n,~t) onto (n,~.).

< X s a.s.

The

Definftions and General Properties

79

V, T6-T8

T6 THEOREM Let (Xt ) be a martingale (respectively, a supermartingale) relative to a family of (J-jields (ff t ) , and let f be a concave (respectively, concave increasing) function defined on R and such that the random variables f 0 X t are integrable. The process (f 0 X t ) is then a supermartingale relative to the family (ff t ). Proof Let sand t be two elements ofT such that s in both cases

< t. Set Y u =

fo X u for u ET. We have

Write then Jensen's inequality (11.47, property 4),

This establishes the theorem. A corollary of this result is quite commonly used. Let (Xt ) be a martingale, and let A. be a number > 1 such that the random variables IXtl A are integrable. The process (IXtI A) is then a submartingale. The following two theorems generalize those which have been proved. We borrow them from Dubins (64), where they aid in establishing several interesting inequalities. They will not be used in this book. T7 LEMMA (Generalized Jensen's inequality) Let (n,ff,p) be a probability space, X an integrable random variable on this space, and rg a sub-(J-jield of ff. We shall denote by Ya version of E[X rg]. Let f be a measurable mapping ofn X R (with the naturalproduct (J-jield) into R, such that (a) The mapping w J\,/\/+ f(w,t) is rg-measurable for each t ER. (b) The mapping t Jo..N+ f(w,t) is convex for each WEn. Suppose that the random variable w Jo..N+ f(w,X(w» is integrable. We then have the inequality

I

E[f{-,X(o»

I rg] > f(o, Y(o»

a.s.

(7.1)

This theorem could be proved by first supposing X was elementary, and then deducing the general case by using a passage to the limit (which is a little more delicate than usual). We will not enter into the details here. T8 THEOREM For each integer n > 1, let qn be a measurable mapping of Rn into R. Suppose that these functions have the following properties: (1) For each n > 1, and each system ofvalues Xl' ... , Xn-b the function Xn J\,/\/+ qn(X I' ... , Xn- b x n) is concave increasing. (2) qn(x I, ... , x n) > qn+I(Xb ... , x n- l , x m x n). Let then (Xn)n?-l be a supermartingale relative to afamilyof(J-jields(ffn)n?-l' Suppose that the random variables

are integrable. The process (Yn) is then a supermartingale relative to the family (ffn)' Proof The inequality E[ Yn+ll ff n] < Y n a.s. is established by using the reasoning which led to Theorem 6, Lemma 7 replacing Jensen's inequality.

V,T9

Generalities and the Discrete Case

80

2. Fundamental Inequalities The inequalities established in this section will be used later to treat the countable case and the continuous case. The reader will find other interesting inequalities in the article by Dubins (64). Doob's optional sampling theorem The vocabulary and the notation are those of Chapter IV, Nos. 35, 47, and 53. .....

T9 THEOREM Let (Xn )n=I.2•...• k be a supermartingale (respectively, a martingale) relatit'e to a family of a-jields (:F n)n=I •...• k' Let (J:)i=l. .... 21 be a system of stopping times relative to these a-jields. The process (XT )i=I•...• 21 is then a supermartingale (respectively, a martingale) relative to the a-jields (:F T)i=l. ..•• 21' Proof The inequality

Ell X TIl =,~L~,) 1X ,I dP ~ i,E[1 X ,11 < 00 shows that X T is integrable for every stopping time T. We prove only the statement concerning supermartingales: The case where (Xn ) is a martingale then follows by considering the two supermartingales (Xn ) and (- X n ). On the other hand, the definition of supermartingales requiring the consideration of only two instants at a time, it suffices to consider a system reduced to two stopping times 8 and T such that 8 ~ T, and to establish the supermartingale inequality:

J,x

s dP

>

L

X T dP

(A

E j>s)·

(9.1)

Let us assume this inequality has been established in the case where the difference T - S is at most equal to I, and show that the general case can then be deduced. Set Rn = TA (8 + n), n = I, ... ,k. These random variables are stopping times (IV.T36), so we have A E:FRn for every n (IV.T40) and consequently, from the particular case, R n+1 - Rn being less than I for every n, we have

L

XsdP

>

L

X R, dP

> ... >

L

XR.dP

=

L

X T dP,

Le., inequality (9.1). Suppose thus that T - 8 is at most equal to I. We have fA

(X s - XT)dP=If

(X s - XT)dP

n=1 A n{S=n} n{T> S}

=if

n=1 A f"\{S=n} f"\{T> n}

(X n - Xn+JdP.

The event A n {S = n} belongs to :Fn from the definition of the a-field :Fs; the event {T > n}, the complement of {T < n}, belongs to :F n from the definition of stopping times. We are thus integrating X n - Xn+l in the second member over an element of :Fn' which gives a positive result from the supermartingale inequality.

81

V, T1D-T12, 13

Fundamental Inequalities

TI0 COROLLARY Let (Xn)n=l ..... k be a supermartingale, and let T be a stopping time. We then have the inequality (10.1) E[X1 ] > E[XT ] > E[Xk ]· [Apply the preceding theorem to the system of stopping times (I,T,k).]

Tll COROLLARY Let (Xn)n=l ..... k be a supermartingale, and let T be a stopping time. We then have the inequality E[IXTIl

< E[X1 ] + 2E[X;] < 3 sup E[/Xnll.

( 11.1)

n

Proof We have E[IXTI] = E[XT] + 2E[Xp ]. E[XT] is less than or equal to E[X1 ] from TI0. On the other hand, the process (Xn A 0) is a supermartingale from T4, so that (X;) is a submartingale, and we have E[Xp] < E[Xk"] (TI0).

-+

Two fundamental inequalities T12 THEOREM Let (Xn)n=l ..... k be a supermartingale, and A a non-negative constant. We then have the inequalities AP {sup X n > A } < E[X1 ] n

-J:

r~~<~

X k dP

< E[X1 ] + E[X;]

(12.1)

and

Proof To establish the first relation, set T(w) = inf {n: Xn(w) set is empty. It is clear that T is a stopping time. We thus have

E[X1 ]

+J:

> E[X T ] > AP{SUp X n > A}

> A}, or

e~PXn
n

T(w)

=k

if this

X k dP,

an inequality which implies (12.1). To establish the second relation, set T(w)

or T( w)

=k

= inf {n:

Xn(w)

< -A},

if this set is empty. We have E[Xk ]

< E[XT ] < -J.p(inf X n < -J.} + n

Jre~f x

X k dP, n

> -;.}

which implies (12.2).

13 Example ofan application Let (Xn)n=l..... k be a martingale. Suppose that the random variables X n are square integrable: The process inequality (12.2) gives

(X~)

is then a submartingale (T6) and

A2Pb~ IX.I > A} < E[X=l. Suppose in particular that the X n are of the form X n = Y1

+ +... + Y2

Ym

(13.1)

V, TI4

82

Generalities and the Discrete Case

where the random variables Yn are independent, square integrable, and of mean zero. Inequality (13.1) is then well known under the name of Kolmogorov's inequality. The above proof is borrowed from Doob's book. . We shall now use the notions defined in IV.2t. Given random variables Xl' ... , X k, we denote by V(w;[a,bD [respectively, D(w;[a,bD] the number of upcrossings (respectively, downcrossings) by the function n ~ Xn(w) (n = 1, ... ,k) of the interval [a,b]. We then have the following theorem, which we state for a submartingale. It is due to Doob in the case of martingales, to Snell in that of submartingales. We follow here Hunt's method of proof, which appeared in Doob's paper (62).

-+

Tt4

(Doob's inequality) Let (Xn)n=l, ....k be a submartingale relative to a family of a-jields (§'n)n=l. ... .k' and let a, b be two real numbers such that a < b.We then have the inequalities THEOREM

E[U(';[a,b]) 13',]

< E[(X

k -

a)+ ~~'~ - (X, - a)+

(14.1)

and

(14.2) Proof In fact, we are going to establish these inequalities for the numbers of upcrossings or downcrossings of the open interval ]a,b[, which we abbreviate by V' and D': They evidently majorize the corresponding numbers for [a,b]. We may replace (Xn) and ]a,b[ without affecting V' and D', by the process (Yn ) = «Xn - a)+) (which is a submartingale from T4), and the interval ]0, b - a[. Define inductively the stopping times T I , ... , Tk+l as follows: TI(w) = 1; T 2(w) is the first index i for which Yi(w) = 0, or k if there exists no such index; Ta(w) is the first index i > T 2 (w) for which Ylw) > b - a, or k if there exists none. We alternate thus up to Tk+l = k. We can write YI(w) = [YT 2(W) - Y Tt(W)] + [YTa(w) - Y T2 (W)] + ... + [YTk +1(W) - YTk(W)]. Consider in this sum the even-numbered terms starting from the left: (YTa - YT 2 )' ( YTs - Y T)' .... We encounter first those which correspond to upcrossings by the path n ~ Yn(w) of the interval ]0, b - a[. The number of these is equal to V'(w), and their contribution to the sum is at least equal to (b - a)V'(w). We encounter next either all zero terms, or an "incomplete" upcrossing followed by zero terms, so that the contribution of this part to the sum is non-negative. We hence have, by taking conditional expectations with respect to §'l' Yk(w) -

I

E[Yk §'l] -

YI

> (b -

I

a)E[U' §'l]

+ 1

n~k

E[YTn +1

-

I

YTn §'l]'

nodd

Inequality (14.1) is then obtained by noting that each of the terms in the second member is non-negative from TtO. Inequality (14.2) is proved in an analogous manner, but we denote this time by T2(w) the first index for which the path attains the value b - a, by Ta(w) the first index for which it returns to the value zero, etc. Theorem to then gives us the inequality E[(YTa -

Y T2 )

+ (YTs

-

YT)

+ ... \§'l] > 0.

83

Fundamental Inequalities

V,15

Now this sum is composed of terms that correspond to downcrossings of ]0, b - a[ by the path n.A.N+ Xn(w), of which the total contribution is less than -(b - a)E[D' ~2]' followed by a single nonzero term at most equal to (Yk - (b - a))+ = (Xk - b)+. We thus have

I

E[(Xk

-

b)+ I ~l]

> (b -

a)E[D' I ~l]'

from which (14.2) follows. 15 Remarks (a) The presence of conditional expectations in formulas (14.1) and (14.2) is a refinement without great utility: the truly useful formulas are those which are obtained by integrating these. It would, in fact, have been possible, in the same manner, to introduce conditional expectations into inequalities (12.1) and (12.2). (b) Let (Xn ) be a supermartingale. By applying the inequalities 14 to the process (-Xn ) and to the interval (-b,-a), the following inequalities are obtained, given only in their integral form:

(15.1) and (15.2) This last inequality is probably a little easier to remember than the others. The case of positive supermartingales 16 The case of positive supermartingales is particularly important in potential theory; thus let us give two inequalities, due to Dubins (64), which improve inequalities (14.1) and (14.2). The notation is the same as in the preceding statements, but the process (Xn)n=l ..... k is a positive supermartingale, and the numbers a, b are non-negative. Here are Dubins' inequalities (p is an integer > 1): P{ U(.

·[a b]) -> p;") -< E[Xb

a](~)p-l

(16.1)

> p} < E[X~ A bl(~r-l.

(16.2)

1 A

"

b

and P{D(';[a,b))

Let us sketch the proof of inequality (16.1), for example. As in the proof of Theorem 14, set T 1 = I, call T2 the instant when the path first has a value b, etc. The event {U > p} is identical to the event {XT2 P+l > b}. We write the following inequality which follows from Theorem 10:

Jr{T2p
X T2p dP

> p}, the second is dominated by

aP{T2p

< k} < aP{T

2 P-l

< k}

(16.3)

V, TI7

Generalities and the Discrete Case

if P > 1. We obtain, for p

> 1,

P{U We thus have P{U

> p} < WP{T•.-, < k} < WP{U > p -

> p} < (ba) P-lP{U > I}.

84

I}.

If P is equal to 1, we can dominate the

second integral of formula (16.3) by E[Xl A a], and inequality (16.1) then follows immediately. Dubins has shown that these inequalities cannot be improved.

3. The Countable Case. ConverBence Theorems This section, in which the results are almost all borrowed from Doob's book, does not pretend to exhaust the whole subject. The reader will find other interesting convergence theorems in Doob (56), (62) and Chow (45). We continue to denote by (Q,~,P) the base space of the process.

-+

T17

(a) Let (Xn)neN be a supermartingale relative to an increasing family of sub-a-jields of ~. Suppose that

THEOREM

(,p;n)neN

sup E[X-;;]

< 00.

(17.1)

n

The random variables X n then converge a.s. to an integrable random variable X oo . (b) This condition is satisfied in particular when the X n are positive. We then have E[Xoo ] < limn E[Xn], with equality if and only if the random variables X n are uniformly integrable. The process (Xn)neNv{oo} is a supermartingale. (c) Suppose that the X n are uniformly integrable. Condition (17.1) is then satisfied, the process (Xn)neNv{oo} is a supermartingale, and the convergence of X n to X oo takes place in the Ll norm. (d) Suppose that the X n are uniformly integrable, and that the process (Xn)neN is a martingale. The process (Xn)neNV{oo} is then a martingale. Proof (a) Consider an

wE

Q such that

lim sup Xiw)

> lim inf Xn(w). n-+oo

n-+oo

There can then be found, between these two limits, two rational numbers a, b such that a < b. We then have U(w;[a,bD = 00, denoting by this the number of upcrossings of the interval [a,b] by the path n .A./II'+ Xn(w). Thus X n converges a.s. if and only if U(w;[a,bJ) < 00 a.s. for every pair of rational numbers a, b such that a < b. To prove this we use inequality (15.1): E[U(.;[a,bD] ~ sup E[(X n - b)-] . n b- a The second member is finite, by virtue of the hypothesis (17.1) and the inequality (Xn

-

b)-

< X;;+ b+.

We have on the other hand (No. 11), for every kEN, that

E[\Xkll

< E[Xo] + 2 sup E[X-;;]. n

Thus X oo is integrable, from Fatou's lemma.

V, T18

Fundamental Inequalities

85

(b) Suppose that the X n are non-negative. Property (17.1) is then clear, and the second

assertion of the statement is a simple repetition of Fatou's lemma and of II.T21. To show that the process (Xn)neNU{oo} is a supermartingale [relative to the family of a-fields (ff n)neNU{oo}-see IV.30 for the definition of ff 00]' consider two integers m, n such that n < m, and an element A of ff n; the inequality

L

X n dP

>

L

X m dP

(17.2)

I

passes to the limit when m ~ 00 from Fatou's lemma. Thus we do have X n > E[Xoo ff n] a.s. (c) Suppose that the Xn are uniformly integrable. The inequality sUPn E[IXnlJ < 00 (11. T19) then implies (17.1). The convergence of X n to X 00 takes place in the sense of the Lt norm from 1I.T21, and this justifies passing to the limit under the integral sign in formula (17.2) above. (d) To treat the case where (Xn ) is a uniformly integrable martingale, it suffices to apply (c) to the two supermartingales (Xn ) and (- X n ). The following theorem (due in part to Paul Levy) develops statement (d). ~

Let (Xn) be a stochastic process adapted to the family of a-fields (ffn)' In order that (Xn) be a uniformly integrable martingale [with respect to the family (ff n)] it is necessary and sufficient that there exist an integrable random variable Y such that X n = E[ Y ff n] a.s. for every n EN. There then exists a random variable Yo ' essentially unique, which has this property and which is ff oo-measurable. We have a.s. Yo = limn~oo X n , and this equality takes place in the sense of the Lt norm.

T18

THEOREM

I

Proof Suppose that the X n are uniformly integrable, and use T17(d). We have X n = E[Xoo ff n] for every n, and X oo can be chosen ff oo-measurable. We have X oo = limn X n in the sense of convergence a.s. and in the sense of the Lt norm. Let Y be an ff oo-measurable random variable such that X n = E[ Y ff n] for every n. Denote by vii the collection of events A E ff 00 such that

I

I

L L YdP=

XoodP.

Since vii is closed under passage to a monotone limit, and contains the union r:c of the a-fields ff m we have vii = ff 00 (I.T19). We thus have Y = X oo a.s., from II.T9. Conversely, we show that every martingale of the form X n = E[ Y ff n] is uniformly integrable. Let A. be a positive number. We have IXnl < E[I YII ff n] (by Jensen's inequality) and consequently

I

f

IXnl dP

{lX.. 1>A}


IYI dP.

It remains to show that the second member tends to zero uniformly in n when A. ~ we have

P{IXnl

(18.1)

{lX.. 1>A}

00.

Now

> A.} < E[lXnl]/A. ~ E[I YIl/A..

It then suffices to apply property II.19(b) to the uniformly integrable collection consisting of the single random variable Y. The preceding reasoning does not use the order structure of the time set. It yields, in fact, the following result, which merits being made explicit.

86

Generalities and the Discrete Case

V, T19, 20, T21

T19 THEOREM Let Y be an integrable random variable. The collection of random variables of the form E[ Y ~], where ~ ranges over the collection of sub-a-fields of ~, is uniformly integrable.

I

Here is another proof of this theorem. We can restrict ourselves to the case where Y is positive. Choose a positive, increasing, convex function g defined on R+, such that lim t ---+ +00 g(t)/t = + 00, and E[g 0 Y] < + 00 (such a function exists from 1I.T22). From Jensen's inequality, we have g 0 E[ Y ~] < E[g 0 Y ~], and consequently sup E[g 0 E[Y ~]] < + 00.

I

I

I

<§

I

This implies, from 11. T22, that the random variables E[ Y ~] are uniformly integrable.

20 Remark It is interesting to know that Theorem 18 remains true for martingales whose time set is any ordered set filtering to the right, with the exception of the assertion relative to a.s. convergence, which can be false even when the index set is countable [see Dieudonne*]. We shall sketch this generalization [due to Helms (73)]. Let T be a set with an order relation <, filtering to the right for this relation, let (~t)teT be an increasing family of sub-a-fields of ~, and let (Xt)teT be a uniformly integrable martingale with respect to this family. We show by contradiction that the X t converge in Ll to a random variable Xoo (along the directed set T). In fact, if this were not so, the Cauchy criterion would not be satisfied. There thus could be found (for a suitable e > 0), for every t ET, two elements rand s of T such that t < r, t < s, and 11 X r - Xsll > e. This would permit the extraction from T of an increasing sequence (tn)neN such that the X tn do not constitute a Cauchy sequence. This would contradict Theorem 18, since the process (Xt)neN is a uniformly integrable martingale. Denote by ~ 00 the a-field generated by the union of the ~t, t ET. It can evidently be supposed that X oo is ~oo-measurable. The relation X t = E[Xoo ~t] for every t ET is very easily verified, and it can be shown as in No. 18 that Xoo is the only ~oo-measurable random variable which has this property (up to an a.s. equivalence). Conversely, let Y be an integrable random variable. It follows from No. 19 that the random variables E[ Y ~ t] are uniformly integrable. The simplest convergence theorem holds in the case of an index set filtering to the left. We limit ourselves to considering the negative integers.

I

I

-+

T21 THEOREM Let (Xn)ne-N be a supermartingale relative to a family of a-fields (~ne-N' Denote by ~ _ 00 the n-field ne -N ~n' Suppose that

n

sup E[X n ]

< 00.

(21.1)

n

We then have the following properties: (a) The random variables X n are uniformly integrable. (b) The random variables X n converge a.s. to an integrable random variable X -00 when n -+ - 00; the convergence also takes place in the sense of the Ll norm. (c) The process (Xn)ne{ -oo}u( -N) is a supermartingale with respect to the a-fields (~n)ne{ -oo}U( -N)'

(d) Suppose that the process (Xn)ne -N is a martingale; condition (21.1) is then satisfied, and the process (Xn)n e {_ OO}U ( _ N) still is a martingale. • "Sur un theoreme de Jensert," Fund. Math., 37 (1950), 242-248.

87

V,21

Fundamental Inequalities

Proof (a) We show first that the X n are uniformly integrable. Fix an e negative integer k such that

< e.

lim E[Xil - E[Xkl i-+-oo

We then have 0 < E[Xnl - E[Xkl We show that the integral

< e for

all n

> 0, and choose a

< k.

Let A be a non-negative constant.

(21.2) is less than e for every n when A is large enough. It suffices to prove it for values of n less than k. This integral is equal to

-f{Xn<-~

X n dP

+ E[Xnl

-

r

J{Xn~~

X n dP.

(21.3)

This expression is dominated, from the supermartingale inequality (1.1), by

-f

{X n <-~

X k dP

+ E[Xnl

-f

{Xn::;~

Xk dP.

(21.4)

We can now replace E[Xnl by E[Xkl, without changing this quantity by more than e, from our choice of n and of k. We obtain

f

{IXnl >A}

\Xkl dP.

(21.5)

It remains simply to show that this integral tends to zero uniformly in n when A -+ Since the process (X;) is a submartingale, we have E[IXnll = E[Xnl

+ 2E[X-;;l < sup E n

[Xnl

00.

+ 2E[Xi)l.

The probability P{I Xnl > A}, since it is dominated by (I/A)E[I Xnll, tends to zero uniformly in n when A -+ 00. The same property then holds for the integral (21.5) from II.TI9(b), and we have completed the proof of uniform integrability. We do not detail the rest of the proof: The a.s. convergence of the X n follows as in No. 17 from Doob's inequality and from the relation sUPn E[IXnll < 00, which we have just established. The convergence in Ll norm follows from the a.s. convergence and uniform integrability (II.T21). Assertions (c) and (d) are immediate consequences of the uniform integrability, which justifies passing to the limit under the integral sign. We leave to the reader the task of generalizing this theorem to ordered sets filtering to the left, in the manner of No. 20. A theorem of Doob's We have avoided mentioning convergence in LP(1 < P < (0) in the statements of Theorems 17 and 21, in order to avoid overburdening them. The study of convergence in LP rests on the following theorem, due to Doob, which we have occasion to use later. We merely reproduce Doob's proof here.

V, T22, 23 ~

T22

THEOREM

Generalities and the Discrete Case

88

Let p and q be two conjugate exponents, * distinct from 1 and from

00.

Let (Xn)neN be a non-negative submartingale such that sup E[X~] n

< 00.

(22.1)

The random variable sUPn X n then belongs to .pP, and we have


Ilsup Xnll v n

(22.2)

Proof The random variables X n are uniformly integrable (II.T22), so that the limit X 00 exists, and belongs to'pPfrom Fatou's lemma. Denote it simply by X, and put sUPn X n = Y. From inequality (12.2) we have, for every A > 0,

AP{Y > A} Set P{ Y

> A} =

< J{Y~)J r X dP.

F(A). We have E[YV]

=_J:ooo Av dF(A) =J:oo F(A) d(AV) 0

lim h~oo

[AVF(A)]~

<100 F(A) d(AV) <100 .!( r X dP) d(A V) A J
0

=1 x(l n

Y

1 d(AV ») A

0

dP = -.L E[Xyvp-1

1 ].

We hence have

E[YV ]

<

P E[Xyp-l] = qE[XYP p-1

I

].

The second member is dominated by q 11 XII v 11 YV-111 q (from HOlder's inequality). Replacing 11 YP-111 by its value (E[ yP])l/q, we obtain q (E[ YP])l-l/q

< q 11 XII v'

This inequality is equivalent to (22.2). 23 Remarks (a) This result is not valid for p = 1. Doob has shown by a similar method [(63) p. 317] that the condition sup E[ X n log+ X n]

<

00

(or

X 00 log+ X 00

E 'pI)

n

implies the integrability of sUPn X n . This result is extremely useful, but we do not establish it here. Blackwell and Dubins (11) have shown that there exists no condition more general than the latter, depending on X 00 only, and sufficient to imply the integrability of sUPn X n • (b) Here is a simple consequence of Theorem 22, which is often used. Let (Xn)neN be a martingale such that sUPn IIXnll v < 00. The submartingale ClXnl) then satisfies the hypothesis of Theorem 22, the X n are dominated in modulus by an element of.pP, and the convergence of the X n to X 00 takes place in LP. 1

1

P

q

*-+-=1.

89

Fundamental Inequalities

V, D24, T25

Riesz decomposition

D24 By analogy with the classical theory of harmonic and superharmonic functions (an analogy pursued a great deal further in what follows), we pose the following definitions: (a) Let (Xn)neN be a supermartingale relative to a family of a-fields (:F n)neN' such that sUPn E[X;l < 00. The random variable X 00 = lim n-+ 00 X n will be called the stochastic boundary function associated with the supermartingale (Xn ). (b) The martingale X~ = E[X00 :Fnl is called the stochastic Dirichlet solution associated with the supermartingale (Xn ) (or with the random variable X (0)' (c) A potential is a positive supermartingale (Xn) such that lim n_ oo E[Xnl = 0 (which amounts to saying that X n ~ 0 when n ~ 00, in the sense of convergence a.s. and in the Ll norm). Let (Xn) and (Yn) be two real-valued processes. We say that (Yn) dominates (Xn) if we have X n < Y n a.s. for every n.

I

T2S THEOREM ("Riesz decomposition") Let (Xn ) be a supermartingale. The following two conditions are equivalent; (a) (Xn ) majorizes a submartingale. (b) There exists a martingale (Yn) and a potential (Zn) such that X n = Y n + Zn for every n. These two processes are then unique (up to a modification). The martingale (Yn) majorizes every submartingale (Y~) dominated by (Xn ).

Proof Condition (b) clearly implies condition (a), since the martingale (Yn ) is majorized by (Xn ). Suppose that (a) is satisfied, and that (Y~) is a submartingale majorized by (Xn ). Set

(p We have

= 0, I, ...).

I:F

I

E[Xn+1I-tl n+pl :Fnl < E[Xn+p :Fnl = Xn,p so that Xn,p decreases a.s. when p increases. On the other hand, X n,1I-tl

=

a.s.

I

X n. p > E[Y~+p :Fnl > Y~ a.s. Denote by Yn an :Fn-measurable random variable equal a.s. to limv-oo Xn,p' The process (Yn ) is majorized by (Xn ), and majorizes (Y~). We have

I

I

E[Yn+l1 :Fnl = lim E[X n+l,p :Fnl = lim E[Xn+l+ p :Fnl = Yn a.s., p-+ 00

p-+ 00

so that (Yn) is a martingale. It only remains to show that the process (Zn) = (Xn - Yn), which is evidently a positive supermartingale, is a potential. But we have, from the definition of ( Yn), that lim E[Zn+p :Fnl = 0 a.s. for every n.

I

p---. 00

It suffices to apply Lebesgue's theorem in order to obtain the relation limv- 00 E[Zn+Pl = U, which shows that (Zn) is a potential. We establish finally the uniqueness. Let X n = Y: + Z: be a decomposition of the same type. We have E[X n+p :Fnl = E[Y:+ p :Fnl + E[Z:+PI :Fnl·

I

I

The first term of the second member is a.s. equal to Y:; by letting p tend to infinity, it follows that Y n = Y: a.s., and the theorem is established.

V, 26, 27, T28 26

Remark

90

Generalities and the Discrete Case

Let (Xn) be a positive or uniformly integrable supermartingale. Set

I

Sn = E[ X 00 § n],

and T n = X n - Sn'

The first process is the stochastic Dirichlet solution associated with (Xn). It is a uniformly integrable martingale. The second process is a positive supermartingale whose stochastic boundary is zero a.s. (T18). We have thus obtained a decomposition, similar to the preceding, whose uniqueness is easy to verify.

4. The Optional Sampling Theorem in the Countable Case 27 We have established Doob's "optional sampling" theorem in No. 9, in the finite case only, in order to deduce from it the fundamental inequalities of martingale theory. We now extend this theorem, still following Doob, to the case of a supermartingale (Xn)neN relative to an increasing sequence of a-fields (§ n)neN' We make the following hypothesis about the supermartingale (Xn ): There exists an integrable random variable Y such that X n a.s. for every n EN

> E[ Y I§

n]

(27.1)

[i.e., we require that the supermartingale (Xn)neN can be extended to a supermartingale (Xn)neNu{oo} (put X oo = Y)]. The stopping times we consider will all be allowed to take the value stopping time T, we set XT(w) = Yew) on the set T = + 00.

+ 00. For every such (27.2)

We call Y the "random variable at infinity." The case of an arbitrary system of stopping times reduces to that of a system of two stopping times Sand T such that S < T. We then have the following statement.

-+

T28 THEOREM Suppose that the supermartingale (Xn)neN satisfies condition (27.1). The random variables X s and X T are then integrable, and we have the supermartingale inequality Xs

> E[XT I §s]

a.s.

(28.1)

Proof Suppose that we have been able to establish the theorem in the following two special cases: (a) The supermartingale (Xn ) is non-negative, and the random variable at infinity is taken to be O. (b) (Xn) is a martingale of the form E[ Y § n], and the random variable at infinity is equal to Y. It will then suffice to write the decomposition:

I

X n = (Xn - E[YI §nD

+ E[YI §n],

in order to deduce formula (28.1) in its full generality. We thus treat cases (a) and (b) separately.

91

The Optional Sampling Theorem in the Countable Case

V, T29

Case (a) We denote by Sk' T k the bounded stopping times S A k, TA k (k EN). We have E[XTkl < E[Xol for every kEN from TI0, and X T < limk ->-oo X Tk • Fatou's lemma thus implies the inequality E[XTl < E[Xol, so that X T (and in the same way X s ) is integrable. Let A be an element of :Fs. The set A n {S < k} then belongs to :FSk (IV.T39), and we have, from T9,

1

A f"'I{S:Sk}

XSk

dP

>1

A f"'I{S:S k}

The second integral is diminished by replacing {S

1

Xs dP

A f"'I{S:Sk}

X T dP. k

< k} by {T < k}. It thus follo,:\,s that

>1

A f"'I{T:Sk}

X T dP.

Letting k tend to infinity, it follows that

1

Af"'I{S
Xs dP

>1

Af"'I{T
X T dP,

an inequality equivalent to (28.1), since we have Xs = 0 (respectively, X T = 0) on the set {S = oo} {respectively, {T = oo}). Case (b) We show that X T = E[YI :FTl for every stopping time T. This implies (28.1) from the inclusion :FS c :FT (IV.T40) and the equalities

= E[Y I:Fsl = E[Y I:FT I:Fsl = E[XT I:Fsl. Let A be an element of :FT. We have X T = Yon the set A n {T = oo}, so that it suffices Xs

to verify the relation

f

A f"'I{T<

Since the event A

n

{T

oo}

X T dP

=1

(28.2)

Y dP.

Af"'I{T
< k} belongs to:FT k (IV.T39), it follows from Theorem 9 that

1

A f"'I{T:Sk}

X T dP k

=1

A f"'I{T:Sk}

X k dP

=i

~ f"'I{T:Sk}

Y dP.

Relation (28.2) is then obtained by letting k tend to infinity, and by using the uniform integrability of the random variables X T k = E[ Y :FTkl, established in No. 19. The case where the process (Xn) is a uniformly integrable martingale, which we treat now, merits being stated separately.

I

-+

T29 THEOREM Let (Xn)neN be a uniformly integrable martingale, and let Xoo be a random variable equal a.s. to limn->-oo X n • Let T be a stopping time,jinite or not; set XT(W)

=

Xoo(w)

on the set

{T = oo}.

We then have the relation

(29.1) (It suffices to apply the preceding proof, taking Y = Xoo , which is legitimate from TI8.) The following theorem does not hold for supermartingales with continuous time sets, as we shall see in Chapter VI.

V, T30

92

Generalities and the Discrete Case

T30 THEOREM Let (Xn)neN be a uniformly integrable supermartingale, and let .r be the collection of all finite stopping times relative to the family (§' n)' The collection of random variables (XT)Te.r is then uniformly integrable. Proof The decomposition in No. 26 permits us to consider the process (Xn ) as the sum of a uniformly integrable martingale (Yn), and a positive supermartingale (Zn) , which satisfies the condition lim n--+ oo Zn = 0 a.s. It follows from Theorem II.T20 that the vector sum of two uniformly integrable subsets of 'pI still is uniformly integrable, so that the supermartingale (Zn) is uniformly integrable. In view of this result, we can limit ourselves to verifying the statement separately for the processes (Xn) and (Zn). This verification is evident for the martingale (Yn), from Nos. 29 and 19. On the other hand, the process (Zn) is uniformly integrable, so that the relation lim n--+ oo Zn = 0 a.s. implies lim n--+ oo E[Znl = O. Fix a number e > 0, and choose an integer k large enough so that E[Zkl < e. It then follows, for every stopping time T and every number A > 0, that

i

ZT dP

{ZT>A}

= 2k

i=l

i

Zi dP

{T=i}n{Zi>A}

r

The second integral is dominated by event {T > k} belonging to

§'k)

+

i

ZT dP.

{T>k}n{ZT>A}

ZT dP, which is, in turn, dominated (the

){T>k}

by the integral

J:

Zk dP

< E[Zkl < e.

The sum in

{'l'>k}

the second member is, on the other hand, dominated by

2k

i=l

f

{Zi > A}

Zi dP

which is independent of T and tends to 0 when A ~ established.

00.

The uniform integrability is thus

CHAPTER

VI

Continuous Parameter Martingales

This chapter consists principally of rather easy extensions of the results in the preceding chapter. We limit ourselves to the case in which the time set is the half-line R+, and we consider almost exclusively martingales whose paths are a.s. right continuous-we call them right-continuous supermartingales for abbreviation. The reader can find several complementary results in Doob's book. In this chapter all supermartingales and all stopping times will, unless explicitly mentioned to the contrary, be defined on a single probability space (0, ~,P), and relative to a single increasing family (~t)teR+ of sub-<1-fields of~. We suppose that the probability space is complete, and that each <1-jield ~t contains all of the P-negligible sets.

1.

Regularity Properties

cif Paths

Fundamental inequalities Let (Xt)teR+ be a supermartingale, let H be an arbitrary subset of R+, and let a and b be two numbers such that a < b. We denote by U(w;H;[a,b]) the number of upcrossings of the interval [a,b] by the function t -¥I'+ Xt(w), considered on H (see 11.21); the notation D(w;H;[a,b]) is used for the number of downcrossings. The notion of separability is not indispensable for this chapter (nor for what follows). The reader can, if he so desires, limit himself to reading that which deals with rightcontinuous supermartingales, and simply ignore the statements concerning separable supermartingales.

...

Tl

Let (Xt ) be a supermartingale. We denote by S a countable dense subset of R+, by I = [r,s] a compact interval ofR+, and by A a positive constant. Then; (1) We have the following inequalities; THEOREM

AP( AP(

sup

Xt > A) < E[X + E[X;-]

(1.1)

inf

Xt < -A) < E[lXsl]

(1.2)

teSf"'II

teSf"'II

r]

93

VI, 2, T3

Continuous Parameter Martingales E[U(·; S

n

I; [a,bD]

~ E[(~s_-ab)-]

< E[(Xr

94

(1.3)

b) - (X s A b)]. (1.4) b-a (2). We can replace S n I by I in these formulas if the supermartingale (Xt) is rightcontinuous, or more generally if it is separable. E[D(.; S

n

I; [a,bD]

A

Proof Let u be any finite subset of S n I. The inequalities obtained by replacing S n I by u in (1.1), (1.2), (1.3), and (1.4) reduce, respectively, to inequalities (12.1), (12.2), (15.2), and (15.1) of the preceding chapter. The inequalities relative to S n I are then obtained by letting u increase to S n I. Suppose now that the supermartingale (Xt ) is right continuous. The inequalities relative to I are obtained by writing the inequalities above for a countable dense set S that contains s, and then noting that the left-hand sides are not changed if S n I is replaced by I. The case where (Xt ) is separable is treated similarly, choosing for S a universal separating set which contains sand r (see Nos. IV.I8 and IV.24). 2 Remark Theorem V.22 extends in the same way to a submartingale (Xt) which is positive and right continuous (or more generally separable). Let p be a number such that 1 < P < 00, and let q be the exponent conjugate to p. Then

Ilsup tel

Xtll p

< q sup 11 Xtll = q IIXsll tel p

p•

(2.1)

T3 THEOREM (1) Let (Xt ) be a right-continuous supermartingale (or only a separable one); the following properties are satisfiedfor every W E Q; (a) The path of w is free of OScillatory discontinuities. (b) The path of w is bounded on every compact interval. (2) Let (Xt) be any supermartingale, and S a countable dense set in R+. The restriction to S of the mapping t -A./II'+ X t(w) then has, for every w E Q, a right limit and a left limit at every point of R+. Proof Suppose that the supermartingale (Xt) is right continuous or separable. We have, for every n EN and every pair of rational numbers a, b such that a < b, E[U(·;[O,n];[a,bD]

< 00

from inequality (1.3). This implies U(w;[O,n];[a,bD

< 00

a.s.

Property (a) is then deduced from Theorem IV.22. Property (b) follows similarly, using inequalities (1.1) and (1.2). Now consider the case of an arbitrary supermartingale. Denote by H n.a.b the set {w: U(w; S

n

[O,n]; [a,bD = oo}.

It follows from inequality (1.3) that H n •a •b is negligible, and the same holds for the union H of the sets H n •a •b • The remark that follows Theorem IV.T22 then shows that the mapping t -A./II'+ xtCw) has, for every w ~ H, a right limit and a left limit (through S) at every point. To be precise, we put

Xt+(w) = lim XsCw); • This last definition for

t

> 0 only.

Xt_(w) = lim XsCw)*

s~t

s~t

seS

seS

s>t

s
95

VI, T4

Regularity Properties of Paths

for every t

E

R+ and every w

i

H. We complete these definitions by putting, for

Xt+(W) = Xt_(w) = 0

for every

t

E

WE

H,

R+.

We note that the P-negligible set H belongs to every a-field ~t (see the assumptions in the introduction to this chapter), which implies that Xt+ is measurable with respect to the a-field ~t+ = ns>t ~s· We also note that the paths of the process (Xt+) are right continuous. The existence of right-continuous modifications The random variables Xt+ and X t- that figure in this statement are those defined above. We denote here by ~t- the a-field generated by the union Us
T4

Let (Xt ) be a supermartingale; then: (1) We have the following inequalities: THEOREM

> E[Xt+ I~tl X t_ > E[Xt I ~t-l Xt

(4.1)

a.s.

(4.2)

a.s.

(2) The process (Xt+)teR+ is a supermartingale relative to the family of a-jields (~t+)teR+ [in fact, a martingale if the process (Xt ) is one]. (3) Suppose that the family (~t) is right continuous. The supermartingale (Xt) then admits a right-continuous modification if and only if the function t ~ E[Xtl is right continuous.

>

Proof Consider a sequence (tn)neN of elements of S such that t n t, which decreases to t. The random variables X tn are uniformly integrable from Theorem V.T21. Let A be an

event in ~ t. The supermartingale inequality

LX,dP>Lx,.dP

(nEN)

passes to the limit as n ~ 00, and (4.1) then follows from No. II.9(a). Consider similarly a sequence (tn)neN of elements of S, such that t n < t, which increases to t. Put

I

This process is a posItive supermartingale, which converges to X t_ - E[Xt ~t-l as n ~ 00 (see V.T18). This last random variable is thus positive, yielding inequality (4.2). It is clear that X t+ is ~t+-measurable. Let sand t be two instants such that s < t, let (sn) be a sequence of elements of S such that t > Sn > s, which decreases to s, and let (tn) be a sequence of elements of S such that In > t, which decreases to t. Since the random variables (Xs ) and (Xt ) are uniformly integrable from V.T21, the inequality

Lx.. dP > Lx,. passes to the limit as n ~ II.9(a).

00,

tiP

(AE

~>t)

I

which yields the inequality X s+ > E[Xt+ ~s+l from No.

VI, 5, T6

96

Continuous Parameter Martingales

Suppose the family (~t) is right continuous. Inequality (4.1) then becomes simply X t > Xt+ a.s., and these random variables are a.s. equal if and only if they have the same expectation. Let (tn)nEN be a sequence of elements of S that decreases strictly to t. The random variables (Xt ) are uniformly integrable from V.T2I, and so we have

E[Xt+l = Hm E[Xtnl. n-+oo

Thus X t = Xt+ a.s. if and only if this limit is equal to E[Xtl. Since the function s ~ E[Xsl is decreasing, this amounts to saying that there is right continuity at the time t. Let (Yt) be a right-continuous modification of the supermartingale (Xt); then E[Xtl = E[ Ytl for every t, so that the function t ~ E[Xtl is right continuous, from the preceding work. Conversely, if the function t ~ E[Xtl is right continuous, the process (Xt+) is the desired right-continuous modification. Assertion (3) fails if the family (~t) is not right continuous.

5 Remark * Let us apply Fatou's lemma to the positive supermartingale Ytn X tn - E[Xt I ~tJ considered above. We obtain

I

E[Xt_ - E[Xt ~t-]]

< n-+oo lim E[Xtn -

=

I

E[Xt ~tn]]

with equality if and only if the Ytn are uniformly integrable (II.T2I). Alternatively, we have

E[Xt-l

< lim E[XtJ n-+oo

with equality if and only if the X tn are uniformly integrable (the random variables E[Xt ~tJ are uniformly integrable from V.TI9). Let (Xt) be a supermartingale such that the function t ~ E[Xtl is continuous, and let I be a compact interval of R+. The random variables Xlt E I) are then uniformly integrable. Indeed, if they were not, it would be possible to find a monotone sequence (tn)nEN of elements of I such that the X tn would not be uniformly integrable. This would contradict Theorem V.TI9 if the sequence were decreasing, the remarks above if it were increasing.

I

Convergence theorems We limit ourselves to stating the theorems corresponding to Theorems V.I7 and V.2I. The proofs present in fact no new difficulties. ~

T6 THEOREM Let (Xt ) be a right-continuous (or only separable) supermartingale. (a) Suppose that sup E[X;l < 00.

(6.1)

t

The random variables X t then converge a.s. to an integrable random variable X 00 as t ~ 00. (b) This condition holds in particular when the X t are positive. The process (Xt)tER+V{oo} is then a supermartingale. (c) Suppose that the X t are uniformly integrable. Condition (6.1) is then realized, the process (Xt)tER+V{oo} is a supermartingale, and the convergence of X t to X oo takes place in the Ll norm. (d) Suppose that the X t are uniformly integrable and that the process (Xt ) is a martingale. The process (~)tER+ v{oo} is then a martingale.

* This remark is due to Johnson and Helms, and will be used only in No. 21.

97

Regularity Properties of Paths

VI, T7, 8, 9, TIO

The following consequence of Theorem 6 is often used. Let Y be an :Foo-measurable, integrable random variable, and let (Yt ) be a right-continuous modification of the martingale (E[YI :FtD· We then have Y = limt~oo Yt a.s. Indeed, the random variables Yt are uniformly integrable from V.T19: The martingale (Yt ) thus converges a.s. from (c) and the limit is equal to Y from V.T18. Here is the theorem corresponding to V.T21. ...

T7 THEOREM Let (Xt)t>o be a right-continuous (or only separable) supermartingale relative to afamily of a-fields (:Ft)t>o. Let:Fo be the a-field ns>o :Fs • Suppose that

(7.1) sup E[Xtl < 00. t>O The random variables X t then converge a.s. to an integrable random variable X o when t ~ 0, the process (Xt)teR+ is a supermartingale relative to the family (:Ft)teR+, and the convergence of X t to X o takes place in the Ll norm.

8 We now introduce certain language conventions, which facilitate our work through the remainder of this chapter and Chapter VII. Let (At)teR+ and (Bt)teR+ be two real-valued stochastic processes with a.s. right-continuous paths (we henceforth simply call them right-continuous processes, as with the supermartingales). We say that (At) is dominated by (B t), or that (B t) dominates (At) and we write (At) < (B t), if At < B t a.s. for each t E R+, or even only for each rational t E R+. It then follows from right continuity that At(w)

< Bt(w)

for every

t

E

R+,

except for a negligible subset of Q. Suppose in particular that the right-continuous processes (At) and (B t) are modifications of one another. They then have essentially the same paths, and there is no reason to distinguish them. We shall speak in what follows as if they were the same process. Finally, we suppose henceforth that the family of a-fields (:Ft)teR+ is right-continuous. The Riesz decomposition 9

Let (Xt ) be a right-continuous supermartingale. We say that (Xt ) is (a) a potential if the random variables X t are positive, and if lim E[Xtl = O. t~oo

(b) a stochastic Dirichlet solution if the process (Xt ) is a uniformly integrable martingale (the random variable X 00 = lim n -+oo X n is then the corresponding boundary function). We then have the following theorem:

TIO THEOREM ("Riesz decomposition") Let (Xt) be a right-continuous supermartingale. The following two conditions are equivalent: (a) There exists a submartingale dominated by (Xt ). (b) There exists a martingale (Yt) and a potential (Zt) such that X t = Y t + Zt a.s. for each t E R+. These two processes are then unique up to modification. The martingale (Yt) dominates every submartingale (Ye') such that (Ye') < (Xt ).

VI, 11, 12, T13

98

Continuous Parameter Martingales

The proof is identical to that for the discrete case (V.T25). 11 Let (Xt ) be a right-continuous supermartingale, positive or uniformly integrable. Denote by (St) a right-continuous modification of the martingale (E[X <Xl §'tD, and set Tt = X t - St· The supermartingale (Tt) is positive and we have

I

lim Tt = 0

a.s.

t -+ <Xl

Indeed, we have lim t --->- <Xl St = X <Xl a.s., from T6 and V.T18.

2.

The Optional Sampling Theorem in the Continuous Case

12 We continue to suppose that the family (§'t) is right continuous. We consider a rightcontinuous supermartingale (Xt ) which has the following property: There exists an integrable random variable Y such that

Xt

> E[Y I§'t]

for each

t

E

R+.

(12.1)

The stopping times we consider will all be allowed to take the value + 00. We set XT(w) = Yew) on the set {T = oo}, as in V.27, for every stopping time T. The function X T was defined in IV.47, and was shown to be measurable with respect to the a-field §'T in IV.49. We call Y the "random variable at infinity," as we did in No. V.27. .....

T13 THEOREM Suppose that the supermartingale (Xt) satisfies (12.1). Let Sand T be two stopping times such that S < T. The random variables X s and X T are then integrable, and we have the supermartingale inequality

> E[XT I §'s]

Xs

a.s.

(13.1)

Proof Let n be a positive integer. Denote by D n the collection of numbers of the form kj2 n , and consider the discrete supermartingale (Xt)teD , relative to the family of a-fields " and to the stopping times SIn) (§'t)teD,,' Apply Theorem V.T28 to this supermartingale, and T
§'s'

> E[XT1n) I §'Sln)]

we have A

L,

Xs'o, dP

E §'SIn)

>

L,

a.s.

and

X p'O' clP.

Now we have

X s = lim Xs1n); n

X T = lim XT1n) n

from the right continuity of the supermartingale. The theorem will thus be established if we show that the random variables Xs1nl, X TIn) are uniformly integrable (II.T21). We show this, for example, for S. The stopping time SIn-I) is greater than SIn) and takes its values in D n • We thus have, from V.T28,

Xs1n) Set then, for every n EN,

> E[Xs1n-l) I§'Slnl].

99

The Optional Sampling Theorem in the Continuous Case

VI, 14, T15, TI6

The process (Yk)kE-N is a supermartingale relative to the family of a-fields (~k)kE-N' On the other hand, we have E[Y_ n ] < E[Xo] from V.T28. Theorem V.T21 then shows that the random variables Yk are uniformly integrable. 14 Remarks (a) Suppose that (Xt) is a uniformly integrable martingale. The processes (Xt ) and (-Xt ) then satisfy condition (12.1), from T6(d). The inequality in (13.1) can hence be replaced by an equality. (b) Theorem 13 is true, with no restriction on the right-continuous supermartingale (Xt ), if the stopping times Sand T are bounded by a constant a. We have, in fact, used only the order structure of the time set, and the sets [O,a] and [0,00] are order isomorphic. Condition (12.1) is satisfied, on the other hand, if for Y one takes the random variable X a • Applications The following theorem is a weakened translation of the "minimum principle" from potential theory, according to which a positive superharmonic function cannot be zero at a point without being zero everywhere. T15

THEOREM

Let (Xt)tER+ be a right-continuous, positive supermartingale. Set T(m)

= inf {t:

=

XtCm)

0 or Xt_(m)

=

O}.

The function t ~ X t(m) then has the value 0 from the time T(m) on, for almost every mEn. Proof Denote by T n the stopping time

T.(w)

= inf (t: X,(w) <

M.

We have X T (m) < I/n for every m such that Tn(m) < 00, from the right continuity of the paths. It then follows that the stopping times T n increase with n. We know, on the other hand, that T n < T. Set S = sUPn T n ; we are going to show that the path of almost every mEn has the value 0 from the instant S on. To do this it suffices, from right continuity, to show that we have Xt(m) = 0 a.s. for each rational number t, on the set {S < t}. To show this, we apply Theorem 13 to the pair of stopping times T n and S V t, giving the value o to the random variable Y in formula (12.1). We obtain, for every n,

Since the first expectation is less than I/n, we have a.s. that X SVt = O. This establishes the theorem. In potential theory it is shown that the upper envelope of an increasing sequence of superharmonic functions is again superharmonic. The following theorem is the translation of this result into martingale theory. ...

T16 THEOREM gales. Set

Let ((XntER)nEN be an increasing sequence ofright-continuous supermartinXt(m) = sup X:(m). n

The paths ofthe process (Xt) are then a.s. right continuous andfree ofoScillatory discontinuities.

VI, T16

Continuous Parameter Martingales

100

Proof It follows from the right continuity that X 1t (w)

2 < _ X t(w) -< ... -< Xn(w) t -< ...

for every

t E

R+,

and for almost every wE Q. We begin by supposing that the random variables X t are positive and integrable. The process (Xt ) is then a supermartingale, but it cannot be affirmed a priori that (Xt ) is separable. Keeping the notation of Theorem 1, and denoting by U(w;I;[a,bD and Un(w;I;[a,bD the number of upcrossings by the path of w for the processes (Xt ) and (X;), respectively, we have, from IV.21, that U(w;I;[a,bD

< lim inf Un(w;I;[a,bD. n

The expectation of the second member is finite, from inequality (1.3). It then follows that the first member is finite, and (from IV. T22) that the paths of the process (Xt ) are almost all free of oscillatory discontinuities. It will be simpler to reason as if they all were, which is possible by means of a trivial modification of the processes (X~). Put then Xt+(w) = Hm XsCw) s-+t s>t

The theorem will be established (under the particular hypotheses considered above) if we can show that, for every w E Q, for every t

E

R+.

Note first that the function t JW+ Xi w) is the upper envelope of a sequence of rightcontinuous functions. It is hence "lower semicontinuous from the right," and we have the inequality (16.1) for every t and every w. We show first that, for every stopping time T, a.s. on the set {T

< oo}.

(16.2)

Put T n = T + Iln, and apply T13, taking all random variables at infinity equal to O. The inequality E[X~ n ] < E[X~] passes to the limit from lebesgue's theorem, and we find that E[X T +] < Hm infE[X Tn ] < E[X T ] n

by virtue of Fatou's lemma. The comparison of this inequality and (16.1) shows that relation (16.2) is verified. Note then that the procc;;eses (X;), (X,+) are progressively ffi.easurable with respect to the family (:Ft), from IV.T The same thus holds for the process (Xt ) = (suPn X;), and for the process (Yt ) = (Xt . - X t ). Let e be a number >0; the function T(w)

= inf {t:

Yt(w)

> e}

is thus a stopping time from IV.T52. Since the paths of the process (Yt ) are "lower semicontinuous from the right," we have YT( w) = X T+( w) - X T( w) ~ e on the set {T < oo}. This is compatible with (16.2) only if T = 00 a.s., and this whatever be e > o.

101

The Optional Sampling Theorem in the Continuous Case

VI, D17, 18

We now must remove the restrictions we made at the beginning of the proof. Let a and k be two numbers >0; denote by (M t)o5,t5,a a right-continuous modification of the martingale (E[X~ §"tD, and by (Y~)o5,t5,a the process defined by

I

Y: = (X: - M t ) A k. Let Y t be the upper envelope of the Y;. The processes (Y;), ( Yt) satisfy the hypotheses of positivity and integrability on the interval [O,a], so that the paths of the process (Yt ) are a.s. right continuous. We have only to let a tend to infinity, then k to infinity. The preceding proof was adapted from Doob's reasoning concerning decreasing sequences of excessive functions. Remark The following consequence of Theorem 16 is often used. Let (X;) be an increasing sequence of right-continuous supermartingales, and let (Yt ) be a right-continuous supermartingale such that for each t:

Yt

= sup X:

(16.3)

a.s.

n

We then have a.s. ~(w) =

sup X:(w)

for every t

E

R+,

(16.4)

n

and in particular, for every stopping time T, YT(w) = sup X;(w)

a.s.

(16.5)

n

Consider, indeed, the process (Xt ) = (suPn X7), which is right continuous from T16. We have a.s. for every rational t, from (16.2). The two processes (Xt ) and (Yt ) being right continuous, this relation implies (16.4). Uniform integrability properties D17 DEFINITION Let (Xt)tER+ be a right-continuous supermartingale relative to the family (§"t)tER+' and let .r be the collection of all the finite stopping times relative to this family (respectively, .ra the collection of all stopping times bounded by a positive number a); (Xt ) is said to belong to the class (D) (respectively belong to the class (D) on the interval [O,a]) if the collection of random variables X T , T E.r (respectively, T E .ra) is uniformly integrable. (Xt) is said to belong to the class (DL), or locally to the class (D), if(Xt ) belongs to the class (D) on every interval [O,a] (0 < a < (0).

18 Remarks (a) There exist uniformly integrable supermartingales that do not belong to the class (D), as Johnson and Helms (80) have shown (see No. 21). This is a fundamental difference between the discrete case and the continuous case (cf. V.T30). (b) Let (Xt ) be a supermartingale of the class (D). It is then uniformly integrable, and Theorem 6 implies the existence of the random variable XC() = limt-+C() Xt. We can thus define the random variable X T for every stopping time T, finite or not. The relation X T = limn-+C() X TAn and Theorems II.T21, II.T20 then imply the uniform integrability of all the variables X T' whether T is finite or not. Here are several criteria for belonging to the class (D).

VI, T19, T20

Continuous Parameter Martingales

102

T19 THEOREM (a) Every right-continuous martingale belongs to the class (DL); every right-continuous and uniformly integrable martingale belongs to the class (D). (b) Every negative right-continuous supermartingale belongs to the class (DL).

Proof Let (Xt ) be a right-continuous martingale, and a an instant; for every stopping time T < a we have

from T13. It then suffices to apply V.T19. The reasoning is the same in the case where (Xt ) is uniformly integrable, replacing X a by X oo • Let (Xt ) be a negative right-continuous supermartingale, let a be an instant, and T a stopping time bounded by a. We have

from T13. We also have E[XTl > E[Xal, and consequently ).P{lXTI > ).} < E[lXTIl < E[IXall. The probability of the set {IXTI > ).} thus tends to 0 uniformly in T as ). ~ 00. It then follows from II.T19 that the third integral tends to 0 in the same manner, and this implies the desired uniform integrability. Remark A uniformly integrable supermartingale that belongs to the class (DL) belongs also to the class (D). The proof of this result is very much like that of V.T30. T20

THEOREM

*

Let (Xt ) be a positive, right-continuous supermartingale, such that Hm X t = 0 t-+

a.s.

00

Let us set, for every stopping time T, XT(w)

We denote by Rn(n

E

=0

on the set

{T

= oo}.

(20.1)

N) the stopping time

RnCw) = inf {t: Xlw)

> n}.

The following conditions are then equivalentt : (a) X t belongs to the class (D). (b) limn _ oo E[XTJ = 0 for every increasing sequence (Tn)nEN of stopping times, which converges to + 00. (c) lim E[XRnl = o. n-+oo

Proof We establish the implications (a) => (b) => (c) => (a). Suppose (a) is satisfied. The random variables X Tn are then uniformly integrable, from remark 18(b). On the other hand, we have lim n _ oo X Tn = 0 a.s. Thus we do have lim n _ oo E[XTJ = 0, by virtue of II.T21. The implication (b) => (c) is clear, since the stopping times Rn increase to infinity, from T2 (they even tend to infinity by jumping to infinity). • Johnson and Helms (80). t If the above convention (20.1) is not made, E[XTn] must be replaced in (b) by E[XTi{Tn
103

The Optional Sampling Theorem in the Continuous Case

VI, 21, 22

We establish finally the implication (c) => (a). Let T be an arbitrary stopping time, and T' the stopping time defined by T(W) if XT(w) > n T'(w) = { .. + 00 otherwIse We have Rn

< T', and consequently E[XRnl > E[XT'l, from T13. In other words, E[XRnl

> ){X r "2n} X T dP, T

for every stopping time T. The random variables X T are thus uniformly integrable. 21 Remark We outline now, following Johnson and Helms (80), for the attention of the readers who are familiar with the theory of Brownian motion, the construction of a uniformly integrable supermartingale that does not belong to the class (D). Let (B t ) be a Brownian motion in the space R3, starting from a point ~ other than the origin. Let r(x) be the distance of the point x E R3 from the origin, and let h be the superharmonic function l/r(x) (the Newtonian kernel). The process (Xt ) = (h 0 B t ) is a positive supermartingale with continuous paths. The expectation E[Xtl is easily computed and is easily verified to be a continuous function of t. This implies that the supermartingale (Xt ) is uniformly integrable, from Remark 5. But, on the other hand, for a continuous supermartingale (Xt ),

XR n = n

on the set

{Rn

< oo},

so that condition (c) of Theorem 20 can be written simply

< oo} = O. hex) > n (the ball

Hm nP{R n n-+oo

(21.1)

Let D n be the set of points x such that of center 0 and radius l/n). The probability P{R n < oo} is equal to the probability of hitting D n in a Brownian motion starting at ~, which is 1

if

1

I.f

nr(~)

r () ~

>-n1

Expression (21.1) thus maintains the constant value l/r(rx) for large enough n, and the supermartingale (Xt ) cannot belong to the class (D). 22 Remark Let us return to the notation of T20: The supermartingale obtained by stopping (Xt ) at time T n = Rn 1\ n then belongs to the class (D), since it is dominated by the integrable random variable X Tn V n.

CHAPTER

VII

Generation of Supermartingales

The conventions and hypotheses of the preceding chapter will be used again in this one (see the introduction to Chapter VI, and No. VI.8). The stopping times we consider will be allowed to take the value + 00, unless explicitly mentioned to the contrary. The use without further explanation of such a notation as X T' where (Xt ) denotes a process and T a stopping time, already implicitly supposes that the limit X co = lime_co X t exists a.s., and that X T = X co on the set {T = oo}. Recall that a stochastic process (Xt ) can be considered as a function of the two variables t, co; this identification allows the use of such notations as (X;) (sum of a series of realvalued processes), etc.

In

1. The Discrete Case 1 Let (!F n)neN be an increasing family of sub-a-fields of !F, and let (Xn)neN be a supermartingale relative to the family (!F n). Define the random variables Yn' An' by induction, in the following manner:

Yo = X Q YI = Y o + (Xl - E[XI I!FoD

+ (Xn -

I

Ao = 0

I

Al = X o - E[XI !Fo]

E[Xn !F n-l]) The following properties are easily verified: (a) X n = Y n - An for every n. (b) The process (Yn ) is a martingale. (c) An is obtained from A n- l by adding a positive quantity; i.e., the paths of the process (An) are increasing functions of n. (d) A o = 0; An is !F n_I-measurable for every n, and integrable. We call any process (Rn), adapted to the family (!F n) and having the following properties, an increasing process: (~) Rn is integrable for every n; R o = O. (fJ) The paths of the process (Rn) are increasing functions of n. Y n = Y n- l

104

105

Increasing Processes

VII, 2, D3, 4, D5

The preceding construction shows that every discrete supermartingale (Xn) is equal to the difference of a martingale and an increasing process. This remark has been used by Doob, who has posed the problem of the existence of such a decomposition in the continuous case. We solve this problem in this chapter, and see that the decomposition (which we call the "Doob decomposition") is then possible only for certain supermartingales. We also study the possibility of decomposing a supermartingale by means of an increasing process with continuous paths. 2 Consider now the uniqueness of such decompositions. Starting from an increasing process (B n) and a martingale (Zn), form the supermartingale (Xn) = (Zn) - (Bn), and construct the process (Yn) and (An) as above. We then have B n = Anfor every n if and only if B n is ~ n_l-measurable for every n. There thus exists only one decomposition of (Xn ) by means of an increasing process which satisfies property (d). We have an analogous uniqueness theorem in the continuous case, but the condition defining the "natural" increasing process that enters into the decomposition will be much more complicated. We hope that these few remarks will be of help to the reader in understanding this chapter, which begins in a rather abrupt manner with a certain number of "technical lemmas" on increasing processes (which have some independent interest). The fundamental results are contained in Sections 3 and 4, which the reader can peruse first, if he so desires.

2.

Increasing Processes

D3 DEFINITION Let (At)teR be a real-valued stochastic process, adapted to thefamily + We say that (At) is an increasing process if (1) The paths t ~ At(w) are a.s. zero for t = 0, increasing, and right continuous. (2) The random variables At are integrable. We say that the increasing process (At) is integrable if sup E[Atl t

< 00.

(~t).

(3.1)

4 Remarks A process adapted to the family (~t) that satisfies condition (1), but not necessarily condition (2), is said to be increasing in the broad sense. Condition (1) implies the existence of the random variable lim t _ oo At = A oo • An increasing process is integrable if and only if E[Aool < 00. D5 DEFINITION Let (Xt) be a right-continuous supermartingale. We say that (Xt ) admits a Doob decomposition if there exists a right-continuous martingale (Mt) and an increasing process (At), such that X t + At = M t for every t E R+. (5.1) Suppose in particular that (Xt ) is a uniformly integrable potential; since the expectations E[Mtl and E[Xtl are bounded, condition (3.1) is satisfied, and A 00 is integrable. The random variables At, being dominated by A oo ' are then uniformly integrable; so are the random variables (Xt) by hypothesis, and hence so is the martingale (M t). We thus have M t = E[M 00 ~tl a.s., from VI.T6(d). Now X 00 = 0, and formula (5.1) then takes the form

I

I

X t = E[A oo ~tl - At a.s. This leads us to the following definition.

VII, 06, T7, 08

Generation of Supermartingales

106

D6 DEFINITION Let (At) be an integrable increasing process, and let (Mt) be a rightcontinuous modification of the martingale (E[A oo §'t)); the process (M t - At) is called the potential generated by At.

I

The modification (M t ) considered exists from VI.T4. The expression "the potential generated by (At)" conforms to the convention of No. VI.8 in not distinguishing between right-continuous modifications of a single process. We now need only to justify the definition as a "potential." ~

T7 THEOREM Let (Xt) be the potential generated by the integrable increasing process (At). (1) (Xt ) is a potential of the class (D). (2) For every stopping time T we have XT = E[A oo

I§'Tl -

AT

a.s.

(7.1)

Proof The process (M t) is a right-continuous martingale, and the process (At) a rightcontinuous submartingale, so that (Xt ) is a right-continuous supermartingale. We have, for each t E R+,

The paths of the process (Xt ) are thus a.s. positive, in view of the right continuity. We finally have Hm E[Xtl = E[Aool - Hm E[Atl = 0, t-+oo

t-+oo

from Lebesgue's theorem; (Xt ) is thus a potential. Let .r be the collection of all stopping times. The random variables M T(T E:Y) are uniformly integrable from VI.TI9. The random variables AT(T E:T) are dominated by A oo ' and hence are uniformly integrable. It then follows that (Xt ) belongs to the class (D). To establish assertion (2), it suffices to note that

and M T = E[A oo

I§'Tl

a.s.

(from VI.TI3 and 14).

We show later that the converse of this theorem is true: Every potential of the class (D) is generated by an integrable increasing process (not necessarily unique). This will allow us to find a necessary and sufficient condition for a supermartingale to admit a Ooob decomposition. The rest of this section is devoted to the study of integrable increasing processes. We begin with some elementary remarks, not worth being stated as theorems. Strong order properties D8 DEFINITION Let (At) and (B t) be two increasing processes. We say that (B t) dominates ,(At) in the strong sense, and we write (At) (B t), if the process (B t - At) is increasing. Let (Xt ) and (Yt ) be two right-continuous supermartingales. We say that (Yt ) dominates (Xt) in the strong sense, and we write (Xt) (Yt), if the process (Yt - X t) is a positive supermartingale.

« «

107

Increasing Processes

VII, 9-11

«

9 Remarks (a) Let (At) and (B t) be two integrable increasing processes such that (At) (B t), and let (Xt) and (Yt) be their respective potentials. We then have (Xt) (Yt). (b) There is evidently an analogous definition for increasing processes in the broad sense. (c) Let (A~) (n EN) be a sequence of increasing processes that increases in the strong sense. Suppose that sUPn E[A~] < 00 for every t E R+, and put At = sUPn A~. We will show that (At) is an increasing process. Since the paths of the process (At) are a.s. increasing, and zero for t = 0, and each random variable At is integrable, we need only show that the functions s ~ AsCw) are a.s. right continuous. Now for s < t we have

«

Since the random variable At is a.s. finite, there is a.s. uniform convergence on the interval [O,t], which implies the desired right continuity. We note also, in view of later applications, that continuity a.s. of the processes (A~) implies the same property for the process (At). The continuous and discontinuous parts of an increasing process 10 Let (At) be an increasing process, and e a number >0. Define by induction the stopping times: T~+I(w) =

inf {t: t

> T~(w), AtCw) -

At_(w)

> e}

(see No. IV.44). Then, for every t E R+, set A:(w) =

L

(AT~(w) - AT~Jw))

T~(co)~t

(the sum of the jumps larger than e). It is clear that the processes (A~) are increasing processes dominated in the strong sense by (At), and strongly increasing as e decreases. They thus converge, when e ~ 0, to an increasing process (A ~), called the purely discontinuous part of (At). The increasing process (A~) = (At - A~) then has a.s. continuous paths; it is called the continuous part of (At). 11 The decomposItion of the discontinuous part can be pursued further. Given a decreasing sequence (en) of strictly positive numbers, which converges to zero, put B tn --

A£n+l _

t

A£n

t .

The increasing process (A~) is the sum of the processes (B~). Each path of the processes (B~) has a finite number of discontinuities on every compact interval; thus denote by T nm( w) the instant when the mth discontinuity of the function t .A.f'.I+ B~t( w) occurs, and by a nm ( w) the size of the jump at this instant. It is easily verified that T nm is a stopping time, and that anm is §'T nm -measurable. The process defined by

is thus an increasing process, with paths having at most one discontinuity, and we have

n,m

VII, T12, D13

Generation of Supermartingales

108

Change of time associated with an increasing process

This notion will be an important tool for us, both in martingale theory and in the study of Markov processes. , The following lemma was already known to Lebesgue. T12 THEOREM Let a be a function defined on R+, with positive values, not necessarily finite, which is increasing and right continuous. For every t E R+ put c(t) = inf {s: a(s)

> t}.

(12.1)

The function c is then increasing, right continuous, and such that a(s) = inf {t: c(t) Suppose that a(O)

= 0, and let f

> s}.

(12.2)

be a positive Borelfunction on R+: then

f/(l) da(l) = (OO)/(C(I»

dl.

(12.3)

Proof The function c clearly is increasing, and right continuous at every point t such that c(t) = 00. Suppose that c is not right continuous at a point t such that c(t) < 00. There then exists a number h such that c(t) < h < c(t + e) for every e > o. These relations imply, respectively, the inequalities a(h) > t and a(h) < t + e for every e > 0, leading to a contradiction, and it follows that the function c is right continuous. Note that c(a(s» > s for every SE R+, and consequently c(a(s + e» > s + e > s for everye > o. We thus have a(s + e) > inf {t: c(t) > s}.

Since the function a is right continuous, it follows that a(s)

> inf {t: c(t) > s}.

Let t be a number such that c(t) > s. The definition of the function c implies the inequality a(s) < t. We hence have also that a(s) < inf {t: c(t) > s}. Relation (12.2) is thus established. Suppose that a(O) = 0 and, to simplify things a little, that the function a is bounded. We take forfthe indicator function of an interval [O,s], and verify relation (12.3). The left side is equal to a(s); the right side is equal to the length of the interval Is = {t: c(t) < s}, also equal to inf {t: c(t) > s} = a(s), from (12.2). Denote by.Ye the vector space of bounded Borel functions f such that relation (12.3) holds, and by C(j the collection of indicator functions of intervals of the form [O,t]. Theorem I.T20 shows that .Ye contains all of the bounded Borel functions. Formula (12.3) is then verified for all positive Borel functions by means of a passage to the limit. D13 DEFINITION Let (At) be an increasing process in the broad sense. The system of stopping times (Ct)teR+ defined by clw) = inf {s: AsCw) > t} is called the change of time associated with (At). Let (Xt) be a stochastic process progressively measurable with respect to the family (~t). The process (XCt)teR+ is called the transform of(Xt) by the change of time (c t).

109

Increasing Processes

VII, 14, TI5

The relations C t < a and Aa > t are equivalent for every a E R+. Since Aa is ~a-measur­ able, the random variables C t are clearly stopping times, as stated in the definition. The paths of the process (c t ) are right continuous. Right continuity of process is thus preserved under this transformation. Integration with respect to an increasing process 14 Let (At) be an increasing process in the broad sense (we suppose only, for simplification, that the random variables At are a.s. finite). Let (Xt) be a measurable process with positive values (see IV.D45). Since each function t ~ Xt(w) is measurable from II.TI4, we can consider for each WEn the Lebesgue-Stieltjes integral on R+:

f.oo Xt(w) dAt(w). This integral is an ~-measurable function of w, from Fubini's theorem (II.TI4). Suppose in particular that the process (Xt ) is progressively measurable with respect to the family (~t). Consider the process (Yt) defined by

t f.t X

Y =

s

dA s

(the point t being included in the interval of integration): The same reasoning as above shows that Yt is ~t-measurable for every t E R+. The process (Yt ) admits, on the other hand, right-continuous paths. It is hence progressively measurable with respect to the family (~t), from IV.T47. Let T be a stopping time; the random variable

T= f.T X

Y

is

S

dAB

~t-measurable

from IV.T49. We shall consider only positive random variables in the following theorems, in order to avoid considerations of integrability. The symbols E[·I .] will denote generalized conditional expectations.

T15 THEOREM Let (Xt ) and (Yt ) be two measurable stochastic processes with positive values [not necessarily adapted to the family (~t)] such that, for each stopping time T, (15.1)

E[XTI{T< oo}] = E[YTI{T< oo}]· We then have,for every increasing process (At) and every t

< 00, (15.2)

Proof We begin by treating the case t = + 00. Introduce the change of time (cs) associated with the increasing process (As). It follows, from Theorem 12 and Fubini's theorem, that

E[f.oo X. dA.] = E[f.oo X"I{,.
= f.OOE[X'/l"

<

oo}) ds.

The process (Ys ) satisfies an analogous relation, so that it suffices to note that for each s, since Cs is a stopping time,

VII, T16, T17

110

Generation of Supermartingales

The case where t is finite is treated by applying the preceding result to the process (Bs ) defined by for s t. Remarks Let T be a stopping time, H an element of :FT; a new stopping time THis defined by setting T

W _ (T(W) H( ) 00

for for

WEH W ~ H.

Apply formula (15.1) to the stopping time T H' where H ranges over !FT' We obtain the following formula (which is apparently stronger):

I

I

E[XT1{T< oo} :FT] = E[YTI{T< oo} !FT]

a.s.

(15.3)

Formula (15.2) can be strengthened in an analogous manner. T always denoting a stopping time, H an event which ranges over !FT' apply (15.2) (for t = + (0) to the increasing process (Bs) defined by B s = IH(A s - AT)I{s>T}'

We obtain the general formula

E[f

X,

dA,1 ffTJ = E[f Y, dA,1 ff TJ

a.s.

(15.4)

Here is a simple, often useful corollary of the preceding theorem.

T16 THEOREM Let (Yt ) be a positive right-continuous martingale and let (At)bean increasing process. Then for every t E R+ we have

E[A,Y,] = EU: Y, dA.J. This inequality also holds for t =

(16.1)

+ 00 if the martingale (Yt) is uniformly integrable.

Proof Let us establish this last point first. Since the martingale (Yt) is right continuous and uniformly integrable, we have YT = E[Yoo :FT] a.s. for every stopping time T (VI.T13 and 14). It then suffices to apply the preceding theorem to the process ( Yt) and the constant process equal to Y00' Equality (16.1) is easily established in the same way, but it is still simpler to reduce to the case already treated, by noting that the ordered sets [O,t] and [0,00] are isomorphic. The following statement applies in particular to two integrable increasing processes which generate the same potential. The use of the expression "increasing processes" supposes implicitly that they are adapted to the family (:Ft). The reader will note that this hypothesis is not used in the proof.

I

T17

THEOREM

Let (At) and (B t) be two increasing processes such that

I

I

E[B t - B s :Fs] = E[A t - As :Fs] a.s.

(17.1)

for every pair of numbers s, t such that 0 < s < t < 00. Let (Y t ) be a process with positive values, adapted to the family (:Ft), and having left-continuous paths a.s. We then have, for every t < + 00,

(17.2)

111

VII, Dl8

Uniqueness of the Doob Decomposition

Proof It suffices to treat the case where t is finite and where the random variables Y s are bounded by a constant. The general case can then be deduced by passing to a monotone limit. Suppose then that the process (Ys) is bounded, and set, for every integer n > 0,

Y: = Y o = Y!£t

Y:

J

k

k+l

SE;; t, ~ t

for

n

J

(k EN).

Since the paths of the process (Ys) are left continuous, it will suffice to establish (17.2) for the process (Ysn), and then to let n tend to infinity, using Lebesgue's theorem. Now we have

!FkJJ.

I -t -t -t -t n n n n We also have an analogous relation for the process (Bs). It then is sufficient to note that from (17.1) = ni1E[Yk E[Ak+l - A k o

E[Ak+ln t-

A!£t n

I!F~tJ n

=

E[Bk+ln t-

B~tn I!F~tJ n

a.s.

Remarks (a) In formula (17.2), the expectations can be replaced by conditional expectations E[- !Fo]· (b) Formula (17.2) extends to all positive processes (Yt ) such that the mapping (t,w) .J\,/I/'+ YtCw) is measurable with respect to the a-field on R+ X n generated by the processes with left-continuous paths.

I

3 - Uniqueness

cif the

Doob Decomposition

We begin by establishing the uniqueness theorem, whose proof requires less technique than that of the existence theorem. Natural increasing processes The definition of "natural" increasing processes, which we now give, will doubtless appear particularly artificial. We shall see later that the "natural" processes are, roughly speaking, the limits of increasing processes with continuous paths. We shall also see that they appear "naturally" in the proof of the existence theorem. D18 DEFINITION process if

Let (At) be an increasing process. We say that (At) is a natural increasing (18.1)

for every t

E

R+ and every positive, bounded, right-continuous martingale (Yt ).

This definition is simplified in the particularly important case where the increasing process is integrable. We then have the following theorem.

VII, T19, T20

Generation of Supermartingales

112

T19 THEOREM Let (At) be an integrable increasing process. (1) The process (At) is natural if and only if (19.1) for every positive, bounded, right-continuous martingale (Y t ). (2) We then have also,for every stopping time T,

(19.2) (3) Under the same conditions the increasing process (B t ), defined by

B t = Al{t
+ A T I{f2:T}'

Proof Property (18.1) implies immediately (19.1), from Lebesgue's monotone convergence theorem. Conversely, suppose that (19.1) is satisfied. We show first that

(19.3) where the notations (Ys), (Bs) are as in the statement. Denote by (Zs) the process defined by Zs = Yl{s
Equality (19.3) is then deduced by subtracting from both sides the finite quantity

E[J,

]T.oo[

Zs dAsJ =

E[J:

Zs- dAsJ.

]T.oo[

Note that the equalities (19.3) and (19.2) have the same meaning; (19.1) hence implies (19.2), which implies in particular [by taking T = t in (19.2)] that (At) is natural. Relation (19.3) means, on the other hand, that the process (B t ) satisfies (19.1). The process (B t ) is hence natural itself. The following extension of (19.1) is often useful. T20 THEOREM Let (At) be a natural, integrable, increasing process, and let (Yt ) be a positive, right-continuous, uniformly integrable martingale. We then have

(20.1) Proof For each nE N, denote by (Y;) a right-continuous modification of the bounded positive martingale (E[Yoo " n :FtD -such a modification does exist, from VI.T4. We then have, since the process (At) is natural,

I

(20.2)

113

Uniqueness of the Doob Decomposition

VII, T21

for every nE N. It follows from Theorem VI.TI6 that, for almost every w EO, lim Y:(w) =

~(w)

for every

t E R+.

n-+oo

The left side of (20.2) thus tends to E [LOO Y t dA t ] , from Lebesgue's monotone convergence theorem. Similarly, for every

W

Y;_(w)

EO we have

<

y~~l(W)

for every t and every n.

Put Zt(w) = lim n-+ oo Y;_(w). We can apply the same reasoning to the right side of (20.2) if we show that, for every w E 0,

ztCw) = Now we have, for every

E

Y t-(w) for every t

E

R+.

> 0 and every n EN,

p(s~p (~_ - Zt-) > E} < p(s~p (~ - Y:) > E} =

P(i~f(Y: - ~) < -

E} < ~ E[~ -

Y~],

from inequality VI.l.2. It then follows that the left side is zero, and the theorem is established. The uniqueness theorem .....

T21 THEOREM Let (Xt ) be a right-continuous supermartingale. There exists at most one natural increasing process (At) such that the process (Xt + At) is a martingale. Proof * Let (B t ) be another natural increasing process, which has the same property. We shall show that, for each t E R+, At = B t a.s. This is indeed the desired result, since we do not distinguish two right-continuous modifications of the same process. It suffices to show that E[YA t ] = E[YB t ]

for every bounded, positive, ~t-measurable random variable Y [see remark 11.9(a)]. Denote by (Ys) a right-continuous modification of the martingale (E[ Y ~s])' We have

I

E[YA,] = E[f:Y. dA.]

E[f:¥.- dA.] E[YB,] = E[J:¥. dB.] = E[f Y.- dB.], =

from Theorem 16 and the fact that the two processes are natural. Since the process (B t - At) is a martingale, the hypothesis of Theorem 17 is satisfied, and so we have

This establishes the theorem. • This proof communicated by P. Courrege.

VII, 22

114

Generation of Supermartingales .

4. The Existence Theorem We begin by establishing the existence of an integrable increasing process generating a potential of the class (D) (T29). The necessary and sufficient condition for the existence of a Doob decomposition will be given in No. 31. The crucial point of the proof is the theorem concerning uniform integrability (T2S). We prove it directly for all natural, integrable, increasing processes, but the reader will note that this general form is not necessary for the existence proof. The latter uses it, in fact, only for increasing processes of the form

At = {H, ds, where Ht is a process adapted to the family (§'t), with positive values and right-continuous paths. This remark might lead to a more elementary proof of Theorem 29. Uniform integrability properties

22 We begin by proving a general formula for integration by parts, which seems not to be entirely classical. Let f and g be two increasing and right-continuous functions on R+, such that f(O) = g(O) = O. For simplicity, we suppose also that the quantities f(oo) = limt-.oof(t) and g( (0) = limt -. oo g(t) are finite. We then have

=f.

f( 00 )g( (0)

df(x) dg(y).

R+xR+

Denote by D+ the set of points of R+ X R+ situated above, but not on, the diagonal, and by D- the complement of D+. By applying Fubini's theorem to the integrals

f

n+

j(oo)g(oo)

Irn - d/(x) dg(y),

df(x) dg(y),

= flg(oo) -

g(u)] dj(u)

we obtain

+ flj(oo)

- f(u-)] dg(u),

(22.1)

where f(u-) denotes the left limit off at u. The general formula f(oo)g(oo) = fg(U) df(u)

+ ff(U-) dg(u)

(22.2)

is then easily deduced. This formula is not symmetric with respect to f and g. Its "symmetrization" yields the formula for integration by parts given by Hewitt (75). We point out the following formula, where p is an integer> 0, and where the functionf is assumed to be increasing, continuous, and zero for t = 0: j(oo)' =

Pl( o

dj(u,)

r ··r dj(u,)'

df(u,).

(22.3)

U~-l

Ul

The following identity is the basis for the theory of energy, which we shall present in Section 6.~It ,was,~sed f~r th~ tst time~b Volkonski (12~in a less general form. C'\!

v ",J..

~

J

(L\

\" (". .

\») - ""... .n} j

l.. f,. \!

-:.,

v,.... ..J.

,.t

{j

~1 P I ' (,..

i /_ \. '.

L-'.'::' i,.n ; l

i 0. '\ i IJ ') J) .

"" '1

..>;.. ~

(j (t.) .

.

~•. :'J ~, \

,./

.U -

t".

0

I

The Existence Theorem

115

VII, T23

T23 THEOREM Let (Xt ) be the potential generated by an integrable, natural, increasing process (At). We then have

E[A:') = E[f(X' + Xt-> dA'l

(23.1)

Proof Denote by (Af) the bounded integrable increasing process (At A n)(n EN), and by (Xf) the potential of (A f). The process (X~ + A~) is a right-continuous modification of the martingale (E[A~ I ~t]), and the increasing process (At) is natural. We thus have, from T16 and T20,

E[AooA~) =

E[(X; + A:) dA,J

=

E[f(X~ + A~) dA,J.

and consequently

2E[AooA~) = E[( (X: + X:_ + A: + A:_) dA'l The left side is also equal to

from formula (22.2). Since the second integral is finite, we are led to the relation

w~

then let n tend to infinity. The right side above tends to

E[(A' + A.-> dA,J = E[A:'] [formula (22.2)]. On the other hand we have, from VI.T16, Hm X:(w) = X t ( w) for every t

a.s.

n

The random variables X;_ increase with n. They thus have a limit as n ~ denote by Yt. We show that

ye<w) = Indeed, we have for every c > 0, p(suP (X t_ - Yt)

X t _(w)

for every t

00,

which we

a.s.

> cl < p(s~p (X t - x:) > cl < P(i~f (x: + A: -

X t - At)

< -cl

< ! E[A oo - A~] c

from formula VI.l.2. Since this last expectation tends to zero when n ~ 00, we see that the left side is zero. It then only remains to apply Lebesgue's monotone convergence theorem.

116

Generation of Supermartingales

VII, T24, T25

T24 COROLLARY Let (At) be an increasing natural integrable process whose potential (Xt) is dominated by a constant c. We then have

(24.1) Proof We have

E[A:'l = E[f<x. + X._) dA.] < 2cE[A ro l = 2cE[Xol < 2c'. Remark It is possible to establish Theorems 29 and 37, which are the fundamental results of this chapter, by using relation (23.1) only for an increasing continuous process (At). This relation is then almost immediate. In fact, we have, from (22.1),

A:'

f

=2

[A ro

A,l dA,.

-

Taking the expectation of both sides, and noting that X T = E[A oo every stopping time T, we can apply T15, which yields

E[A:'l = 2E[f
A,) dA,]

-

-

AT

I.:FT] a.s. for

= 2E[f X, dA,J.

which is equivalent to (23.1) when the process (At) is continuous. T25 THEOREM Let (Yt ) be a right-continuous potential belonging to the class (D), and let .91 be the collection of all integrable increasing processes that generate a potential dominated by (Yt). The collection of random variables A oo associated with the processes (At) E.9I is then uniformly integrable.

Proof Let c be a positive constant. Denote by T c the stopping time such that

= inf {t:

Tiw)

YtCw)

> c},

and by r(c) the expectation E[ Y T J. We have seen in VI.T20 that r(c) tends to zero as c tends to 00. For every process (At) E.9I, denote by (A~) the integrable increasing process defined by A~ = AtI{t
uniform integrability will follow without difficulty. Indeed, let e be a number >0, let us choose c large enough so that r(c) < e/2, and show that

1 +1

Aoo

dP

{Aoo>a}

~e

independently of Aoo ' if a is large enough. This integral is dominated by r(c)

{A oo>a}

A~ dP

from property (1), and it remains to show that the second integral can be made less than e/2 for a large enough. Since the random variables (A~) are uniformly integrable, from

117

The Existence Theorem

VII, 26, 27

property (2) (see II.T22), it suffices to show that the probability P{A oo > a} is dominated by a function of a which tends to zero as a tends to infinity (II.T19). Now we have

P{A oo

> a} < -1 E[A oo ] = a

1

- E[Xo]

a

< -1 E[Yo]. a

Let us now establish properties (1) and (2). The first one follows from

E[A oo

-

A~]

= E[A oo - AT) = E[E[A oo = E[XTJ < E[YT ) = r(c).

-

I

ATe ~TJ]

To establish the second, note that the process (A~) is natural (T19). It hence suffices to show that the process (XD is dominated by c, and to use T24. Note that the inequality YtCw) > c implies TC<w) ~ t, and hence A~(w) - A~(w) = O. Consequently,

X~

= E[A~

I ---,= E[(A~ - A~)I{l't:S;c} I~t] A~ I~]It{Y,:s;c} < Yi{y,:s;c} a.s.

- A~ §"t]

= E[A~ -

We then conclude by using the right continuity of the process (X~). 26 In No. 1 we constructed the increasing process (An) associated with a discrete supermartingale (Xn ). We had n-l

An =

!

(Xk

k=O

-

I

E[Xk+l ~k])'

It is natural to seek an analogous formula in the continuous case, in which the summation is replaced by an integration, and the difference operator under the sign "~" is replaced by a "derivative." This last notion is, however, difficult to define. We hence use a difference quotient and pass to the limit. It is possible to use another construction procedure, the passage from the discrete case to the continuous case by means of finer and finer subdivisions. The proof is then simpler, but the procedure unfortunately does not always lead to a natural increasing process. 27 DEFINITION Let (Xt)teR+ be a right-continuous potential of the class (D), and let h be a number >0. We denote by (PhXt)teR+ a right-continuous modification of the supermartingale Yt

= E[XHh I ~t]·

(27.1)

(a) This definition must be justified. Let sand t be two instants such that s

I

< t. We have

I

E[ Yt §"s] = E[XHh I §"t I §"s] = E[XHh ~s] = E[XHh I §"s+h I §"s] ~ E[XS+h ~s]

I

= Ys

a.s.

The process (Yt ) thus is a supermartingale. Since the function t -¥I'+ E[ Yt ] = E[XHh ] is right continuous, Theorem VI.T4 establishes the existence of a right-continuous modification of (Yt ). We have Yt ~ X t a.s. for each t. This implies that the supermartingale (PhXt) is dominated by (Xt ); it is hence also a potential of the class (D). (b) Let T be a stopping time (finite or not); we have

I

PhXT = E[XT+h ~T]

a.s.

(27.2)

VII, T28

Generation of Supermartingales

118

Indeed, denote by (Tn ) a sequence of elementary stopping times which decreases to T-the stopping times T
f.lh X T.

dP

=

Let n tend to infinity; the left side tends to

L

X T.+, dP.

L

PhXT dP, from the right continuity of the

supermartingale (PhXt) and the uniform integrability of the random variables Ph X T n [note that this supermartingale belongs to the class (D), or use V.T21(a)]. The right side tends similarly to

LX + dP, and formula (27.2) is thus established. T h

Definition 27 extends to any right-continuous supermartingale [but it is then necessary to take a few precautions in the use of formula (27.2)]. The constant h can also be replaced by a stopping time. These generalizations will be of no use to us. Remark

T2S THEOREM Let (Xt ) be a right-continuous potential of the class (D), and let (A~) be the increasing process defined by

h At =

it o

X s - PhXs

h

ds (t

E

R+).

(28.1)

(1) The increasing process (A~) is integrable and generates a potential (X:) dominated by (Xt ). For every stopping time T we have

X~ = ~ E[fX ";+. ds I j " J T

a.s.

(28.2)

(2) The potentials (X~) increase as h decreases: for every stopping time T we have

Hm X~ = X T

in the L 1 norm.

(28.3)

h-+O

Proof We have

hEIA:1 = E[f.'<X. - Ph X.) dsJ = E[J:<X. - X>+h) dsJ = E[f X. dsJ - E[r'x. dsJ 0. We have

t (A~+.

- A})

dP = fJf." ~ (XT+. -

Ph X T+.) dS) dP.

But from (27.2) we have

t

Ph XT+.

dP = LEl XT+>+h I T+.l dP = LX T+.+, dP. j"

119

The Existence Theorem

VII, T29

and consequently

In

(A'T+u - A'T) dP

=

fJf ~

(XT+' - X T+H ,) dS] dP

Let u tend to infinity, noting that

E[J.U+AXTHdS]

= fE[XT+uHl ds < hE[XT+ul.

This last quantity tends to zero as u tends to infinity, since (Xt ) is a potential of the class (D). We thus have

L

X~ dP = L (A~ - A~) dP = L(~fXT+' dS) dP.

which is equivalent to (28.2). We then have

The function

S.A.N+ fB X T+s dP is decreasing, right continuous, and tends to fB X TdP as

s -+ O. The integral of the first member thus increases as h decreases. This implies [1I.9(a)] that the random variables X~ increase a.s. when h decreases. Since their integrals remain bounded, they admit a limit in Ll, which can only be X T • Let (h n ) be a sequence that decreases to zero. The process (X:n) forms an increasing sequence of right-continuous supermartingales. We hence have, for almost every WE Q,

xtCW) =

lim X~f1(w) for every t

E

R+,

n-+oo

from VI.T16. We now arrive at the existence theorem.

-+ T29

Let (Xt ) be a right-continuous potential of the class (D). There then exists an integrable, natural, increasing process (At), which generates (Xt), and this process is unique. For every stopping time T we have THEOREM

AT = Hm A~

(29.1)

h-+O

in the sense of the weak topology a(Ll,L(0). Proof The potentials (X:) generated by the increasing integrable processes (A~) are dominated by (Xt ). The random variables A~ are thus uniformly integrable from T25. Let U be an ultrafilter* on R+""-{O}, which converges to O. It follows from the DunfordPettis compactness criterion (II.T23) that the mapping h .A.N+ A~ has a limit along U, in the

* The reader can avoid the use of ultrafilters if he so desires, and reason with sequences which tend to 0, due to assertion (3) of II.T23.

VII, T29

Generation of Supermartingales

120

topology U(Ll,L (0). Denote this limit by Aoo ' and a right-continuous modification of the martingale (E[A oo ~t)] by (Mt). Finally, set

I

At = M t - X t for every t E R+.

(29.2)

Let us prove that (At) is a natural, integrable, increasing process that generates (Xt). Note first that the random variables A~ are ~ oo-measurable; thus the same holds for their limit Aoo [see 1I.9(b)]. We hence have limt-+ oo M t = Aoo a.s. (VI.T6 and V.T18). On the other hand, limt-+ oo X t = 0 a.s., and consequently lim At t-+ 00

= A oo

(29.3)

a.s.

Let T be a stopping time; we can write the relation

I

A~ = E[A~ ~T] - X~.

The conditional expectation operators are continuous in the topology U(Ll,Loo) (see 11.48). The first term of the right side thus converges along U to E[A oo ~T] = M T a.s. The second term on the right tends strongly to X T from T28. We hence have

I

lim A~ = M T - X T = AT u in the sense of the weak topology. In particular, for every two instants sand s < t, we have As = Hm AZ < lim A~ = At a.s.

u

(29.4) t

such that

u

(see Remark 1I.9(a)). It is clear that A o = 0 a.s. Since the paths of the process (At) are right continuous, these properties show that (At) is an increasing process, integrable from (29.3), which, from (29.2), generates (Xt). It remains to show that (At) is natural. - Let (Yt ) be a bounded, positive, right-continuous martingale. Since the process (A~) is continuous, we have (29.5)

We shall study the limits of both sides along U. We have first

IW'E[f

Y,

dA~] = 1i,T E[YwA~] = E[YwAwl =

E[fY. dA.J.

from T16 and the definition of the weak topology (Y00 being bounded). Considering the right side, suppose we are given an e > 0, and construct inductively the functions Tn+l (w) = inf {t: t

>

To = 0, T n (w); I Y t (w) -

YTn (w)1

> e}.

The reader can easily verify that these functions are stopping times (e.g. see IV.44 and IV.53). We have I YTn+l - YTnl > e on the set {Tn+l < oo}, which, since the paths of (Yt ) have left limits a.s. at every point, including t = + 00, implies that T n = 00 a.s. for large enough n. We then have

E [LOO Ys- dAs] = E o

[1 r

n J]T n .Tn +1]

~_ dA

s] •

VII, T29

The Existence Theorem

121

I

On the other hand, Ys- - YTJ < e for every SE ]Tm T n +1]. The sum under the expectation in the right-hand side thus differs in modulus from "~- n YTn (AT n+l - ATn ) by at most e A oo . We therefore have

where (l is a quantity dominated in modulus by e E[A oo ] = e E[Xo]. Let C be a constant that dominates the martingale (Yt ), and let p be a positive integer. We have

E[f Y.-

dA,]

= E[~: YT.(A T•., -

AT'>]

+ E[n~p YTn(ATn+l =

E[~: YTn(ATn+l -

AT)]

where (3 is a quantity dominated in modulus by CE[A oo way,

E[f Y.- dA~] = E[~: YT.(A~'H where (lh is dominated in modulus by e E[X:] CE[XT 21 ]. On the other hand, we have

liff E[:~:YTn(A}n+l -

-

AT)]

-

+ lX

+ fJ +

(l,

A T21 ] = CE[XT ). In the same

A~.)] + P' + (X',

< e E[Xo], and fJh is dominated by CE[X~ ] <

A~,.)]

21

=

E[~:YTn(ATn+l -

A Tn)] ,

from (29.4) and the definition of the weak topology. The difference between

JiN' E[f l',- dA~]

and

is hence at most equal to 2e E[Xol + 2CE[XT 21 ], a quantity which can be made arbitrarily small, since e is arbitrary and E[XT 21 ] tends to 0 when p ~ 00. Equality (29.3) thus passes to the limit along U, and Theorem 19 then shows that (At) is a natural increasing process. We can now complete the proof. We know in fact (T21) that the natural increasing process (At) generating (Xt) is unique. We thus have AT = lim A~ (in the weak sense in Ll) u

for every stopping time Tand every ultrafilter U on R+"'{O} which converges to O. Hence we also have The theorem is thus completely established.

VII, T30, T31

Generation of Supermartingales

122

The proof of Theorem 29 can serve as a model for a large number of analogous proofs, and we will have occasion to return to it several times. The following theorem is established in exactly the same manner; we state it only for a sequence of potentials, although it applies to convergence along any filter. T30 THEOREM Let (Yt ) be a right-continuous potential of the class (D), and let (X~)(n EN) and (Xt ) be right-continuous potentials dominated by (Yt ). Suppose that for every stopping time T we have lim X~ = X T in the sense of the topology a(L1, LOO).

(30.1)

n--+OO

Denote by (A~) [respectively, (At)] the integrable, natural, increasing process which generates (X;) [respectively, (Xt )]. We then have,for every stopping time T,

lim A~

=

AT in the sense of the topology a(L1 , LOO).

(30.2)

n--+ OO

This theorem applies in particular to increasing sequences of potentials dominated by a potential of the class (D), from VI.T16. Existence of the Doob decomposition ...

T31

THEOREM

A right-continuous supermartingale (Xt ) has a Doob decomposition

(31.1)

where (M t) denotes a right-continuous martingale and (At) an increasing process, if and only if(Xt ) belongs to the class (DL) (see VI.Dl7). There then exists a decomposition (31.1)for which the process (At) is natural, and this decomposition is unique. Proof Suppose that (Xt ) has a decomposition of the form (31.1). The process (Mt ) belongs to the class (DL) from VI.T19, and so evidently does (At). The supermartingale (Kt) thus belongs to the class (DL). Conversely, suppose that (Xt ) belongs to the class (DL). Let n be an integer >0, g a continuous, strictly increasing mapping from [O,n[ onto [0,00[, and h its inverse mapping. For every t E [0,00[, put

The supermartingale (Yt)with respect to the family (f§t) belongs to the class (D); decompose it into a uniformly integrable martingale and a potential (Y;) of the class (D) (VI.Tll), and apply the existence theorem T29 to (Y;). We thus obtain a natural, integrable, increasing process (B t ) such that the processes (Y; + B t), and hence (Yt + B t), are martingales. Finally, put t E [0, n[. for It follows immediately from the uniqueness theorem (T21) that the process (A~+l) coincides with (A~) on the interval [O,n[. There thus exists a natural increasing process (At), which coincides with (A~) on [O,n[ for every nE N. The process (Xt + At) is then a martingale (Mt ), and this decomposition is unique from T15.

The Existence Theorem

123

VII, T32

Remark Let us say that a positive right-continuous supermartingale (Xt ) admits a generalized decomposition if there exist two right continuous processes (At), (M t) and a sequence (Tn ) of stopping times such that: (a) The T n are a.s. finite and increase to + 00.

+

.

(b) X t At = Mt. (c) The processes obtained by stopping (At) at times T n are integrable and natural

.

mcreasmg processes. (d) The processes obtained by stopping (M t ) at times T n are martingales. Using Remark VI.22 (which is due to them), K. Ito and S. Watanabe* have proved the existence and uniqueness of generalized decompositions for all positive supermartingales.

Potentials of the class (D) and bounded potentials The following theorem is a consequence, often useful, of the existence theorem. T32 THEOREM (1) Let (X?) be a sequence of right-continuous potentials of the class (D), and let (Xt ) be the process (X~). Then (Xt ) is a potential of the class (D) if and only if E[Xol < 00. (2) Every right-continuous potential ofthe class (D) is equal to the sum ofa series ofbounded, right-continuous potentials.

2n

Proof Suppose that E[Xol < 00. It is already known (VI.T16) that (Xt ) is a right-continuous supermartingale. Let (A~) be an integrable increasing process that generates (Xf), and let (At) be the process 2n (A~). We have

It then follows from property 9(c) that (At) is an integrable increasing process, and the relation X t = E[A<X) ~tl - At a.s.

I

follows immediately from Lebesgue's theorem. The process (Xt ) is thus a right-continuous potential of the class (D), from T7. We next establish property (2). Let (Xt ) be a right-continuous potential of the class (D), and (At) an integrable increasing process that generates (Xt). For every nE N, denote by (A~) the integrable increasing process (At A n), which has a potential bounded by n, by (B~) the integrable increasing process (A~+l - A~), and finally by (X;) the potential generated by (B~). It is clear that each of these potentials is bounded, and that

Regular potentials and continuous increasing processes Since the natural increasing process (At) that generates a potential (Xt) is uniquely determined, we may ask ourselves which property of (Xt) implies that (At) has some given property. This problem will be solved now for the continuity of the paths of (At). Our • See their paper: Transformation of Markov Processes by Multiplicative Functionals (Ann. Inst. Fourier, Grenoble, 15 (1965), p. 13-30).

Generation of Supermartingales

VII, 033, T34

124

v

method rests on the fundamental lemma, which allowed Sur (118) to treat the analogous problem for Markov additive functionals [the absence of this lemma in Meyer (95) was compensated for by the introduction of a supplementary hypothesis, in reality unnecessary]. We speak, for abbreviation, of "continuous processes" in place of "processes with a.s. continuous paths". D33 DEFINITION Let (Xt)teR+ be a right-continuous supermartingale of the class (DL). We say that the supermartingale (Xt) is regular if, for every increasing sequence (Tn)neN of stopping times, which converges to a bounded stopping time T,

lim E[XTn ] = E[XT ].

(33.1)

n

Remarks (a) Every right-continuous martingale is regular. (b) Suppose that the regular supermartingale (Xt ) belongs to the class (D). Relation (33.1) is then true for every increasing sequence of stopping times (Tn ) , finite or not, which converges to a stopping time T, finite or not: Decompose (Xt ) into a uniformly integrable martingale (Mt ) and a potential (Yt ) of the class (D). Since the expectation E[MTJ is constant, equal to E[MT] (VI.TI3 and 14), it suffices to verify that lim E[YTn ] = E[YT ]. n

Now for every t

E

R+ we have lim E[YTnAt ] = E[YTAt ] n

from (33.1). We then note that the four integrals

f

{Tn=t}

YtdP,

f

{Tn>t}

YT dP, n

r

J{T=t}

~dP,

and

f{T>t} ~dP

are dominated by E[Yt ] (VI.TI3), and this last expectation is as small as one may wish when t is large enough. The theorems that follow are stated only for potentials of the class (D). The extension to supermartingales of the class (DL), which is easy, is left to the reader. We first give several elementary results for regular potentials, which will be useful in proving the main theorem. T34 THEOREM Let (Xt) be a potential of the class (D), which is right continuous and regular. (1) Every right-continuous potential (Yt) such that (Yt) (Xt ) (No. 8) is then regular. (2) There exists a sequence of regular, bounded, right-continuous potentials (X~), such that (Xt ) = (X~).

«

!n

Proof (1) There exists a potential (Zt) such that (Xt) = (Yt) + (Zt). Let T be a stopping time, and (Tn ) an increasing sequence of stopping times, which converges to T. For every n E N we have, from VI.T13,

and consequently also,

lim E[YTn ] > E[YT ] n

and

lim E[ZTJ n

> E[ZT].

The Existence Theorem

125

VII, T35, T36

These two inequalities must, in fact, be equalities, since an equality is obtained by adding them, from the regularity of (Xt ). To establish (2), it suffices to consider the bounded potentials (X;) constructed in No. 32 and to note that each of them is regular from (1). T35 THEOREM Let (Xt ) be a potential of the class (D), which is right continuous and regular. The potential (PhXt) (No. 27) is then regular.

Proof Let (Tn ) be an increasing sequence of stopping times, which converges to a stopping time T. From formula (27.2) we have E[PhXTJ = E[E[XTn+h TJ] = E[XTn+h], and similarly E[PhXT] = E[XT+h]. It then suffices to note that the stopping times T n + h increase to T + h. v Here now are the two results borrowed from Sur. Assertion (2) will be generalized later, in the section on energy (cf. No. 63).

I§"

T36 THEOREM Let (Xt ) be a regular, right-continuous potential of the class (D). (1) Let (Xr) be an increasing sequence of right-continuous potentials, which converges to (Xt ). For every e > 0 put (36.1)

The stopping times T: then increase with n and we have

lim P{T:

< oo} = o.

(36.2)

n-+ 00

(2) For each n EN, let (A7) [respectively, (At)] be the natural, integrable, increasing process, which generates (Xr) [respectively (Xt)]. Suppose that the process (Xt) is bounded. The random variables A oo , A~ then belong to 2 2 , and we have 1im E[(A oo

-

A~)2]

= O.

(36.3)

n-+ 00

Proof The process (Xt - Xr) is right continuous. It follows from IV.44 that T: is a stopping time, and it is clear that Ten increases with n. Put T = limn ---+ 00 T;-. By applyip.g VI.T13 to the process (Xr), we have Hm E[X~] > E[X~] n-+ 00

e

and also, from the regularity of (Xt ), lim E[X~] = E[XT ].

n-+ 00

e

Thus, for every p, we have

E[XT - X~] On the other hand, for p

> lim E[X~ -

X~~].

n-+ 00

< n, we have from (36.1)

XTne - X~nE

> X Tn E

XTnE

> e I{Tn< oo}' E

which implies, for every pEN,

E[X T

-

X~]

> e Hrn P{T: < oo}. n-+ 00

VII, T37

126

Generation of Supermartingales

Formula (36.2) can then be deduced by noting that the left side tends to zero when p ~ 00, from VI.T16 and Lebesgue's theorem. Suppose next that the process (Xt ) is bounded by a constant c. The same then holds for each of the processes (X;), which implies the relations E[A~]

< 2c 2

and

E[(A~)2]

< 2c2

(36.4)

from T24. Denote by (Yt) the process (Xt - X;), and by (B t) the process (At - A~). Since the random variables A oo and A~ belong to 2 2 , we can apply the proof of formula (23.1) to (B t ), which is the difference of two natural increasing processes. We obtain

E[(A oo

-

A::'l'l = E[B:'] = E[f(yu + Yu-l dB u ]'

We separate this last expression into the two integrals

E[J[O.T~[ r (Yu + Yu-) dB u]

and

E

We have Yu(w) < e, Yu_(w) nated in absolute value by

[J:

[~.oo[

(36.5)

(Yu + Yu-) dB u] .

(36.6)

< e for every u E [O,T:(w)[. The first

2eE[J[O.T~[ r d(A

u

integral is hence domi-

+ A:)] < 2eE[A oo + A~] < 4eE[Xo].

The second integral is dominated in absolute value by

fI{,F,
< 4cE[I{~< oo}(A oo + A~)]. This expectation is dominated, from Schwartz's Inequality, by

4c(E[A~])1/2(p{~n <

00 })1/2

+ 4c(E[(A~)2])1/2(p{T: <

from (36.4). This last quantity tends to 0 when n ~ Hm E[(A oo

-

A~)2]

n .... oo

00,

00 })1/2

< 8~2 c(p{~n < 00})1/2

from (36.2). Finally, we obtain

< 4eE[Xo],

which implies (36.3), since e is arbitrary. The following theorem is as important as Theorem 29. The notations same significance as in Nos. 28 and 29.

-+

(A~), (X~)

have the

T37 THEOREM * Let (Xt) be a right-continuous potential of the class (D), and let (At) be the natural, integrable, increasing process, which generates (Xt ). (1) The process (At) is continuous if and only if the potential (Xt) is regular. (2) For every stopping time T we then have Hm E[lA T h .... O

• Meyer (97).

-

A~I] = O.

(37.1)

VII, T37

The Existence Theorem

127

Proof Suppose that the potential (Xt ) is generated by the continuous integrable increasing process (At). Denote by (Tn ) an increasing sequence of stopping times, which converges to a stopping time T. We have n-+oo

from Lebesgue's theorem and the continuity of (At). This implies the equality lim E[X T,.] n-+oo

= lim E[A oo

-

ATnl

= E[A oo

-

AT]

=

E[XTl

n-+oo

The potential (Xt ) is thus regular. We consider the converse first in the case where the process (Xt ) is bounded. Denote then by (M t) [respectively, (M~)] the martingale (Xt + At) = (E[A oo I §"t]) [respectively, (X: + A~) = (E[A~ ff t ])]. It follows immediately from (36.3) that

I

lim E[(A oo

-

A~)2]

= O.

h-+O

Now strong convergence in L2 is stronger than strong convergence in Ll; we thus have for every e > 0, from VI.(I.I) and (1.2), lim p(suP t

h-+O

IMt -

M~I > el =

O.

In other words, the random variable SUPt IMt - M~I converges to 0 in probability as h ---+ O. Similarly, formula (36.1) shows that SUPt IXt - X~I tends to 0 in probability as h ---+ O. It follows that SUPt IA t - A~I tends to 0 in probability. We can then find a sequence (h n ) that decreases to 0, such that lim sup IA t - A:n/ = 0

n-+ 00

a.s.

t

(cf. 11.10). Since the paths of the processes (A~n) are continuous, and converge a.s. uniformly to the paths of the process (At), the latter are continuous, and assertion (1) is established in the case where (Xt ) is bounded. * Now suppose that (Xt ) belongs to the class (D) and is regular, but not necessarily bounded. We know then that (Xt ) is equal to the sum of a series of bounded regular potentials (X;) (cf. T34). Each of these is generated, from the case considered above, by a continuous, integrable, increasing process (A~). We have

Zn

(A~) so that remark 9(c) applies to the series (A~), and the integrable increasing process is continuous. This increasing process evidently generates (Xt). It is thus equal to (At) from the uniqueness theorem, and assertion (1) is completely established. To establish relation (37.1), we denote by (Yt) [respectively (Zt)] the potentiall~=l (Xf) [respectively, 1:=n+l (Xn], by (B t ) [respectively, (C t )] the integrable, natural, increasing process which generates (Yt) [respectively, (Zt)], and we denote by (B~), (C~) the increasing

• This proof is simpler than that of Meyer (97). It has also been used by Blumenthal and Getoor in their work on the theory of Markov processes.

Generation of Supermartingales

VII, 38

128

processes constructed, as in No. 28, from the potentials (Yt), (Zt). The potential (Yt) is bounded, so that

from T36. Since L2 convergence is stronger than Ll convergence, and the conditional expectation operators diminish norms, we have Hm E[B~ I§"Tl

h-+O

= E[B oo I§"Tl

in L1 norm.

Similarly, from T28, we have Hm Y~ = YT in L1 norm, h-+O

and finally by subtracting, lim E[IB';. - BTIl = 0, h-+O

all this for any stopping time T. Note then that (37.2) Given an e

> 0, choose n so large that 00

~E[xgl <~; 3

n+l

we then have

and, similarly, E[ChT l

e < E[Zolh < E[Zol < 3"'

The left side of (37.2) is then less than e whenever h is small enough so that E[IBT e/3. Relation (37.1) is thus established.

5· The Classification

of Stopping

-

B~ll

<

Times

Definition 18 does not permit us to recognize easily whether an increasing process is natural. In this section we give a more manageable criterion, thanks to a classification of the stopping times which "carry" the discontinuities of the increasing process under consideration. 38 NOTATION The following conventions will spare us certain artificial difficulties. We put §"t = §"o for every t < 0; if (At) is an increasing process, we put At = 0 for every t < 0; if (Xt ) is a right-continuous supermartingale, we put X t = Xo for every t < O. The time set for these processes will hence be the entire line, and the point 0 will not play an exceptional role.

129

The Classification of Stopping Times

VII, D39, D40, 41

Let (Sn)neN be an increasing sequence of stopping times. We denote by V !Fs" the n a-field generated by the union of the a-fields !FS". Let T be a stopping time. We denote by f/ T the collection of all increasing sequences (Sn)neN of stopping times less than T. Let A be an element of !FT; we have already used the notation TA at the end of No. 15 to denote the random variable defined by TA(w) = T(w) for w EA; TA(w) = +00 for w $A. This notation will be used systematically in this section. From IV.T38, the random variable TA is a stopping time. Times of discontinuity The theory we develop in this section takes a particularly simple form when the family (!Ft) has the following property. D39 DEFINITION The family (!Ft) is said to be free of times of discontinuity if, for every increasing sequence (Sn)neN of stopping times, !F(limS,,) = n

V !FS".

(39.1)

n

This property holds in most applications of the theory to Markov processes. We shall need the definition of the times of discontinuity themselves. D40 DEFINITION Let T be a stopping time,. T is said to be a time of discontinuity for the family (!Ft) if there exists an event A E!FT and a sequence (Sn)neN E f/ T A such that the event {w: lim Sn(W) = TA(w)} n

(40.1)

does not belong to the a-jield Vn !FS". *

41 It is necessary to verify that Definitions 39 and 40 are compatible. To do that, we need the following remark: Let (Sn) be an increasing sequence of stopping times, and let H be an element of !F 00" The event H n {limn Sn = oo} then belongs to V n !FS". Since this a-field contains all of the negligible sets, we can reason up to a negligible set. Put S = lim n Sn and denote by (Yt ) a right-continuous modification of the martingale (E[IH I !Ft]); we have I H = lim t __ oo Y t a.s., from VI.T6, and consequently IHn{s=oo} = lim n __ oo Y s " I{s=oo} a.s. It then is sufficient to note that this last function is measurable with respect to Vn !FS". Supposing then that the family (!Ft) is free of times of discontinuity in the sense of D39, we show that it has no times of discontinuity in the sense of D40-i.e. that the event {limn Sn = TA} belongs to V n!FS". Indeed, this event belongs to !F(lim"S,,) from IV.T41, and it suffices to apply D39. Conversely, supposing that there exists a sequence (Sn) for which (39.1) does not hold, we put lim Sn = S, and show that S is a time of discontinuity in the sense of D40. Indeed, let A be an event that belongs to the left side of (39.1), but not to the right side. The remark above shows that A n {S < oo} does not belong to Vn!FS . Now we have " A n {S < oo} = (li~ = n {S < oo}.

Sn SA)

The second event of the right side belonging to desired result.

Vn !FS,,' the first event must not. This is the

• An example of a time of discontinuity will be given in No. 54.

VII, D42, T43

Generation of 8upermartingales

130

Accessible stopping times. Elementary properties D42 DEFINITION (a) Let T be a stopping time; T is said to be totally inaccessible not a.s. infinite and if, for every sequence (8n) E !/T'

P{w: lim Sn(W) = T(w), Siw)

< T(w) < 00 for every nE N} =

O.

if T

is

(42.1)

n

(b) The stopping time Tis said to be inaccessible if there exists a totally inaccessible stopping time 8 such that P{w: T(w) = 8(w) (c) A stopping time is said to be accessible

< oo} > O.

(42.2)

if it is not inaccessible. *

The significance of D42 can perhaps be made clearer by the following considerations. We have seen in No. IV.32 that each stopping time T can be interpreted as the time at which a certain physical phenomenon [which we denote by e(T)] occurs for the first time. The relation 8 < Tbetween two stopping times then expresses the fact that the phenomenon e(8) is a "forerunner" of e(T). Let (8 n) be a sequence of stopping times dominated by T; the relation P{limn 8 n = T,8 n < Tfor every n} = 1 says that it is always possible to foresee the phenomenon e(T), before it occurs, by observing a succession of forerunners. The relation P{limn 8 n = T < 00, 8 n < T for every n} ~ 0 says that this prediction is possible in certain cases. Finally, the existence of totally inaccessible stopping times corresponds to that of completely unpredictable phenomena. T43 THEOREM (a) Let T and T' be two totally inaccessible (respectively, accessible) stopping times; the stopping times TAT' and T V T' are then totally inaccessible (respectively, accessible). (b) Let T be a totally inaccessible (respectively, accessible) stopping time, and let A be an element of ofFT such that P(A) > O. The stopping time TA is then totally inaccessible (respectively, accessible). (c) Let (Tn)llEN be an increasing sequence of accessible stopping times; the stopping time sUPn T n is then accessible.

Proof Suppose that Tand T' are totally inaccessible and consider a sequence (8 n) E !/TAT" We then have (8 n ) E !/T' (8 n ) E !/T" and

{li~ 8 n =

TAT' =

< 00,

{li~ 8 n = U

8 n < TAT' T

< 00,

{li~ 8 n =

for every nJ

8 n < T for every n, T'

<

00,

8 n < T'

T = TA T'J for every n, T' = TAT').

These last two sets are negligible, so that the first is also, and thus TAT' is totally inaccessible. • The reader will find some examples in No. 54.

131

VII, T44

The Classification of Stopping Times

Similarly, consider a sequence (Sn) E !/TVT', and put T n = Sn A T, T~ = Sn A T'. We have (Tn ) E!/ T' (T~) E!/T', and

(li~ Sn =

Tv T' =

< 00,

Sn

(li~ Tn =

< Tv T'

T<

u

Tn

00,

(li~ T~ =

T

for every n)


for every n,

< 00, T~ < T'

T= Tv T')

for every n,

T' = Tv T).

Since these last two sets are negligible, we see that T V T' is totally inaccessible. Suppose next that Tv T' is inaccessible; there exists a totally inaccessible stopping time S such that the event {S = Tv T' < oo} has nonzero probability. The same must then hold for at least one of the events {S = T < oo}, {S = T' < oo}. At least one of the stopping times T, T' is thus inaccessible. The reasoning is similar for TAT', and assertion (a) is established. The same reasoning also shows that the accessibility of the stopping time T implies that of TA for every A E ~T. Suppose that T is totally inaccessible, consider a sequence (Sn) E !/T A' and put T n = Sn A T; we have (Tn) E !/T and

(li:," S. = TA <

00,

S.

< TA

for every = A n

n) (ll~ Tn =

T<

00,

Tn


for every

n).

This last set is negligible since T is totally inaccessible. It then follows that TA is totally inaccessible. We finally establish assertion (c), reasoning by contradiction. Suppose the stopping time T = lim n T n is not accessible: there then exists a totally inaccessible stopping time T'such that P{T = T' < oo} > O. Let us denote this event by A and set T" = T~: this stopping time is totally inaccessible according to (b). The sequence (Tn ) belongs to !/T and the probability P{limn T n = T" < oo} is greater than o. There thus exists an integer n such that P{Tn = T" < oo} > 0, which contradicts the accessibility of T n. H ,

Let T be a stopping time. There exists an (essentially unique) partition of the set {T < oo} into two elements of :y;T' A and A', such that the stopping time TA is accessible, and the stopping time TA' is totally inaccessible. *

T44

THEOREM

Proof Associate with each sequence (Sn) E!/T the set K[(S.))

= (li:," S. = T < 00,

S.

< Tfor every ni,

and denote by $' the collection of countable unions of events of this form. There exists in $' an event A with maximal probability. Let A' be the complement of A relative to {T < oo}; the stopping time TA' is then totally inaccessible. Indeed, if it were not, there would exist a sequence (Sn) E !/TA' such that

P(li~ 5 n = • Or

+ 00 if T was accessible.

TA'

< 00,

5n

< TA' for every n) > O.

VII, T45

Generation of Supermartingales

132

This probability is also equal to P[A' () K[(Tn)]], where the sequence (Tn) = (Sn belongs to !7T' We thus have P[A U K[(Tn)]

A T)

> P[A],

contradicting the maximal character of P[A]. Consider next a totally inaccessible stopping time R. We show that P[TA = R < 00] = O. This event is, indeed, contained in A; if its probability were not zero, there would exist a sequence (Sn) E !7T such that P[K[(Sn)] n {TA = R < oo}] > O. By putting Rn = Sn A R, a sequence belonging to !7R would be obtained, which would contradict the total inaccessibility of R. The task of verifying the uniqueness is left to the reader. Remark The proof just given leads to the following characterization of accessible stopping times: T is accessible if and only if T = TA' i.e., if the set {T < oo} is the union (up to a negligible set) of a sequence of events of the form K[(Sn)]' In particular, the existence of a sequence (Sn) E !7T such that lim Sn = T a.s., Sn < T for every n, implies that T is accessible. The next theorem shows that such a sequence exists for every stopping time when the family (jZ="t) has no time of discontinuity. We have already indicated that this situation is frequently encountered in applications. ~

T45 THEOREM Let T be an accessible stopping time, which is not a time of discontinuity for the family (jZ="T)' There then exists a sequence (Sn)nEN E !7T such that

pili: Proof Let

Sn

=

T,

Sn

< Tfor every nE N) =

1.

(45.1)

be the collection of events A E jZ="T with the following property: there exists a sequence (Sn) E !7T' which converges a.s. to T, and such that SnCw) < T(w) for every n and almost every wE A. We start by showing that every countable union of elements of f§ belongs to

f§

f§.

Consider, indeed, a sequence (AP)PEN of elements of f§, and for each of them a sequence (S~)nEN E !7T' which has the above property relative to AP. For each pair of integers (m,p) choose an integer k mp such that, putting d(x,Y) =

x

l+x

Y l+y

we have

We can also suppose that k mp increases with m for each p. Now set Sm(w)

= inf S:mp(w). P

The stopping times Sm increase with m, are no larger than T, and are strictly less than T on A = Up AP. They converge a.s. to T, since

133

The Classification of Stopping Times

VII, T46

It then follows that A belongs to f§; f§ thus contains an event Bwith maximum probability, I.e.,

P[B] = sup P[A]. Ae~

The theorem will be established if we prove that P[B] = 1. Suppose instead that P[B] we show that this leads to a contradiction. Note first that B contains the set {T = oo}. In fact, we have, putting T n = TAn, {T=

oo} =

(li~ Tn =

T,

Tn

< 1;

< Tfor every nJ,

so that {T = oo} belongs to f§. Let B' be the complement of B; since the stopping time T is accessible, and B' is contained in {T < oo}, there exists a sequence (Rn) E [ / T such that (using the notation K[(R n )] of No. 44) we have

P[B' n K[(R n )]]

> 0.

(45.2)

Let e be a number >0. The event {lim n Rn = T} belongs to the a-field V n:FRn from D40 and the hypothesis made about T. Since this a-field is generated by the union U n:FRn' which is closed under (UJ: f), we can find, from IV.24, a set C, which is the intersection of a decreasing sequence (Ck)kEN of elements of U n:FRn' and such that

n

Choose an increasing sequence of integers nk (k E N) such that C k E :FR nk for every k, and put Qk(W) = Rnk(w) for wE C k, Qk(W) = 00 for WiCk' and Sk = Qk A T. The sequence (Sk) of stopping times is increasing, and converges everywhere to T. We have Sk(W) = T(w) for large enough k and W i C, Sk(W) = Rnk(w) for every k when wE C. Consequently, we have K[(Sn)] c K[(R n)], and P[[K(Rn)]""-K[(Sn)]]

<

p[C""-(li~

Rn =

TJJ < e.

We thus have from (45.2), if e has been chosen small enough, P[B' n K[(Sn)]]

or P[B U K[(Sn)]]

> 0,

> P[B].

Since the set B U K[(Sn)] belongs to f§, this contradicts the maximality of P[B], and the theorem is established. Accessible stopping times and martingales T46 THEOREM Let T be a totally inaccessible stopping time. There then exists a uniformly integrable, right-continuous martingale (Yt ), whose only discontinuity is a jump of size 1 at the time T.

VII, T47

Generation of Supermartingales

134

Proof Consider the integrable increasing process (Ut) defined by Ut(w) =

o (1

for for

< T(w) t > T(w).

t

Let (Zt) be the potential generated by (Ut), and let (Sn) be an increasing sequence of stopping times. Put S = lim n Sn. The event {limn USn ¥: Us} is the same as

(li~ SnAT =

T

< 00,

SnAT < Tfor every

n)

which has probability zero, since T is totally inaccessible. We thus have lim n E[UsJ = E[Us], hence lim n E[ZsJ = E[Z8], so that the potential (Zt) is regular. This potential is thus generated by a continuous, integrable, increasing process (Vt ), from T37. We hence have E[Uoo

-

I

Voo §"t] = Ut - Vt·

It then suffices to put Y t = Ut - V t to obtain the desired martingale. This theorem leads to a simple characterization of accessible stopping times, when the family (§"t) is free of times of discontinuity.

....

T47 THEOREM Suppose that the family (§"t) is free of times of discontinuity, and that T is a stopping time. Then T is accessible if and only if

YT = YT -

(47.1)

a.s.

for every bounded right-continuous martingale (Yt ). Proof Suppose that T is accessible. There exists (from T45) an increasing sequence of stopping times Sn such that a.s. lim Sn n

=

for every n.

and

T

Let (Yt ) be a uniformly integrable, right-continuous martingale; we have, from VI.14 and V.T18 YT

= E[Yoo I§"T] =

E[Yoo

I Vn §"8 n] = lim E[Y I§"8 n ] = lim YSn = YT n n oo

a.s.

Conversely, suppose that T is inaccessible. There then exists a totally inaccessible stopping time S such that P{S = T < oo} > 0. Let (Xt ) be a right-continuous uniformly integrable martingale, with its only discontinuity a jump of size 1 at time S (see T46). This martingale is not bounded, but let c be a constant >0, and let Rc be the stopping time Rc(w)

= inf {t: IXlw) I > cl,

and denote by (Yt ) the martingale obtained by stopping (Xt ) at the instant Rc. It is clear that this martingale is bounded in absolute value by c + 1, and that

whenever c is large enough.

135

The Classification of Stopping Times

VII, D48, T49

Accessible stopping times and natural increasing processes D48

DEFINITION

(At) charges T

Let (At) be an increasing process, and T a stopping time. We say that

if P{A T =;!= A T -}

-+

> O.

Let (At) be an integrable increasing process. Then (At) is natural if and only if the following two properties are satisfied: (1) For every sequence of stopping times (Sn)neN' which increases to a stopping time S, the random variable As is measurable with respect to the a-field Vn §"Sn' (2) (At) charges no totally inaccessible stopping times. T49

THEOREM

Proof Supposing first that (At) is natural, we show that property (1) is satisfied. We constructed in No. 29 continuous, integrable increasing processes (A~) such that AT = lim A'T (in the topology a(Lt,LC1J » h ..... O

for every stopping time T. Now each random variable A~ is measurable with respect to the a-field Vn §"Sn' in view of the continuity of the process (A~). This property is preserved under passage to the weak limit (see II.9(b», and thus As is measurable with respect to

Vn§"s· 11

Now consider property (2). Let T be a totally inaccessible stopping time, and let (Yt ) be a uniformly integrable martingale with its only discontinuity a jump of size 1 at the instant T (see T46). Denote by (Yf) the martingale obtained by stopping (Yt ) at the stopping time

Rn = inf {t: IYtl

> n}

(n EN).

Since this martingale is bounded in absolute value by n implies the equality

E[f.C1J (Y7 -

Y~-) dAtJ

= E[(A T

-

+ 1, the fact that (At) is natural

AT-)I{T ~ Rn}] = O.

The relation E[A T - A T -] = 0 then follows when n --+ 00, so that (At) does not charge T, and property (2) is verified. Conversely, suppose that (At) satisfies properties (1) and (2); we show that (At) is natural. It clearly suffices to consider the case where (At) is purely discontinuous. We begin by giving, based on property (1), a procedure for "extracting" natural increasing processes from (At). Let (Sn) be a sequence of stopping times increasing to a stopping time S; put

Blw) =

o {li~ [As(w) -

< S(w) for t > S(w). for t

Asn(w)]

The process (B t) is an increasing process strongly dominated by (At) (see No. 8). We show that (B t ) is natural. Let (Yt ) be a bounded right-continuous martingale; then

E[f.C1J (Yt - Ye-) dBtJ = E[(Ys - Ys-)] , where denotes the random variable lim n (As - Asn ), which is measurable with respect to the a-field Vn §"S1l from property (1). It is evident that Ys - Ys- = limn (Ys - Ys) on

VII, 51

Generation of Supermartingales

136

the set { #: O}, so that the last expectation can be written as E[ lim n (Ys - Ys )] = lim n E[( Y s - Y s )]. Let n be the conditional expectation E[ :FsJ ; n is then orthogonal to Ys - YSn (from the martingale property), so that

I

The right side is dominated in absolute value by 11<1> - nlll 11 Ys - YsJ,m a quantity that tends to 0 when n ----+ 00, since the martingale (Yt ) is bounded and n converges to in Ll norm from V.TlS. The increasing process (B t ) is thus natural. Now let e be a number >0, and let Tl be the time when (At) first makes a jump larger than e. Property (2) implies that this stopping time is accessible. There thus exist sequences of stopping times (S~)n~1 (p > 1), belonging to [/Tl, such that the union of the sets K[(S~)n~l] is a.s. equal to the set {Tl < oo} (see No. 44). We can then apply the above procedure to the process (At) and the sequence (S~), obtaining a natural increasing process (BD, which satisfies properties (1) and (2) from the first part of the proof. The increasing process (A~) = (At - BD thus satisfies these properties, and we can extract from it a natural increasing process (B~) by means of the sequence of stopping times (S~). Put (A~) = (A~ - B~), and continue iterating this procedure. The increasing process (CD = ziBf) it is evidently natural, and it is easy to see that every jump of (C~) is a jump of (At), and that (At - CD is free of discontinuity at time Tl. In the same manner we define natural increasing processes (CD inductively, (C~H) being the process constructed as above from (At - C~). Let us put (A~) = Zk( C~). This process is still natural, and the process (At - A~) is free of jumps bigger than e. Consequently (An increases in the strong sense to (At) when e ----+ O. The increasing process (At) is thus natural, establishing the theorem. '" !

!

Remarks (a) A comparison of this result with the construction in Nos. 10 and 11 shows that a purely discontinuous natural increasing process is equal to the sum of a series of natural increasing processes with paths having at most one jump. (b) Suppose that the family (:Ft) is free of times of discontinuity. Property (1) is then satisfied by every increasing process, and natural increasing processes are identical to those which charge no totally inaccessible stopping times. In particular, every increasing process strongly dominated by a natural increasing process is itself natural. We shall see in No. 54 that this result is false when the family (:Ft) has times of discontinuity.

51 Let (At) be a natural, integrable, increasing process, and let (Xt) be the potential generated by (At). Let (Sn) be a sequence of stopping times increasing to a stopping time S. The process (Xt + At) is a martingale, so that

for every event HE:FSk and every integer k infinity, we obtain

> n.

By letting first k, and then n, tend to

(51.1)

The Classification of Stopping Times

137

VII, T52, 53, 54

The left side is equal to lim k (As - As) from property (1) of No. 44. This equality thus gives a means of calculating the jumps of the natural increasing process, which generates (Xt )·

Suppose, in particular, that the family (~t) is free of times of discontinuity, and that S is an accessible stopping time. There then exists an increasing sequence (8 n ) of stopping 6mes such that a.s.lim n Sn = S, Sn < Sfor every n (T45). Relation (51.1) then takes the following very simple form: (51.2) As - A s - = X s - - X s a.s.

It is thus seen that the accessible discontinuities of (Xt ) are negative jumps, opposite to the jumps of (At). Here is another consequence of T49, of interest only when the family (~t) admits times of discontinuity. T52

THEOREM

Let T be a stopping time. The increasing integrable process (At) defined by A,(w) =

is natural that a.s.

if and only if there exists an

o (1

< T(w) for t > T(w) for t

(52.1)

increasing sequence (TJneN of stopping times, such

Hm Tn = T, n

Tn

< T for every n.

(52.2)

Proof Suppose first that T satisfies this condition; T is then accessible, so that condition (2) of Theorem 49 is satisfied. To show that (1) is also satisfied, consider an increasing sequence of stopping times (Sn)' and put S = lim n Sn' We have

As

= I{s< CX).S~T} = I{s< CX)} [i~f s~p I{Sn> Tp}] ,

so that As is measurable with respect to the a-field Vn ~Sn' Conversely, suppose that the process (At) is natural. The stopping time Tis then accessible, and we can return to the proof of Theorem 45, wherein the crucial point was showing that the event {limn Rn = T} belonged to the a-field Vn ~Rn' from formula (45.2). We show this point now, using the fact that (At) is natural. It suffices to note that [{urn,. Rn=T
E[Yo]

= E[YT -]

for every bounded right-continuous martingale if and only if, keeping the notation of (52.1),

In other words, the above holds if and only if the process (At) is natural, or if the stopping time T is the limit of a strictly increasing sequence of stopping times. 54 Examples (a) Consider a space Q, consisting of two points wand w', with the a-field ~ = ~(Q), and the law P defined by P({w})

= P({w'}) = 1.

VII, 54

Generation of Supermartingales

138

Consider the following family of sub-a-fields of :F, which is evidently right continuous: t -

<1 for t > 1.

{0,Q} for

ClZ'_ 07"

{:F

t

The two integrable processes (At) and (B t) given by

o {1

Aiw) = At(w') =

and Bt(w)

o

Biw')

= {2

for

t

for

t

<1 > 1,

=0 <1 for t > 1. for

t

generate the same potential (Xt ), defined by I for t Xiw) = Xiw') = {0 for t

<1 > 1.

The process (At) is natural from T52; the process (B t) does not satisfy condition (1) of the statement ofT49, and hence is not natural [the stopping time T = 1is a time of discontinuity: If one puts T = 1, A = {w}, Sn = 1 - Iln, the event {limn Sn = TA} = {w} does not belong to the a-field Vn:FSn = {0,Q}]. The random variable T defined by T(w) = a, T(w') = a'

(0

< a, a' < (0)

is a stopping time if and only if it is constant (a = a'), or if the numbers a and a' are both > 1. It is then easily deduced that all stopping times are accessible. It will be noted that the natural increasing process 2(A t ) strongly dominates (B t ), without the latter process being natural (see No. 50). This is due to the existence of times of discontinuity for the family (:Ft). (b) Let (Q,:F,P) be a probability space sufficiently rich so that there exists a positive random variable S, defined on Q, and admitting an exponential law : P{S

> t} =

e- t for every t

Put then Xt(w)

I

for

t

o

for

t

={

> o.

< S(w) > S(w)

and denote by :Ft the a-field consisting of the sets, which differ from an element of .9""(Xs , s < t) by a negligible set. The family is evidently increasing, and we shall see later that it is also right continuous. We begin by noting that .9""(Xs , s

< t) = .9""(S At).

Indeed, the random variable SAt is measurable with respect to the first a-field, since SAt is the upper bound of the rational numbers s < t such that X s = 1. On the other hand, each random variable Xis < t) is measurable with respect to the second a-field, since Xs =

I{s<sAt}·

139

The Classification of Stopping Times

VII, 55

The a-field §" 00 is hence equal to !Y(S) (up to negligible sets); every §" oo-measurable and bounded random variable is thus a.s. equal to a function of the form Ho S, where H is a bounded Borel function on R+ (see I.T18). We can then exhibit a specific right-continuous modification of the martingale (E[H 0 §"tD,

Si

5.

00

I

E[H 0 S SAt]

=H

0

S . I{s.~t} '-.

+

I{S;t}

H(x)e- X dx

5.

(54.1)

00

e- x dx

t

Let us show then that the family (§"t) is right continuous. Let A be an element of §"t+; choose the function H so that H 0 S = lA' and denote by (Yt ) the modification constructed above. The martingale (Yt ) has a discontinuity only at the instant S, and S #: t a.s. We thus have Y t = Y t- a.s. Since Y t is equal to lA' and §" t contains the negligible sets, we have A E§"t. We then have, for s < t, E[Xt -

I

Xsl §"s] = -E[l{s<s:5:t} §"s]

=-

[1 - e-
The process (X t ) is thus a bounded and right-continuous supermartingale, and in fact a potential of the class (D) since limt~oo X t = O. Using the procedures of Nos. 28 and 29 to evaluate the integrable increasing process (At) which generates (Xt), we obtain (the symbol "lim" here meaning a weak limit) A t-I' - ~

it

h-+O

0

= lim

1

h-+O

I

X u - E[X U + h §"u] d U -h

- e h

it

h

I{u<S}

du

0

= SAt.

Since this process is continuous, the potential (Xt ) is regular. It then follows immediately that lim X Tn = X T a.s. (54.2) n

for every increasing sequence (Tn ) of stopping times, which converges to a stopping time T. In other words, the stopping time S is totally inaccessible. We show finally that the family (§"t) is free oftimes ofdiscontinuity. Let (Tn) be a sequence of stopping times increasing to T, and consider an event A E §"T' Choose the function H of formula (54.1) in such a manner that Ho S = lA' and denote by (Yt ) the martingale (54.1). We evidently have lim YTn = YT a.s., n

since the martingale (Yt ) has a discontinuity only at the instant S, and the stopping time S is totally inaccessible. The random variable YT = lA is thus measurable with respect to Vn§"T' n

Outline of another classification of stopping times 55 We have published elsewhere [see (97)] a slightly different classification. We limit ourselves here to indicating the definitions and the principal results. The reader will have no

VII, D57

140

Generation of Supermartingales

trouble in recovering the proofs, by modeling them on the preceding work. The two classifications, moreover, coincide when the family (~t) is free of times of discontinuity. Let T be a stopping time. We say that T is totally inaccessible in the weak sense if it is not a.s. infinite and if P{Sn < T < 00 for every n EN} = 0 for every sequence of stopping times (Sn) which increases a.s. to T. We say that T is weakly inaccessible if there exists a stopping time S, totally inaccessible in the weak sense, such that P{S = T < oo} > O. If this does not hold, we say that T is strongly accessible. Let ( Y t ) be a right-continuous and bounded martingale, and let T be a stopping time such that P{ YT #: YT-} > O. A number e > 0 can be chosen such that P[A] > 0, denoting by A one of the events {YT > Y T - + e} or {YT < Y T - - e}. The stopping time TA is then totally inaccessible in the weak sense. The reasoning of Theorem 45 then permits us to establish the following result. 56 Let T be a stopping time; the following four properties are equivalent: (1) T is strongly accessible. ~ (2) T is accessible, and is not a time of discontinuity for the family (~t). (3) P{YT #: Y T -} = Ofor every bounded and right-continuous. martingale (Yt ). (4) There exists an increasing sequence of stopping times (Sn) such that lim Sn = T a.s.,

Sn

n


a.s. for every n,

and ~ T =

V~Sn' n

The following theorem can be deduced from property (3). Let T be a stopping time. There exists a partition of the set {T < oo} into two sets Band B' such that the stopping time TB is strongly accessible, and TB' totally inaccessible in the weak sense. * This partition is essentially unique. (B is the essential union of the sets of the form {YT #: Y T-}' where ( Y t ) is 'a bounded and right-continuous martingale.)

6. A Few Results on EnerBY It can be shown (though we shall not attempt it here) that this section constitutes a

generalization of the classical theory of energy with respect to the Newtonian kernel. The theory, however, loses delicacy as it gains generality, and it is far from being as useful as the classical theory. D57 DEFINITION Let (Xt ) be a right-continuous potential. t If (Xt ) does not belong to the class (D), we say that (Xt ) has infinite energy. If (Xt ) belongs to the class (D), we call the energy of (Xt ) the (possibly infinite) number e[(Xt )] = lE[A~],

(57.1)

where (At) denotes the natural increasing process which generates (Xt ). • Or

+ 00 if T was strongly accessible.

t· All supermartingales considered in this section will be supposed right continuous, unless mentioned to the contrary.

A Few Results on Energy

141

VII, 58, 59, T60

Computation of energy 58 (a) Keeping the notation of the preceding definition, we note that No. 23 gives us the "energy formula"

e[(X,)] =

~E[r(X. + X~)dA.l

(58.1)

(b) Let us check the relation

2e[(X,)]

>

E[f

X._ dA.J

.

> e[(X,)]

(58.2)

The first inequality is an immediate consequence of formula (58.1). To establish the second one, we begin by supposing that the energy of (Xt ) is finite. We have, from T20 and the na'turalness of the increasing process (At),

This relation can also be written, since (Xt ) has finite energy, as

The second inequality then follows from (58.1). To treat the case where the energy of (Xt ) is infinite, introduce the integrable increasing processes (A~) = (At A n) (n EN), and denote by (X:) the potential of (A~). Since the processes (A~) are natural from T49, we have

E[f

X._ dA.J

>

E[f

X::.- dA.J

>

E[f

X:_ dA:J

>

lE[(A::,)'].

This last quantity tends to infinity with n.

59 Remark

Definition 57 can be generalized by setting, for every integer p ep[(Xt )] =

1- E[(Acx')P]. p!

> 1, (59.1)

These quantities have not been used in classical potential theory, and we point out only one result concerning them. If the potential (Xt ) is dominated by a constant c, we have ep[(Xt )] ~ c1J , or in other words, (59.2) /

Here is the idea of the proof of this inequality: First the case where (At) is continuous is considered, using formula (22.3). Next, the case of a natural increasing process is covered by means of a passage to the weak limit in Lp, based on T30, and then proceeding as in Remark 61 below:

TOO THEOltEM Let (Yt ) be a potential withfinite energy and let (Xt ) be a potential dominated by (Yt ). Then (Xt ) ha3 finite energy and e[(Xt)]

~

4e[( Yt)].

(60.1)

VII, 61

Generation of Supermartingales

142

Proof We first establish (60.1), supposing that (Xt) has finite energy. Let (At) and (B t) be the natural increasing processes which generate (Xt ) and (Yt ), respectively; we have -[(X,)] The process (Yu

< E[f X~ dA.J < E[f(y~ + H._) dA.}

+ B u ) is a martingale, and the increasing process (At) is natural; Theorem

20 hence permits us to replace the last expectation by E[f.oo(y.

+ H.) dA.J. an expression

which, from T16, also equals E[BooAcxJ. We thus have, using Schwarz's inequality,

e[(Xt)] < (E[A~]E[B~])1/2 = (4e[(Xt)]e[(~)])1/2. Inequality (60.1) then follows immediately. The restriction made on (Xt ) is next removed in the following manner: Let (A~) be the increasing process (At A n), which is natural from T49. The potential of (A~) is dominated by n, and hence has finite energy from T24. We thus have We then let n tend to infinity.

Remark The same reasoning also gives the following results: (a) A potential (Xt) has finite energy if and only if the random variable Y = SUPt X t belongs to ,22, and we have e[(Xt)] < 2E[Y2] < 16e[(Xt)]. (b) Let (B t ) be an integrable, increasing process which generates (Xt ) (natural or not); if E[B~] is finite, then (Xt) has finite energy and e[(Xt)] ~ 8E[B~]. To prove (a) and (b), assume first that (Xt) has finite energy, and let (At) the natural increasing process that generates it. Then Y is dominated by SUPt E[A oo ~t], which belongs to ,22 (VI.2); formula VI.(2.1) also yields the inequality:

I

E[ Y2]

< 4E[A~] =

8e[(Xt)]. The same reasoning applies to (B t) under assumption (b), giving that E[ Y2] Conversely, to dominate e[(Xt)], assume that (Xt) is bounded. Then

< 4E[B~].

< E[f.oo X u- dAuJ < E[f.oo Y dAuJ < E[YA oo ]. Applying the inequality of Schwarz, one finds that e[(Xt )] < 2E[ Y2]. A passage to the limit e[(Xt)]

as above then extends it to the general case. Monotone convergence and energy

61 We first recall several elementary results on weak convergence in L2. Let (fn)neN be a sequence of elements of L2 such that sUPn IIfnl12 < 00, and which converges weakly in Ll to a functionf;fis then the only possible cluster point of the sequence (fn) in the weak topology of L2. Since, on the other hand, the set offn's is bounded in L2, the sequence (fn) must converge weakly to f in L2. The L2 norm is a lower semicontinuous (1.s.c.) function under the weak topology on L2; thus (61.1) IIfl12 < lim inf IIfn112. n

143

A Few Results on Energy

VII, T62, T63

The fn will converge strongly to fin L2 if and only if

IIfl12

= lim n

Ilfnl12'

(61.2)

This condition is clearly necessary. Conversely, if it is satisfied, lim E[(f - fn)2] n

=

E[f2]

+ lim E[!;] n

2lim E[ffn] n

= 0.

Here is a consequence of these properties. T62 THEOREM Let (Xt ) be a potential which is the upper envelope of an increasing sequence ofpotentials (X;). We then have the inequality e[(Xt )]

< lim inf e[(X~)].

(62.1)

n

Proof It suffices to establish this inequality in the case where the right side is finite. We can then suppose [by extracting a subsequence from the sequence (Xr) if necessary] that the energies e[(Xn] are all finite, and converge as n tends to infinity. Each potential (Xr) then belongs to the class (D), and is generated by a natural, integrable, increasing process (A~). Let A be a weak cluster point in L2 of the sequence (A~) -A does exist, since the expectations E[(A~)2] are uniformly bounded. The relation Xf < E[A~ ~t] becomes in the limit X t < E[A I ~t] a.s. It then follows immediately that the potential (Xt ) belongs to the class (D), and is hence generated by a natural, integrable, increasing process (At). It follows from T30 that A oo = lim n A~ in the weak topology of L\ and relation (62.1) is then an immediate consequence of No. 61. Inequality (62.1) cannot always be replaced by an equality, as we shall see in No. 67. This can be done, however, in two very important cases, which are the object of the following two theorems.

I

T63 THEOREM Let (Xt ) be a regular potential of the class (D), which is the upper envelope of an increasing sequence ofpotentials (Xf). We then have e[(Xt )]

= lim e[(X:)].

(63.1)

n

Proof This equality is trivial, from (62.1), when e[(Xt )] = 00. We can thus limit ourselves to the case where (Xt ) [and hence also each (Xf), from T60] is of finite energy. Let (At) and (A~) be the natural, integrable, increasing processes which generate, respectively, (Xt ) and (Xf). We show that

lim E[(A oo

-

A~)2] = 0,

(63.2)

n

which evidently implies (63.1), and which is equivalent to it from the remarks in No. 61. We have already established this equality in No. 36, in the case where the process (Xt ) was bounded, and we refer to the proof of T36, which remains valid (with only insignificant changes) up to the point where one is trying to dominate the integral (36.6),

144

Generation of Supermartingales

VII, T64

Note first that the process (At) is continuous; this integral is thus increased by replacing the interval of integration by ]T:,oo[, which removes a negative term. Denote then by (B t ) [respectively, (B7)] the integrable increasing process defined by 0 Bt(w)[respectively,

B~(w)]

=

(

t

for

At(w) -

< T:(w)

AT:(w)

[respectively, A~(w) - A~(w)]

for

t

> T:(w).

These processes are natural from T48, and a simple calculation yields the potential (Zt) generated by (B t ),

I~t]I{t
Zt = E[XT=

There is an analogous formula for the potential desire to dominate is thus the same as E[f.OO(Zt

+ Zt_ -

(Z~)

generated by

Z:- - Z:-) deBt -

(B~).

The integral we

B~)J.

This expectation is itself equal to E[(B oo - B~J2] from formula (58.1) (which clearly applies to a difference of natural processes). The integra) (36.6) is thus dominated by E[B~]

+ E[(B~)2] < 5E[B~] =

5E[(A oo

-

AT~)2],

where the < sign is a consequence of T60 and the fact that the potential (Z~) is dominated by (Zt). It only remains to note that tends to infinity with n, and that the last expectation tends to 0 from Lebesgue's dominated convergence theorem. We return in the following statement to the notations (X~'), (A~) of Nos. 28 and 29.

T:

T64 THEOREM * Let (Xt ) be a potential withfinite energy, generated by a natural, integrable, increasing process (At). We then have e[(Xt )] = Hm e[(X~)],

(64.1)

h-+O

and lim E[(A oo

-

A~)2]

= O.

(64.2)

h-+O

Proof Equations (64.1) and (64.2) are equivalent from the remarks of No. 61. We begin by proving (64.1) in a particular case, supposing that (At) is a purely discontinuous process with paths having at most one jump. Let us then denote by T( w) the single value of t for which At(w) - At_(w) > 0 (or infinity if there is no such value). We have seen in No. 53 that there exists an increasing sequence (Sn)neN of stopping times, which a.s. satisfy the properties (64.3) lim Sn(w) = T(w); SnCw) < T(w) for every n. n We also set
• Not proven.

A Few Results on Energy

145

VII, T64

since the set {u: Xiw) :F Xu_(w)} is countable for almost every w. In addition, put

IJ:t+h A duo u

B~ = -

h

t

A simple calculation shows that, for every pair of times (s,t) such that s

I

I

E[B~ - BZ §"s] = xZ - E[X~ §"s] = E[A~ -

< t,

AZ I §"s]·

We also have then, from Tt7 (and the remark which precedes this theorem),

e[(X~)] = E[LOO X~_ dB~ ]

=

E[LOO X~ dB~ ]

E[f X~GJ:+hA. dV) duJ = E[f x~(~rhA.dV) dUJ.

=

since

X~(w)

< Xiw) = 0 for u > T(w). The inside integral can then be evaluated, yielding e[(X:)]

=

E[~J.TX~. <1>. (u + h -

T). I{u
<

E[

T).



~J.T X (u + h u •

Itu
Denote by (Zt) a right-continuous modification of the martingale (E[ §"t)); it follows from the properties (64.3), and the measurability of with respect to the a-field V n §"Sn' that ZT- = a.s. (V.Tt8). We then have, taking account of the relation X t = Zt I{tJ:TX u . (u h-+O h 0

+h-

T). I{u< T
fT Xu.(u = lim-

+h-

T)du

lim XT- fT .(u h-O h T-h

+h-

T) du

h-+O h T-h

=

=

}2.

Now set Z = SUPt Zt; Z is square integrable from (VI.2), so that these random variables are dominated by the integrable random variable Z. We can thus apply Lebesgue's dominated convergence theorem, obtaining lim sup e[(X~)] h-+O

< }E[2] =

e[(Xt)],

which yields equality (64.1), by comparison with T62. Now consider the general case. A natural increasing process (At) is equal to the sum of a series of natural increasing processes (A~) (n EN), where the process (A~) is continuous, and each (An(p > 1) is a natural increasing process with a single jump of the preceding type. * Set (B t) = 2~=o (An, (C t) = 2:=n+l (An, and denote by (Yt), (Zt) the potentials • See No. 50,(a).

Generation of Supermartingales

VII, T65, T66

146

generated, respectively, by (B t ), (C t ). We can also construct the processes (B~), (C~) associated with (B t ) and (C t ) as in No. 28, and consider their potentials (Y:), (Z:). Finally, let e be a number >0. We can choose n large enough so that E[C;,] ~ e, and then h small enough so that

E[(B<Xl - B~)2]

<e

from T63 and the above. We tqe,n have, from T60~ I '\:~ ;'",;/

'

'-.\ --'~ ( "

..

t. t"

"\'

2 ] ". FE"(Ch )2]' ,<" 6 )2Ji <' -f E[(B <Xl - Bh<Xl )2] \j+'\.' E[c <Xl+ E[(A <Xl - A h<Xl, _'\ <Xl'_ e. The proof which led us to assertion (2) of T37, applied anew, gives the following result.

Let (Xt ) be a potential of the class (D), generated by the natural, integrable, increasing process (At). For every stopping time T we then have T65*

THEOREM

Hm E[IA T

-

A~I] =

o.

(65.1)

h-+O

Monotone convergence and martingales

It is shown in classical potential theory that the harmonic functions with finite Dirichlet integral and the potentials of finite energy form two orthogonal spaces under the inner product defined by the Dirichlet integral. This fact cannot be translated exactly into the theory we are developing here, but suggests the existence of irregular phenomena when a potential converges to a martingale. We shall now see that this is indeed the case.

Let ( Y t ) be a positive, uniformly integrable martingale such that E[ Y~] < 00; then (1) Every potential (Xt ) dominated by (Yt ) has finite energy. (2) Let (X;) be a sequence of pot~tials increasing to (Yt ), and let (A~) be the natural, integrable, increasing process which generates (X;); then we have T66

THEOREM

Hm A:

=0

in the 13 norm, for every t

E

(66.1)

R+.

n

lim A~ = Y<Xl

in the weak sense in 13.

(66.2)

n

Proof Let (At) be the natural process which generates (Xt ). Choose a number r and set B,(OJ)

=

(;l _

r.

for

,-u-,,) ¥,(OJ)

t

for

Let (Y;) be the potential generated by (B t ). We have

'() (~(W) Ytw =

e-
for

t
for

t

> r.

Since the process (B t ) is continuous, (Y;) is regular. Similarly set

*

Not proven.

for

t
for

t

> r.

>0

147

A Few Results on Energy

VII, 67

The potential (X;) of (A;) is given by

xtC W )

X'(w) t

for

{e-
-

£' 10r

<..v

r.

t

We have, from T60, since the processes (B t) and (At) are natural, that E[(A:x,)2]

< 4E[B~],

and consequently, letting r tend to infinity, E[A~]

< 4E[Y~].

(66.3)

This establishes assertion (1). To establish (2), define as above increasing processes (A~n), with potentials (x;n), and note that these potentials increase to the regular potential (Y;). We thus have, from T63, lim A~

= Boo

in the L2 norm,

n

or

+ X;) = Yr.

lim (A; n

Equality (66.1) then follows from Lebesgue's theorem. The set of random variables A~ is bounded in L2, and hence weakly relatively compact. If we can show that the sequence (A~) cannot have other cluster points than Y, we will thus have established (66.2). Now let U be an ultrafilter on N which converges to infinity; set U = lim A:' 11

(in the weak sense in I!).

We also have for every t that

I

E[U ~t]

= lim E[A:' I~t] = lim (X: + A:) = 11 11

~,

from (66.1) and 11.48. It then follows that

I

I

E[U ~t] = E[Yoo ~t]

Now U is

~-measurable

a.s. for every t.

from 1I.9(b), and thus U = Y a.s., establishing the theorem.

67 It is natural to inquire whether the weak convergence above can be replaced by strong convergence, as in Theorems 63 and 64. Here, for those familiar with the theory of Brownian motion, is an example that shows that this is, in general, impossible. Let (B t ) be a Brownian motion in R3, starting at the origin, relative to a family of a-fields (~t) satisfying the hypotheses of this chapter. Then for every positive Borel function g defined on R3, we have

E[f.~ goB, ds I jO,] =

(Ug)

0

X, a.s.,

where Ug denotes the Newtonian potential of g, Ug(x)

=1

K

R31x - YI

g(y) dy

VII, 67

Generation of Supermartingales

148

(K a constant, and Ix - yl the Euclidean distance between x and y). Choose for g a positive function with compact support, depending only on lxi, and denote by G the superharmonic function Ug, by (Xt ) the supermartingale (G 0 B t ). The latter is a potential of the class (D), generated by the continuous, integrable, increasing process

The function G clearly depends only on

E[A~1 =

2E[fX dA u

u]

=

lxi, and is decreasing in Ixl. We have

2E[f GOB,. X, dS] = 2U(gG)', gO

where U(gG)° denotes the value of U(gG) at O. For every function f defined on R3, let Tf be the function defined by

Tf(x)

=f(~).

=

u(:g)

The formula TG

is easily verified. Denote by G n the function rnG, by gn the function Tng /4 n, and by (X~) the potential (G n 0 B t ). These potentials increase to the constant martingale G(O). It then follows, denoting by (A~) the continuous increasing process that generates (X~), that lim A~ = G(O) in the weak sense in

L~

n

If this convergence were to hold in the strong sense, we would have G(0)2 = lim E[(A~)2] = lim 2U(gnGn)O = 2U(gG)O n

n

for every function of the type considered. But this equality is not true, as one may see by taking, for instance, g = 1 inside a ball, g = 0 outside. We leave the computations to the reader. It is easy to deduce from this an example of an increasing sequence of potentials of finite energy, which converges to a potential of finite energy., but does not converge in the energy norm. It suffices to look at the potentials (G n 0 B t ) = (Xt ) of the preceding example, to denote by h a continuous, increasing map of the interval [0,1 [ onto [0,00[, and to set

z~ = (:~(t) Z,

= (~(O)

for for for for

<1 t > 1, t <1 t > 1. t

These processes are potentials with respect to the a-fields C§t equal to :Fh(t) for :F 00 for t > 1, and their energies are the same as in the preceding example.

t

< 1 and to

CHAPTER

VIII

Applications of Martingale Theory

In this chapter we have avoided giving applications available in Doob's book (56). The references for several of them, which we think are particularly interesting, are: the theory of sums of independent random variables (p. 334); the strong law of large numbers (p. 341); the regularity of the paths of processes with independent increments (p. 388); Theorem VIII.2.3 on the regularity of the paths of Brownian motion (p. 395).

1.

Applications

e?f the

Convergence Theorems

Symmetric random variables The following theorems are due to Hewitt and Savage [see (76), where they are established analytically]. The proof we give here is probably due to Doob. 1 Let (T,:T) be a measurable space; we denote by (E,tff) the product space (TN,ff N), and by ~ the group of permutations of the set N, which shift only a finite number of integers. For each a E ~ the mapping aCT: (xn)neN.JVII'+ (xCT(n»)neN

is a bijection of E onto itself, which preserves the measurable structure. A probability law P on (E,tff) is said to be symmetric if its image under aCT is equal to P for every permutation a E ~. A random variable f on (E,tff) is said to be symmetric if f 0 aCT = f for every a E ~. It is easy to verify that the symmetric measurable sets (i.e., those with symmetric indicator functions) constitute a a-field [/ on E, and that the symmetric random variables are just those that are f/ -measurable. We denote by (Xn)neN the coordinate mappings, and put tff n = ff(Xb X 2 ,

••• ,

X n),

~ n = ff(Xn+b X n+2' •••).

The a-fields tff n increase with n, and their union generates tff. The a-fields ~ n decrease as n increases, and we denote their intersection by ~. We have the following lemma. 149

VIII, T2-4 T2

LEMMA

Applications of Martingale Theory The a-field

Proof The a-field {XVI E

~n

~

150

is contained in [/ .

is generated by sets of the form

AI' X V2 E A 2 ,

• •• ,

X Vk

E

Ak}

(PI' ...

,Pk

~ n

+ 1, AI, ... , A k E.:T)

which are invariant under every mapping atT such that a shifts only the indices 0, ... , n. This property then extends to ~ n (I.T19). Now let fbe the indicator function of an element of ~, a an element of ~. Let n be an integer such that a shifts no integer p > n; since the functionfis ~ n-measurable, we havef = fo atT from the preceding remarks, and the lemma follows. Here then is the main result of Hewitt and Savage.

T3 THEOREM Let P be a symmetric law on (E,8), and let A be an element of [/; there exists an element B of ~ such that A = B a.s. Proof Set lA = f; we have from V.T18 and V.T19 that

f = lim E[f I 8 n], n-+oo

I = v-+Hm E[f I ~v],

E[f ~]

00

> 0, choose an n so large that (3.1) Ilf - E[fl X o,· •• , Xn]ll l < e (3.2) HE[fl ~] - E[fl ~ n] 111 < e. Denote by a the permutation that interchanges 1 and n + 1, 2 and n + 2, ... , nand 2n,

the limits taken in the Ll norm. Given a number e

and leaves the other integers fixed. Since the mapping atT is measure-preserving and leaves f invariant, (3.1) gives us (3.3) Ilf - E[fl X n +l , ••• , X 2n ] 111 < e,

I

from which, the operator E[ . ~n] being norm-decreasing, we obtain IIE[fl X n +b

•• . ,

X 2n ]

-

E[fl ~n]lll

< e,

and finally, comparing these relations with (3.2), and (3.3) Ilf - E[fl ~] 111

< 3e.

The statement then follows, since e is arbitrary.

T4 COROLLARY Suppose that the random variables X n are independent and identically distributed; the law induced by P on [/ is then degenerate. Proof The law P is the product of identical laws ; it is hence symmetric, and Theorem 3 can be applied. It thus suffices to establish that the law induced by P on ~ is degenerate ("Kolmogorov's zero-one law"), which can easily be done as follows. Letfbe the indicator function of an element of ~; then f = lim n E[f 8 n] a.s. Now f is independent of 8 m so that E[fl tS'n] = E[f] a.s. We thus havef= E[f]. We now establish a theorem (due to de Finetti*) which implies the converse of the preceding result: if P is symmetric, and if the law induced by P on [/ is degenerate, the random variables X n are independent (and evidently identically distributed).

I

* See Loeve (89), Chapter VII, Section 27 (p.

365).

Applications of the Convergence Theorems

151

VIII, T5

T5 THEOREM Let P be a symmetric law. The random variables Xl' ... , X n, ... are then conditionally independent with respect to the a-jield f/. Proof We are going to show (11.49) that E[fl° Xl ... fn

0

I

X n ~] = E[fl 0 f/] ... E[fn

I

X n f/]

0

a.s.,

(5.1)

whatever be the integer n and the indicator functions fl' ... ,fn of elements of or. Denote by f/ m the a-field consisting of the elements of cC, which are invariant under the mappings atT' where a shifts only the first m integers. These a-fields decrease as m increases, and their intersection is equal to f/. On the other hand, put s~

= Ik

It is clear that for every m

0

Xl

+ .... + Ik

(k = 1,2, ... , n).

Xm

0

>n

EfIkoXi If/m]

= S~/m

(k

=

1, ... , n; i

The random variables S~/m thus converge a.s. to E[fk V.T21. We hence have a.s.

=lim m .... oo

1-n

m

=

1, ... , n).

I

Xi f/] when m -+

0

00,

from

~

foX . . . f oX ~ Jl ml In mn' l::;ml::;m

1::; m n ::; m

The number of terms for which the indices m b

••• ,

m n are distinct is equal to

m(m - 1) ... (m - n

+ 1);

its ratio with m n tends to 1 when m -+ 00, which allows us to neglect terms for which two of the indices are equal. The left side above is thus also equal to

Hm 1 n.... oo m(m - 1) ... (m - n

! + 1) ml;!:mt;!:···

11 0 X ml ;!:m

••• In

0

X mn

n

l::;ml::;m, .•. ,I::;mn::;m

= lim E[Il

0

I

Xl' .. In 0 X n f/ m]

m"" 00

I

= E[!1 0 Xl ... In 0 X n f/] from V.T21. Equality (5.1) is thus established.

a.s.,

A theorem of Choquet and Deny 6 We now establish, by means of Theorem 4, an interesting result of Choquet and Deny* [cf. (43)], which Feller has used as the basis for his simplified proofs of renewal theorems [see (70)]. The probabilistic proof we give is borrowed from Doob, Snell, and Williamson (63). Let G be a locally compact Abelian groupt, and let T be a probability law on the a-field • Complete proofs appear in Deny (51).

t Not all of the group axioms will be used.

VIII, T7, T8

152

Applications of Martingale Theory

PAo(G). A real continuous function h defined on G is said to be 'T-harmonic if h(x)

= fa h(x + y) d'T(Y)

for every x

E

G.

More generally we denote by Tf, for every measurable function f such that integrals considered make sense, the function x

J'oI#-

faf(X + y) d'T(Y).

'T-harmonic functions thus are characterized by the relation Th of Choquet and Deny.

= h. Here then is the result

T7 THEOREM Let h be a real function, bounded and continuous on G; h is then 'T-harmonic if and only if it admits every point of the support Of'T as a period. Proof Suppose that h admits every point of the support S of 'T as a period, then

fa hex + y) d'T(Y) = fs hex + y) d'T(Y) = fs hex) d'T(Y) =

hex).

To establish the converse, consider the set Q = GN , on which we denote by X o, ••• , X n the coordinate mappings. Give Q the a-fields /F = ff(Xn, n EN) /Fn = ff(Xo, ... , X n),

and the law P for which the random variables Xn(n E N) are independent, with the same law 'T. Let x be an element of G; set Yn = X + X o + ... + X n • It is easy to verify the formula E[fo Y n+11 :Fnl

=

Tfo Y n a.s.

for every bounded measurable function! The process (h 0 Yn ) is thus a bounded martingale with respect to the family (/F n)' The limit Hi w ) = lim hex

+ Xo(w) + ... + Xn(w))

I

hence exists a.s. from V.T18, and h 0 Y n = E[Ha: /F nl a.s. Now Ha: is clearly a symmetric random variable on GN • We thus have a.s. Ha: = E[Ha:l = hex) from T4, and hence also h 0 Yo = hex + X o) = hex). In other words, hex + y) = hex) for 'T-almost all y. Since the function h is continuous, we have hex + y) = hex) for every yES. The following theorem, a simple consequence of T7, is also due to Choquet and Deny. T8

Let fl be a measure on G (positive or not) of finite total mass. We have if and only if fl admits every point of the support Of'T as a period.

THEOREM

fl = 'T

* flt

Proof It is clear that the indicated condition is sufficient. To establish its necessity, consider a function g E ~%(G), and associate with it the function

h:

x

.A.N+

fag(X + y) dfl(Y).

t p is then called a ,,-harmonic measure. The proof in fact uses a weaker assumption than the boundedness of p.

Applications of the Convergence Theorems

153

From the relation p,

f

VIII, 9

= -r * p, we obtain h(x

+

z) d-r(z) =

G

f

g(x

+

y

+

z) d-r(z) dp,(y)

GxG

= fGg(X

+ u) dp,(u) =

h(x).

The function h is hence bounded, continuous, and -r-harmonic. It therefore admits as a period every point of the support of -r and, since the function g is arbitrary, this also holds for p,. Applications to measure theory 9 Let (Q,§) be a measurable space. We say that a sub-a-field C§ of ~ is separable if C§ is generated by a sequence (An)nEN of elements of ~. Set C§ n = .r(A o, ••• , An); there then exists a partition of Q into a finite number of measurable sets A n.I , A n.2 , ••• , A n.m .. such that every element of C§ n is the union of some of these sets. Let P be a probability law on (Q,§), and let Q be a bounded measure on the same space, absolutely continuous with respect to P [i.e. P(A) = 0 ~ Q(A) = 0]. Set

x () n W

~ Q(An,i) / () = i~ P(A .) A .. ,i W ,

(9.1)

n,t

where ratios of the form % are given the value O. We pointed out in No. V.3 that the process (Xn) constitutes a martingale with respect to the family of a-fields (C§ n)' We are going to deduce from this fact, and the martingale convergence theorems, various theorems of measure theory. For simplicity, we limit ourselves to the case where the measure Q is positive. We did not use the Radon-Nikodym theorem in Chapter V, and it is rather interesting to see that this theorem .can easily be established as a consequence of Theorem V.IS. To this end, note that the following property holds: For every

B

> 0,

there exists a number

'Yj

> 0 such

that the relation A E ~, P(A) < 'Yj, implies Q(A) < B.

(9.2)

The absence of such an 'Yj would, in fact, imply the existence of a sequence (An)nEN of elements of~, such that 1n P(A n) < 00 and Q(A n) > B for every n. Set/A = lim n sup lA.. ; we have A c U ~1> An for every p, hence P(A) < 1:=1> P(A n), and finally peA) = O. On the other hand, Q(A) > lim n sup Q(A n) > B, from Fatou's lemma (applied to the sets Q"'A n). This contradicts the absolute continuity of Q with respect to P. We show next that the martingale (Xn ) is uniformly integrable. It is clear that (9.3)

On the other hand, (9.4)

The left side of (9.4) is thus less than the number 'Yj of (9.2) whenever c is large enough, and therefore the integral in (9.3) is smaller than B, from (9.2), independently of n.

VIII, 10, 11

Applications of Martingale Theory

154

The martingale (Xn ) thus converges to a limit when n -+ 00 in the Ll norm. This limit is evidently a Radon-Nikodym density of the restriction of Q to ~, with respect to the restriction of P to ~. The Radon-Nikodym theorem thus holds for every separable a-field. For each separable sub-a-field ~ of :F, denote by X<§ the density thus constructed. These random variables constitute a martingale indexed by the set (ordered by inclusion) of separable sub-a-fields of :F. This set is clearly filtering to the right, and we see as above that the martingale (X<§) is uniformly integrable. It therefore converges in Ll from No. V.20, and its limit is evidently the desired density of Q with respect to P.

10 We shall need the following result later. Suppose that the a-field :F is separable, that (T,ff) is a measurable space, and that (Qt)teT, (Pt)teT are two .r-measurable families* of bounded positive measures on (.o,:F), such that Qt is absolutely continuous with respect to Pt for each t E T. There then exists a positive function X(t,£O), jointly measurable in the pair (t,£O), such that the function X(t,£O) is, for each t E T, a density of Qt with respect to Pt. Return in fact to the construction of the preceding number, taking the separable a-field ~ equal to :F, and setting, as in formula (9.1),

Xn(t,£O) =

! i=1

Qt(A n.i ) IAn.l£O). PlAn)

(10.1)

This function is clearly jointly measurable in t and £O. We know that X(t,£O) admits P-a.s. a finite limit for each t; denote by H the (measurable) set of (t,£O) E T x .0 such that the limit of Xn(t,£O) does not exist, or is infinite, and set

X(t,£O) =

{Ol. X( ) n t,£O Im

if (t,£O)EH if (t,£O)

~

H.

n~oo

The function thus constructed clearly satisfies the conditions of the statement. The lifting theorem

11 We shall give only a part of this important theorem. Ionescu-Tulcea (72), in fact, established the existence of a lifting endowed with auxiliary algebraic properties, which will not concern us here. See also Maharam (92). Let (.o,:F,P) be a complete probability space, and let JV be the a-field consisting of the negligible sets and their complements. For every real-valued random variable.t: we denote by f'" the equivalence class off under a.s. equality. Let ~ be a a-field on .0 such that JV c ~ c :F; we shall use the notation P both for the law on :F and the law it induces on ~. The space Loo(.o,~,P) can then be considered as a subspace of Loo(.o,:F,P). We say that ~ has the lifting property if there exists a mapping g of the Banach space L 00(.0, ~,P) into the Banach space 2 00 (.0, ~), having the following properties: (a) g is a positive linear isometry; (b) g(a"') = a for every constant a;

(c) [g(x)]'" = x for every x We say then that g is a

*

See H. D13.

E Loo(.o,~,P).

~-lifting.

Here is the main result.

(11.1)

155

TI2

Applications of the Convergence Theorems THEOREM

VIII, T12

The a-jield!F has the lifting property.

Proof Let R be the set of pairs «(§,g), where (§ is a a-field such that JV c (§ c !F, and g is a (§-lifting. Order R by the relation -< defined as follows: «(§,g)

-<

«(§',g'»<=>«(§ c (§' and g' extends g).

It is clear that R is nonempty, since there is clearly an ./V-lifting. We show that R has a maximal element. To do this it will suffice to establish that every well-ordered subset of R has an upper bound in R. Denote then by I a well-ordered set, by i ~ «(§i,gi) a strictly increasing mapping from I into R, and by .Ye the a-field generated by U ieI (§i' We shall construct an .Ye-lifting h which extends each gi' considering two cases. Suppose first that I is not countable. We then have .Ye = U ieI (§i, L oo(O,.Ye,P) = U ieI L 00(0, (§i,P), and it suffices to put, for every x EL oo(O,.Ye,P),

.h(x) = gi(X),

where i is chosen large enough so that x E Loo(O,(§i,P), Now suppose that I is countable. There then exists a sequence (in)neN of elements of I such that every i E I is dominated by one of the in. Let U be an ultrafilter on N, which converges to infinity. For every x EL oo(O,.Ye,P) set

I

h(x) = lim gi (E[x (§i ]) II

n

n

[where the conditional expectation is considered here as an element of L oo(O,(§in,P)], For each x, we have a.s. h(x) = lim gin(E[x (§iJ) = x

I

n-OO

from V.TIS, so that h(x) is indeed an .Ye-measurable random variable. Now h induces gin on Loo(O,(§in,P), since E[xl (§i) =X for every xELoo(O,(§in,P) and every p>n. It remains to verify that h is a positive linear isometry. The first two properties are evident. Moreover, IIE[xl (§]1100 < Ilxll oo for every x, and consequently Ilh(x) 1 < Ilxll oo . On the other hand, the functions Ilh(x) 1 - h(x), Ilh(x) 1 + h(x) are positive, and hence so are their classes. We therefore have x < Ilh(x)1I and -x < IIh(x)1I almost everywhere, thus IIxll oo < Ilh(x)ll, and so h is an isometry. We can thus consider a maximal element «(§,g) of R, and show that the a-field (§ is equal to !F. Let A be an element!F, and A' its complement. Among all the elements of (§ which a.s. contain A, let B be a set whose probability is minimal; the relations C E t§ and A c C then imply B c C a.s. Associate in the same manner with A' an event B' of (§. Let finally .Ye be the a-field generated by (§ and A. Every element x of L oo(O,.Ye,P) can be written where a and b are two elements of L 00(0, (§). Consider a second representation of the same type, x = a:;'(IAY" + b:;'(IA,)''''· We have a = a l a.s. on A, hence a a.s. Put then

= a l a.s. on B, and aIB =

alIB a.s. Similarly, bIB'

=

bllB'

VIII, 13, D14

156

Applications of Martingale Theory

This function is well defined, and it is easily verified that h is a lifting of L OO(Q,dF,P), which extends g. We thus have dF = ~ from the maximal character of (8B,g), and hence A E ~. Since A was arbitrary, we have ~ = §".

2.

cif Processes*

Applications to the General Theory

We are now going to pursue, with the aid of martingale theory, the general study of stochastic processes begun in No. IV.45. We limit ourselves to real-valued processes, although the extension of the results below to processes with values in an LCC space presents no difficulty. Stochastic intervals 13 Let (Q,§",P) be a complete probability space, given an increasing, right-continuous family (§"t)teR+ of sub-a-fields of§". We suppose as usual that §"o contains all the negligible sets. Let (Xt ) and (Yt ) be two stochastic processes such that the set

is negligible. There is then no reason to distinguish the processes (Xt) and (Yt)-this is the point of view we have already adopted in No. VI.8. The set R+ x Q thus has a natural class JV of negligible subsets, consisting of those with P-negligible projection on Q. We should therefore identify two sets that differ only by an element of JV; we leave the task of doing this in the reasonings that follow to the reader to avoid overburdening the text. Let Sand T be two stopping times such that S < T; we denote by [S,T[ ("a half-open stochastic interval") the set {(t,w)

E

R+ x Q: Sew)

< t < T(w)}.

The closed stochastic interval [S,T] is defined in an analogous manner. t An interval of the form [T,T] will be denoted simply by [T]. We denote by f the paving on R+ x Q consisting of the stochastic intervals of the form [S,T[, and by f' the paving consisting of the intervals of the form [S,T] such that T(w) < 00 for every w such that Sew) < 00, and such that the stopping time S is accessible (VII.D42). Every interval [S,T] E f' is the intersection of intervals [S, T + (ljn)[ (n EN) which belong to f; the a-field .r(f') generated by f' is thus contained in .r(f). D14 DEFINITION Let (Xt ) be a process with values in a measurable space (E,C). We say that (Xt ) is well-measurable if the mapping (t,w) .A,/\/'+ xtCw) is measurable with respect to the a-field .r(f).

The elements of .r(f) are called well-measurable sets. • The results of this section are taken from Meyer [6].

_

t The concern in these two cases is with subsets of R+ x n, and not of R+ x n. The cross section of an interval [S,T] at an w such that T(w) =

+ 00 is thus the interval [S(w),oo[ of R+.

157

Applications to the General Theory of Processes

VIII, T15, T16

T15 THEOREM Every well-measurable process is progressively measurable with respect to the family (~t) (IV.D45).

Proof It will suffice to prove that every well-measurable subset of R+ X n is progressively measurable, or equivalently that every stochastic interval [S,T[ in J is progressively measurable. This is established directly with no difficulty, or by applying IV.T47 to the indicator function of [S,T[. Here is an important example of a well-measurable process. T16 THEOREM Let (Xt) be a real-valued process, adapted to the family (~t), with paths right continuous and free of oscillatory discontinuities,. (Xt) is then well-measurable.

> 0, define by induction the following stopping times: Tn+l(w) = inf {t: t > Tiro), IXt(w) - XTn(W) I > e}. (16.1) > e on the set {Tn+l < oo} from the right continuity of the paths.

Proof Given a number e To = 0,

We have IXTn +1 - XTnl The absence of oscillatory discontinuities implies then the relation lim n T n = ~(w) =

!

00.

Set

X Tn(w)I{T.. ~t
n

We have IXt(w) - Yt(W) I < e for every (t,w). The function X Tn is ~ Tn-measurable from IV.T49, and the following random variables are hence stopping times: if ke

< XTn(w) < (k + l)e

(k

E

Z)

otherwise T~n(w) =

Tkn(w)

V

Tn+l(w).

The process (Zt) defined by Zt(W) =

!

keI[Tkn,T'kn[(t,w)

neN keZ

is well-measurable and differs from (Xt ) by less than 2e. The theorem follows immediately, since e was arbitrary. Remar,ks (a) Let (Xt) be a process, with values in a topological space E given the Baire a-field, which satisfies the conditions of the statement. The process (f 0 X t ) is then wellmeasurable for every functionfE ~(E), from the preceding result, and it follows that (Xt ) itself is well-measurable. (b) The indicator function of an interval [S,T[ E J is an adapted process with paths that are right continuous and have left limits. These processes thus generate the a-field .r(J). (c) The statement extends in fact to adapted processes with right continuous paths (i.e., with no assumption on left limits), if two processes that differ only by a set in % are identified. Indeed, let (Xt ) be a process with right-continuous paths. We then use in place of (16.1) the following construction by transfinite induction: To

= 0;

for every ordinal (X; Tp(w) = sup J:(w) 11.
for every limit ordinal {J.

VIII, T17

158

Applications of Martingale Theory

The reader can easily verify the existence of a countable ordinal (J. such that Ta = 00 a.s. The proof is then completed as above, so as to obtain a process (Zt), which differs by less than 2e from (Xt), except perhaps on the set of (t,w) such that Ta(w) < 00, and which is well-measurable. Existence of well-measurable modifications (a) Let (Xt) be a measurable stochastic process (c! IV.45), with values in a compact interval of R. There exists a well-measurable process (Yt) such that for every (finite) stopping time T, (17.1)

T17

THEOREM

(b) Let (Xt) be a real-valued process, progressively measurable with respect to the family (§'t). There exists a well-measurable process (Yt) such that for every (finite) stopping time T, YT

=

XT

a.s.

(17.2)

Proof Let:Ye be the vector space consisting of the bounded measurable processes (Xt) such that there exists a well-measurable process (Yt ) satisfying (17.1). It is clear that :Ye is closed under passage to the limit along monotone uniformly bounded sequences. It

will thus suffice, from I.T20, to show that :Ye contains the processes of the form Xt(w) = a(t)X(w),

where a is the indicator function of an interval of R+, and where X is a bounded random variable on .0. Let then (Zt) be a right-continuous modification of the martingale (E[X §'tD [see VI.T4, (3)]; it suffices to set

I

Yt(w) = a(t)ZtCw ).

This process is well-measurable (TI6) and satisfies (17.1) from VI.14. It follows from (a) that (17.2) is true for every process (Xt ) with values in a bounded interval of R, since X T = E[XT I§'T] a.s. for every finite stopping time T, from IV.T49. The general case is then deduced by considering the bounded process

Assertion (b) remains true for processes with values in a complete separable metric space E. Indeed, it applies to progressively measurable elementary processes with values in E, and every progressively measurable process (Xt ) is the uniform limit of a sequence (Xn ) (n EN) of such elementary processes, with which can be associated wellmeasurable processes (Y;) satisfying (17.2). It then suffices to choose a point e E E, and to set if this limit exists lim Y:(w) ~(w) = n { otherwise. e Remark

A study of the paving of' The next few theorems are not very interesting in themselves. They are the lemmas for the proof of Theorem 21, which is the main result of this section.

159

T18

VIII, TI8

Applications to the General Theory of Processes THEOREM

*

(a) Let S be an accessible stopping time. The interval [O,S[ then belongs

to J~~. (b) Every element of the a-.field :T(J') is J'-analytic. Proof We have previously established (see the remark which follows VII.T45) the existence of stopping times Sn,1J (n EN, pEN), which have the following properties: Sn.1J < Sn.1J+l < S for every n and every p; For almost every w E {S > O}, there exists an integer n such that lim Sn,iw)

=

S(w),

and

Sn.iw)

< S(w)

(18.1)

for every p.

(18.2)

P---+OCJ

It will be convenient to suppose that property (18.2) holds for every w E Q. To accomplish this it suffices to modify the Sn.1J on a negligible set, which is legitimate since the negligible sets belong to §'O by hypothesis. Now set An = {Sn.1J < S for every p, li;n Sn.1J = S}. The union of those sets equals {S > O}. There is clearly no loss of generality in supposing that P(A n ) > 0 for every n. We begin with a construction that will be used several times. Let U be a stopping time with the following properties: U is accessible; (18.3)

U(w)

< S(w) for every WE {U < oo}.

(18.4)

Given an n EN, denote by D(w) the subset of R+ U {oo} consisting of the numbers Sn.iw) V U(w) (p EN). Define inductively the random variables Vk (k EN) as follows: Vo(w) =

< U(w) for every p; t E D(w), t > U(w)} otherwise.

U(w) if Sn 1J(w) ( inf {t:

.

ViW) = U(w) if Viw) = U(w); k V +1(w) = ( inf {t: t

E

D(w), t

> Viw)} otherwise (as usual letting inf 0

=

+ 00).

It is easily verified by induction that the Vk are stopping times. We have by construction Vk+1(w) > Vk(w) on the set {Vk+l '# U, V k+1 '# oo}; it then follows that the stopping time V= lim Vk k-+ OCJ

cannot coincide with a totally inaccessible stopping time T on the set {T < oo}; V thus satisfies (18.3) and (18.4) also, and moreover has the following properties, which we list for later use: (18.5) V(w) = S(w) on the set {U = S} U An. Set U'

=

[U, V[ =

U on {U

Uk [U',

< V}, u' =

V~]

+00

belongs to J~.

on {U= V},

V~

=

(Vk

A k) V U'.

Then (18.6)

• The proof of T18 is not very simple. Since the family (~t) is free of times of discontinuity in most applications, it may be worthwhile to note that (b) then is almost obvious. Let (Tn ) be a sequence of finite stopping times that increases to S, such that T n < Son {S > o} (VII.T45). Then [0, S[ = Un [O,Tn ], and [O,Tn ] = 1J [O,Tn + (lIp)]; now T n + (lip) is accessible for every pEN. Therefore [O,S[ belongs to J~u, and an easy reasoning (see the end of the proof below) establishes the fundamental property (b).

n

VIII, T19

160

Applications of Martingale Theory

Now let p be an integer, and let :Ye p be the collection of stopping times R which satisfy (18.3), (18.4), and in addition the two properties: R(w) = S(w)

for every w

E

The interval [O,R[ belongs to

Ap•

..1"~.

(18.7) (18.8)

:Yep is nonempty: taking U = 0, n = p in the preceding construction yields a stopping time V belonging to :Ye p. The set :Ye p, is moreover, closed under the operation v, and under passage to the limit along increasing sequences [see VII.T43(c)]. There hence exists a maximum stopping time Rp E :Ye p' i.e., one such that

< Rp a.s. We show that R p(w) = S(w) a. s. on the set {Rp < oo}. Suppose that this were not so; there would then exist an integer q such that P[{R < S} n A > 0. Apply the preceding R'

E

:Ye p<;:::> R'

q]

p

construction, taking U = Rp, n = q. It then follows from (18.5) and (18.6) that the stopping time V so obtained belongs to :Yep' and strictly dominates Rp on the set {Rp < S} n A q , contradicting the maximality of Rp. Let ([Sn,TnDneN be a sequence of elements of ..1"' such that [O,R p[ = U neN [SmTn]. We can clearly suppose that

Wv

Riw) = S(w)

for every w

E

{Rp

< oo},

(18.9)

at the cost of modifying Rp, Sm and T n on a set of measure zero. The proof is then nearly complete: We have, in fact, [O,S[ = nPEN [O,R p [, which establishes (a). On the other hand, consider an interval [S,T] E..1"'; its complement is the union of the intervals ]T,oo[ and [O,S[. The first is the union of the intervals [T + (l/n), T + n], which belong to ..1"'; the second belongs to ..1"~;; from (a). Statement (b) then follows from III.T12. The following statement is similar to T16. T19 THEOREM Let (Xt) be a process with values in an LCC space E and with left-continuous paths. The mapping (t,w) .A.f'.I'+ Xt(w) is then measurable when R+ X Q is given the (J-field !T(..1"'). Proof It will suffice to show that {(t,w): Xlw) E U} belongs to !T(..1"') for every open set U c E. Denote by (U p)peN an increasing sequence of open sets, such that U = UP Up,

and, in turn, set

Hp = lim inf H np (equivalently: I Hp = lim inf I HflP )' n

n

and H =

U Hp. PEN

161

Applications to the General Theory of Processes

The reader will easily verify that the set {(t,w): Xt(w) to show that H knp belongs to !T(J'). Now we have H knp = [Tknp , T knp

VIII, T20

U} is equal to H. It thus suffices

E

+ ;n]'

where the stopping time T knp is defined by

k Tkni w ) =

{

2

if

n

+ 00

otherwise.

Let S be a totally inaccessible stopping time; we have P{S = kl2 n } = 0, and thus P{S

=

T knp

< oo} = 0;

Tknp is therefore accessible, H knp belongs to of', and the theorem is established.

T20

Let (Xt ) be a real-valued well-measurable process. There then exists a process (Yt ) with the following properties: (1) The set {(t,w): Yiw) E B} belongs to !T(of')for every Borel set B of the line. (2) There exists a sequence (Tn) of totally inaccessible stopping times, such that THEOREM

{(t, w): Xiw) # yt(w)} =

U [Tn]. n

(3) X T = Y T a.s. for every accessible stopping time T. Suppose in addition that (Xt) is the indicator function of a well-measurable set,. (Yt) can be taken as the indicator function of an element of .r(J'). Proof We saw in No. 16 [remark (b)] that the a-field !T(.f) is generated by the processes,

adapted to the family (~t), having right-continuous paths with left limits. There thus exists a sequence «Z~)tE&)nEN of processes of this type, such that the function (t,w) .A.J\I'+ Xtw) is measurable with respect to the a-field on R+ X Q generated by these processes (1.9). Denote by E the compact space RN, by (Zt) the pro<;ess with values in E defined by Zt(w) = (Z~(w))nEN' The process (Xt) is then measurable with respect to the a-field generated by the process (Zt); there hence exists a Borel function f on E such that Xi w) = f 0 Zt(w) for every t and every w (I.TI8). Denote by (Ut) the process defined by Ut(w)

= f(Zt-(w))

for every t and every w;

this process is measurable with respect to the a-field !T(of'), from T19. Denote by d a metric defining the topology of E, and define inductively the following stopping times, for every integer m > 1: H1,m(w) = inf(t: t H n+1,m(w) = inf(t: t

> 0, d(Zt_(w), Zt<w)) > ~l

> Hn,m(w), d(Zt_(w), Zt(w)) > ~l

(see Nos. IV.44 and IV.53(c)). List these stopping times in a single sequence, which we denote by (Rp)PEN' The stopping times Rp "carry" all of the discontinuities of the process (Zt), in the sense that {(t, w): Zt_(w) # Zt(w)} = U [Rp]. p

VIII, T21

Applications of Martingale Theory

162

Now decompose each stopping time R1J into its accessible part S1J and its totally inaccessible part T1J (VII.T44). Denote by s the set U 1J [S1J]' by s' the complement of this set, and put Yim) = Is,f(Zt-(m))

+ Isf(Zt(m».

The set {(t,m): Yt(m) ¥: Xt(m)} is equal to U 1J [T1J]' a fact which establishes condition (2), and at the same time condition (3), which is an immediate consequence of (2). It remains to establish that the function (t,m) ~ Y t ( m) is measurable with respect to .r(J'). The sets sand s' belong to this a-field, so that it suffices to study the function (t,m) To this end, for every k

E

~

Is(t,m)f(Zt(m».

> 0 set ke < j(Zsp(m» < (k + l)e

Z and every e

Sim) if Sk'£(m) = { 1J + 00 otherwise.

These random variables are accessible stopping times from VII.T43(b); the set

sk,£ =

U [Sk,£] 1J 1JEN

thus belongs to .r(J'), and the function (t, m) ~

1 keIsk.e
is measurable with respect to .r(J'). It only remains to let e tend to zero, noting that this function converges to the function (t,m) ~ f(Zim»Is(t,m). Suppose finally that X t ( m) takes only the values 0 and 1; f can then be chosen as the indicator function of a set, and it follows that Yt(m) also takes only the values 0 and 1. Here now is the main result of this section. Recall that since the probability space (Q,~,P) is complete the projection on Q of a subset of R+ x Q, which is measurable with respect to &l(R+) x ~, belongs to ~ (111.24).

-+

T21 THEOREM Let A be a well-measurable subset of R+ X Q, and let C be the projection of A on Q. Given a number e > 0, there exists a stopping time T such that

(T(m),m)

E

A for every m such that T(m)

< 00;

P{T < oo} ~ P(C) - e.

If A

(21.1) (21.2)

belongs to .r(J'), it can moreover be assumed that T is accessible.

Proof The indicator function of A is a well-measurable process; there thus exists, from T20, a set BE .r(ef') and a sequence (Tn ) of totally inaccessible stopping times, such that n

If A belongs to .r(ef'), we just take B = A. Set: Rn(m) = {

Tn ( m) if

+ 00

(Tn ( m), m)

belongs to A

otherwise.

The set [Rn] is well-measurable, which implies that Rn is a stopping time.

163

VIII, D22

Square-Integrable Martingales

Denote by J; the collection of finite unions of elements of J'. The debut (IV.D5l) of every element of is the lower bound of a finite number of accessible stopping times, and hence is accessible from VII.T43(a). The paving J;lJ is closed under (U f, c). The at every wE Q is a compact set in R+, and the cross section of every element H of debut of H, being the limit of an increasing sequence of accessible stopping times, is accessible from VII.T43(c). Denote by p* the "outer probability" associated with P [P*(U) = infyEoF P(V) for every

J;

n

J;

Y::>U

U c Q]; we saw in No. I1I.24 that p* is a Choquet ~-capacity. Let 'TT' be the projection of R+ x Q onto Q; for every subset H of R+ x Q set I(H) = P*('TT'(H». This set function is a capacity with respect to the paving J;lJ: Properties III.18(a) and (b) are clear, and property III.18(c) is an easy consequence of III.T6. Since the set B is J'analytic from T18, the Choquet capacitability theorem implies the existence of an element J of J;lJ such that e J c Band I(J) > I(B) -

2.

Let S be the debut of J; we have (S(w),w) EJ for every w such that S(w) < 00. Since the stopping time S is accessible, we also have (S(w),w) E A for every w such that S(w) < 00. This settles the case where A E :T(J'). To deal with the general case, set R(w) = R 1(w) A R 2(w) A ••• A Riw),

where the integer p is chosen large enough so that

p{w: R(w)

=

00, i~f R.(w) < 00) < ~.

Again we have (R(w),w) E A for every w such that R(w) this stopping time satisfies property (21.1) and we have

< 00.

Finally, set T = R A S;

P[~{T< oo}] < P[B"{S < oo}] + P[{i~f R. < OO)"{R < oo}] < eo 3. Square-InteBrable MartinBales The martingales, stopping times, etc., we consider in this section are always relative to a family of a-fields (~t), which satisfies the same hypotheses as in the preceding section (No. 13).* D22 have

DEFINITION

We say that a right-continuous martingale (Xt ) is square-integrable

sup E[X~]

< 00.

if we

(22.1)

t

This amounts to saying that (Xt ) is a uniformly integrable martingale of the form (E[Y\ ~t]), where Y belongs to 'p2. We then have E[X~] < E[y2 ] for every stopping time T, from Jensen's inequality. The theory we develop extends to right-continuous martingales (Xt ) such that X t belongs to 'p2 for every t. This immediate generalization is left to the reader. • The results of this section are taken from Meyer (97).

VIII, 23, D24, T25, D26

Applications of Martingale Theory

164

23 Let (Xt ) be a square-integrable martingale, and let (Mt ) be a right-continuous modification of the martingale (E[X~ I §""tD. The submartingale (X:) is dominated by (Mt), and thus belongs to the class (D) (VI.T19). The process (M t - X:) is hence a potential of the class (D), generated from VII.T29 by a unique integrable natural increasing process (At). We say that (At) is the increasing process associated with the martingale (Xt). * Since the process (X; - At) is a right-continuous martingale we have for every pair of stopping times S, T such that S < T,

I

E[A T - As §""s]

D24 DEFINITION if we have

=

I

E[X~ - X~ §""s]

= E[(XT

- X s )21 §""s]

a.s.

We say that a square-integrable martingale (Xt ) is quasi-left-continuous

X um T n = lim X T n

(24.1)

a.s.

n

n

for every increasing sequence (Tn)nEN of stopping times.

Every stopping time T such that XT(w) ~ XT_(w) a.s. on the set

{T < oo}

is then totally inaccessible (VII.D42). t Conversely, the reader can verify that this property implies the quasi left continuity of (Xt ). T25 THEOREM The increasing process (At) associated with (Xt) is continuous if and only if (Xt ) is quasi-left-continuous.

Proof The natural increasing process (At) is continuous if and only if

lim E[X~n ] = E[X~]

(25.1)

n

for every increasing sequence (Tn ) of stopping times which converges to a stopping time T (er. VII.T37). Since the random variables X~ are uniformly integrable, this condition • n can be wntten

Now we have lim n X Tn =

E[X~1 = E[ (Ii:,n X T . ) } E[XT I V §"" TJt (VI.T6). We thus have

E[X~] =

(25.2)

n

E[(lim X T n )2] n

+ E[(XT -

lim X T n )2], n

and we see that (25.2) is equivalent to the relation X T = lim n X T n a.s. D26 DEFINITION We say that two square integrable martingales (Xt ) and ( Yt) are orthogonal if the process (Xt Yt) is a martingale. Suppose that Yo = 0; the martingales (Xt ) and (Yt ) are then orthogonal if and only if E[XTYT ] = 0 for every stopping time T: If the process (XtYt ) is a martingale, we have

* This notion is mainly useful for the theory of stochastic integrals, which we do not develop here. t Or a.s. infinite.

+Notation of No. VII.38.

Square-Integrable Martingales

165

VIII, T27, T28

Conversely, if this property is satisfied we have E[XT AYT) = 0* for every stopping time T and every event AE :FT. This relation can also be written

or even, since E[Xoo Y 00] = 0,

fA XTYT dP = fAXooYoo dP, or finally

I

E[XooYoo ~T] = XTYT a.s. T27 THEOREM Let (Xt) and (Yt) be two square-integrable martingales, and let (At) and (B t) be the increasing processes associated, respectively, with (Xt) and (Yt). Then (Xt) and (Yt) are orthogonal if and only if the increasing process associated with the martingale

(Xt

+ Yt) is equal to (At + ~ ~ k

.s~~

Proof This latter condition !laYs agaifi (since the iHcreasing process (At + B t) is natural) ~ that the process (Xt + Y t)2 - (At + B t ) is a martingale. It then suffices to note that

First decomposition of square-integrable martingales

We are going to decompose every square-integrable martingale into a quasi-left-continuous martingale and a martingale orthogonal to every quasi-left-continuous martingale. We say that two right-continuous martingales (Mt) and (Nt) have no common discontinuities if we have a.s. NtCw) = Nt_(w) for every t E R+ such that MtCw) ¥= Mt_(w). We begin with an auxiliary result. T28 THEOREM Let (Sn)neN be an increasing sequence of stopping times. Let S = lim n Sn and denote by U a square-integrable, ~s-measurable random variable such that

(28.1) then (a) The process (Ut) = (UI{t?-s}) is a square-integrable martingale; (b) (Ut) is orthogonal to every square-integrable martingale (Mt), which has no common discontinuity with (Ut); (c) The increasing process (At) associated with (Ut) is given by

(t ER).,

(28.2)

Proof It will suffice to establish (b), since (a) is then deduced by taking for (Mt) the martingale equal to 1. We have U 00 = UI{s
* The notation TA was introduced in No. VII.38.

166

Applications of Martingale Theory

VIII, T29

since the random variable UI{s
I

I

E[MooUoo ~t] = E[MooUI{t<s} ~t]

I

E[MooUI{sst} ~t]·

+

The first term on the right side is zero. Indeed, denote by H a bounded random variable; we have

Vn ~Sn (28.3)

~t-measurable

I

E[MooUI{t<s}H] = E[E[MooUI{t<s}H ~s]] = E[MsUI{t<s}H], since the random variable UI{t<s}H is ~s-measurable. On the other hand we have Ms = lim n M Sn on the set {U #: O}, since the martingale (M t) has no common discontinuities with (Ut). Now HI{t<s} = lim n HI{t<sn} is measurable with respect to the a-field Vn ~sn; the random variable HI{t<s} lim n M Sn is thus orthogonal to U, and the desired result follows since H is arbitrary. The right side of (28.3) thus reduces to

I

E[M00 UI{sst}l~t] = E[M00 ~t]UI{sst} = MtUt , since UI{sst} is ~t-measurable. The process (MtUt) is hence a martingale, which establishes (a) and (b). The fact that U belongs to 2 2 is not essential; it would suffice to assume that the product MooU is integrable. Assertion (a) thus remains true when U is only supposed integrable (take Moo = 1). Put V = U2 - E[U21 V n ~sJ; the process (VI{t"2S}) is a martingale from above, and it is equal to (U: - At). It thus only remains to verify that the increasing process (At) is natural. For this, note first that the function UI{sn=s} is ~sn-measurable, and hence orthogonal to U. We thus have a.s. U = on the set {Sn = S}, or U2 = U2I{Sn < S for all n}' Let (Yt ) be a bounded right-continuous martingale; we have

°

E[J.oo (~ -

~_) dA t] = E[(Ys -

Ys-)E[U21

~ ~Sn]]

2 =E[(Ys - YS-)E[U I{Sn<Sforann} \ = E[(Ys - Ys-)I{Sn<s

for an

~ ~Sn]]

n}E[ U21 ~ ~Sn]]

= 0, since limn (Ys - Ys ) is orthogonal to every function measurable with respect to the a-field V n ~Sn' The process (At) is thus natural (VII.D18) and the theorem is established.

Let (Xt ) be a square-integrable martingale, and T an accessible stopping time. Set V = X T - X T-. The process (Vt) = (VI{t"2T}) is then a square-integrable martingale, orthogonal to every square-integrable martingale, that has no common discontinuities with (Vt ). T29

THEOREM

Proof There exist increasing sequences (S~)nEN (p EN) of stopping times dominated by T, such that the union of the sets K[(S~)nEN] is a.s. equal to {T < oo} (see VII.T44, remark).

Square-Integrable Martingales

167

Set SP

VIII, T30

= limn S~, and inductively define UO

= Xso

- Hm Xs~ n

first:

(U~) = (UoI{t"2s o})

(X~)

= (Xt UP

then:

-

U~),

= X~P -

lim

X~~

n

(Uf)= (U P I{t"2s'D}) (X~+l) = (X: -

Un.

It follows from T28 that the processes (Uf) are square-integrable martingales without

common discontinuities, and thus pairwise orthogonal. We have

2 E[(UP)2] < E[X~] < 00, and consequently V = 2p UP and (Vt) = Lp (Un, these two series converging in the quadratic mean. The process (Vt ) is thus a square-integrable martingale. Let (Mt ) be a square-integrable martingale having no common discontinuities with (Vt ); (Mt ) then has no common discontinuities with each martingale (Ur): it is thus orthogonal to each (Ur), and hence to (Vt ). Let (Xt ) be a square-integrable martingale; there then exists a unique decomposition of (Xt) into a sum of two square-integrable martingales (Yt) and (Zt) with the following properties: Y o = 0; (Yt) is orthogonal to every quasi-left-continuous martingale,. (Zt) is quasi-leftcontinuous. Then (Yt ) is orthogonal to every square-integrable martingale that has no common discontinuities with it. T30

THEOREM

Proof Let (R1»)1)EN be a sequence of stopping times, which "carries" all of the discontinuities of the martingale (Xt ) (such a sequence was constructed in the proof of T20). Denote by TP the accessible part of the stopping time R1)(VII.T44) and define inductively:

and then

(V~) = ((X TO

-

(X~) = (Xt -

V~,

(Vn =

((X~p

(X~+l) = (X: -

X T~_)I{t"2TO}),

-

X~p-)I{t"2TP}),

Vf).

The martingales (Vn are of the type considered in No. 29, and have no common discontinuities; they are thus pairwise orthogonal. Similarly, the martingale (Xf+l) has no common discontinuities with (V~), , (Vf); (Xt ) is thus the sum of the orthogonal + Vr). We thus have martingales (Xr+l) and (V~ + V} +

L1) E[(V~)2] < E[X~] < 00.

VIII, T31

Applications of Martingale Theory

168

Set Y = !p V~-this series converges in quadratic mean-and denote by (Yt ) a rightcontinuous modification of the martingale (E[ Y ~t]); we have from VI.2 that

I

!~~E[(S~P IY, -i,v:,)j < 4li~E[( y - ~,v:,)j = O. It then follows that the jumps of (Yt ) are the jumps of the (Vf), and that the martingale (Zt) = (X t - Y t) has no accessible jumps-i.e., (Zt) is quasi left continuous, and no quasi left continuous martingale has common discontinuities with (Yt ). The very construction of ( Y t) implies the orthogonality of (Yt) and (Zt), and the validity of the last state-

ment. Finally, the uniqueness of the indicated decomposition is clear. Second decomposition of a square-integrable martingale We are now going to consider the case of a quasi-left-continuous martingale, (Xt ), and show that it is possible to decompose (Xt ) into a square-integrable martingale with continuous paths and a martingale, which is the sum of suitably compensated inaccessible jumps. This decomposition is evidently analogous to the famous decomposition of processes with independent increments, due to Paul Levy. We begin with an auxiliary result, similar to T29. T31 THEOREM Let T be a totally inaccessible stopping time and let Y be a positive, squareintegrable; ~T-measurable random variable which is zero on -the set {T = oo}. Denote by (At) the integrable increasing process (YI{t?T})' There then exists a unique continuous integrable increasing process (Bt) such that the process (Ut) = (At - B t) satisfies: (a) (Ut) is a square-integrable martingale; (b) The martingale (Ut) is orthogonal to every square-integrable martingale (Mt) which has no common discontinuities with (Ut). Proof Let (Pt) be the potential generated by (At), and let (Sn)nEN be an increasing sequence of stopping times. Since the stopping time T is totally inaccessible, we have a.s. A !imn Sn = lim n A sn . It then follows that the potential (Pt) is regular (VII.D33), and is consequently generated by a unique continuous integrable increasing process (B t ). The relation

shows that the process (Ut) is a martingale. It remains to show that Boo belongs to 2 Now we have, from VII.T23 and the continuity of (B t ), that

2

•

this last relation following from VII.T17. On the other hand, we have

and we know that SUPt (At + Pt) belongs to 2 2 (VI.2). Let (Mt) be a square-integrable martingale with no common discontinuities with (Ut); we show that E[M00 U 00 ~t] = MtUt a.s. We can easily reduce to the case where t = O.

I

169

Square-Integrable Martingales

VIII, T32

Using successively VII.Tt5 (and the remark that follows), the absence of common discontinuities, and VII.Tt7 (with the remark (a) following it), we have that

since Ut

= At -

B t , and these processes generate the same potential.

The integrable increasing process (C t) associated with the martingale (Ut) is the natural (continuous) integrable increasing process, which generates the same potential as the process (D t ) = Y2 I Remark

{t!T}'

Indeed, let (ti)i = O.l ..... n be a finite subdivision of the interval [t, 00] (to = t, t n = 00). We have

Expand the parentheses in this last term, and take arbitrarily fine subdivisions; the contribution of the terms (Bt.1+1 - B t1.)2, (Bt.1+1 - Bt.)(A . - At.) tends to 0, and there t1+1 1 1 remains only the sum .2i (Ati+l - A t)2, which is equal to Y2I{t? T}' We thus have E[C oo - C t I§"t] = E[D oo - D t I§"t]· The results we have established clearly extend, by taking differences, to the case of a random variable YE 2 2 which is not necessarily positive. T32

Let (Xt ) be a quasi-left-continuous square-integrable martingale. There then exists a unique decomposition of (Xt ) into a sum of two square-integrable martingales (Yt ) and (Zt) having the following properties: Yo = 0; (Yt) is orthogonal to every continuous square-integrable martingale,. (Zt) is continuous. Then (Yt ) is orthogonal to every square-integrable martingale that has no common discontinuities with (Yt ). THEOREM

Proof We show first that the random variable X T - X T - belongs to 2 2 for every stopping time T. Let (t i)i=O.l ..... n be a subdivision of the interval [0, 00] (t i = 0, t n = 00), such that P[T = t i ] = for every i ~ 0, n. We have

°

Let j( co) be the first integer j such that

It then suffices to note that X tj

ti

> T( co); we thus have

X tj _ 1 tends to X T arbitrarily fine, and to apply Fatou's lemma. -

-

X T - as the subdivision (t i ) becomes

VIII, T32

Applications of Martingale Theory

170

With this point established, consider as in No. 30 a sequence (Tn)neN of stopping times, which "carries" all of the discontinuities of the martingale (Xt ). Since (Xt ) is quasi left continuous the T n can be supposed totally inaccessible (VII.T44). Set

yo = X To

X

-

T 0-

(A~) = (yOI{t?:':T o})' (U~) = (A~ - B~),

where

(B~)

(A~ -

B~)

-is the only difference of continuous integrable increasing processes such that is a square-integrable martingale. Finally, let (X~) = (X t

-

U~).

Now, with the notation (Bf) having the same significance as above relative to define inductively (A~)

= (Y

1J

(A~),

I{t2:T,,}),

(Un = (Af - Bf),

(X:+ 1) = (X: -

un.

The proof is then finished exactly as in No. 30; the martingales (Un, with no common discontinuities, are pairwise orthogonal; the martingales (XfH) and (U~ + ... + U:) are orthogonal for the same reason. The series ~1J U~ converges in the quadratic mean to a random variable Y; the martingale (Y t ) = (E[ Y ~ admits the jumps of (Xt ) as discontinuities, so that the martingale (Zt) = (Xt - Yt) is continuous and orthogonal to (Yt ). We leave the details of the proof to the reader.

I tn

~ Part C

ANALYTIC TOOLS OF POTENTIAL THEORY

CHAPTER

IX

Kernels and Resolvents

1. Kernels. Dispersions Dl

DEFINITION

Let (E,lt) be a measurable space. A kernel on (E,lt) is a mapping (x,A)

JVV+

N(x,A)

of E x It into R+, which has the following properties: (1) The mapping x ~ N(x,A) is It-measurable for every set A Elt. (2) The mapping A .A./II+ N(x,A) is completely additive for every x E E.

The measure A

~

N(x,A) will often be denoted by N(x,dy).

2 The preceding definition in no way limits the "size" of kernels. For example, there exist kernels for which the function N(x,A) takes only the values 0 and + 00. Definition 1 will therefore be completed by the following definitions: (a) A kernel N is said to be sub-Markov (respectively, Markov) if N(x,E)

<1

[respectively N(x,E) = 1]

for every x E E. (b) We shall say that a kernel N is proper if the set E is the union of a sequence (Kn)neN of measurable sets, such that the mappings x JVV+ N(x,Kn) are bounded. * Extensions to functions and measures 3 Let f be a positive measurable function (finite or not). Consider the function x

~ fEN(x, dy)f(y)·

It is easy to show (by approximating f with measurable elementary functions) that this

function is also measurable. We shall denote it by NI In the case where f is not positive we shall continue to use this notation to denote the function Nf+ - Nf-, on the condition that this function be well-defined [not be, at any point, of the form 00 - 00]. • A slightly different definition of proper kernels is obtained by replacing "bounded" by "finite;" these definitions are equivalent for kernels which satisfy the maximum principle.

173

IX, 4-7 4 Let

Kernels and Resolvents

174

~+ be

the collection of all positive measurable functions, finite or not. The mapping f ~ Nf from ~+ into itself is additive, positive homogeneous, and has the following property. Let (fn) be an increasing sequence of elements of ~+; then

N(lim In) n

= lim Nln·

(4.1)

n

Conversely, every mapping from ~+ into ~+ that has these properties is associated with a kernel. Similarly, let ~ 0 be the vector space of bounded measurable functions, and let N be a sub-Markov kernel. The mappingf ~ Nfis a positive operator on ~o of norm ~ 1, which satisfies property (4.1) for every sequence (fn) of positive elements of ~ 0 dominated by a constant. These properties characterize the operators on ~o associated with sub-Markov kernels.

5 Let N be a kernel and f a measurable function such that the function Nf is well defined. The value of the function Nf at a point x E E will be denoted by either N(x,f) (by analogy with the notation N(x,A) used for sets)

or These notations are preferable to the notation Nf(x) , ~hich is unusable in complicated formulas. We can, in addition, always resort to writing (e:x;,Nf). 6 Let ft be a positive completely additive set function defined on 8, finite or not. It is very easy to verify that the set function A

~Lt(dX)N(X,A)

(A E8)

is again completely additive (Lebesgue's theorem). We shall denote this function by the symbol ftN. The measure N(x,dy) can thus be denoted by e:x;N. We shall be able, as in the case of functions, to define the measure ftN in certain cases where the measure ft is not positive. Thus if the kernel N is sub-Markov, the measure ftN is defined and bounded for every bounded measure ft. Let At 0 be the Banach space of bounded measures on (E,8), with the usual norm:

IIA 11

= sup

(A,!)

'E~

1'1

(IIAII is the "total variation" of

~1

A). The mapping ft ~ ftN is a positive operator with

norm less than or equal to 1 on At O' We will not attempt to characterize the operators of this type by a property analogous to (4.1). 7 Let f be a positive measurable function, and ft a positive, completely additive set function. We have the relation This equality evidently extends to measures and functions of arbitrary sign, under the condition that each of the quantities considered be well defined.

175

Kernels. Dispersions

IX, 8, D9

Composition of kernels 8 Let M and N be two kernels on (E,tff). It is easily deduced from Lebesgue's theorem that the mapping (x,A)

~ LM(X, dy)N(y,A)

from E x tff into R+ is also a kernel, which is called the composition of M and N, and denoted by MN. Let f be a positive measurable function. Since f is the limit of an increasing sequence of measurable elementary functions, we have [MN](x,f)

=L

M(x,dy)N(y,f)

= M(x,Nf).

Let then L, M, N be three kernels. From the above relation it follows that [(LM)N](x,f) = [LM](x,Nf) = L(x,MNf).

Composition of kernels is thus an associative operation, which permits us to use some notations without parentheses, such as LMN, flLMN, LMNf, or LMNfX. We observe that the composition of two sub-Markov (respectively, Markov) kernels is still sub-Markov (respectively Markov).

Kernels on a locally compact space Suppose now that E is a locally compact space. There then exist several methods of relating the notion of a kernel to the structure of E, and we will choose only one of them. For others, the reader can consult, for example, Bourbaki (19), Section 3. D9 DEFINITION Let C be a locally compact, a-compact space, with the a-field fJI u(E) of universally measurable sets. A dispersion-kernel (or simply: a dispersion) on E will be a kernel Non (E,fJlu(E», which has the following property: Let fl be a Radon measure with compact support on E. The measure itN is then (the (9.1) canonical extension to fJl u of) a Radon measure on E. In addition we say that the kernel N (or the dispersion N): (a) is continuous if the function Nf is continuous for every function f E ~},' (b) tends to zero at infinity if the function Nf tends to zero at infinity for every function fE~};

(c) is uniform if the function NI is bounded; (d) is strictly positive if all of the measures exN are different from zero.

We shall not be concerned for the moment with properties (b), (c), and (d), which enter later in potential theory; we have borrowed them from Choquet and Deny [see Deny (52), (53)]. The term "dispersion" has a slightly different meaning from Choquet's "diffusion." We ,limit ourselves to a-compact spaces, in order to avoid the difficulties connected with the definition of the universally measurable sets in general locally compact spaces.

IX, 10, TU

Kernels and Resolvents

176

to Remarks (a) Let N be a dispersion, and let p be a positive Radon measure with compact support. To say that the measure pN is the extension to flA u of a Radon measure is equivalent to saying that pN satisfies the condition (pN,IK > is finite for every compact K,

(10.1)

and moreover a certain regularity property (see II.37). Suppose now that p is an arbitrary positive Radon measure. Since the space E is a-compact, p is the sum of a series of Radon measures with compact support, so that flN is the sum of a series of Radon measures. It then follows that pN is a Radon measure if and only if it satisfies condition (10.1). Suppose in particular that the dispersion N is sub-Markov. The measure pN is then a Radon measure for every bounded Radon measure p. (b) Let V be a continuous kernel, and let f be a positive l.s.c. function. Denote by If the collection of functions g E fC} such that g < f The upper envelope of If is equal to f, so that from II.T36 we have Vj= sup Vg. gEl,

Since the functions Vg are continuous, the function Vf is l.s.c. Similarly, it can be verified that the function Vf is u.s.c. when f is U.S.C., positive, and has compact support.

Ttt

Let V be a positive linear mapping of fC%(E) into fC(E). There exists a unique continuous dispersion N on E such that THEOREM

Nf = Vf for every function f

E

fC%(E).

Proof The linear functionalf ~ Vfx on fC% is positive. It is hence a Radon measure N x' which extends canonically to the Borel a-field and then to the a-field flA u • Let, on the other hand, A be a positive Radon measure with compact support, and p the Radon measure defined by (11.1) (p,/> = (A, Vf)

Let .YE be the collection of bounded Borel functions g with the following properties: For every relatively compact open set A (a) The function x ~ Nx(gIA ) is Borel, and A-integrable. (b) The function gIA is p-integrable, and (p,gIA > = fEdA(X) Nx(gIA )·

The set .YE is evidently a vector space, closed under uniform convergence, and the limit of an increasing sequence of uniformly bounded positive elements of .YE still belongs to .YE. Let us prove that the set fC of all bounded, positive, l.s.c. functions is contained in .YE. If g belongs to fC, the function gIA is the upper envelope of a family, filtering to the right, of functions i: E fC}, and the function x ~ NxCgIA)' the upper envelope of the continuous functions Vi:, is hence l.s.c. and consequently Borel. On the other hand, by using again II.T36, we have

(p,g(,) =

s~p (p,!.> = s~p (Je, V!.> =

t

dJe( x)N.(g(,J

Properties (a) and (b) are thus verified. Since fC is closed under multiplication, it follows from I.T20 that .YE contains all bounded functions measurable with respect to the a-field generated by the l.s.c. functions, i.e., the Borel a-field.

177

Kernels. Dispersions

IX, TI2

Suppose now thatfis a bounded, universally measurable function with compact support. Let f1 and f2 be two bounded Borel functions with compact support such that /1
and = ' It follows from properties (a) and (b) that the Borel functions X.JVV'+ N~ (It), X.JVV'+ NifJ

have the same (finite) integral with respect to A. Since they bound between them the function x ~ N~(f), the latter is A-measurable and we have (fl,f) =

L

dJ.(x)N,(1)·

(11.2)

The following facts are thus established: Let f be a bounded, positive, Borel (respectively, universally measurable) function with compact support. The function x ~ Nif) is then Borel-measurable (respectively, A-measurable for every Radon measure A with compact support). This result extends immediately to a Borel (respectively, universally measurable) function witb compact support, but not bounded, by means of passage to a monotone limit. Since the space E is a-compact, we can next remove the restriction on the supports of the measure A and the function f Let A be a universally measurable set. Put N(x,A) =

N~(A).

It follows immediately from the above that the mapping (x,A).JVV'+ N(x,A) is a kernel on (E,81u(E)), and the relation N(x,f) = N~(f)

is established, by means of passage to a monotone limit, for every positive, universally measurable function f Keeping the previous meaning of A and p" relation (11.2) says that

AN = ft. Since the measure ft is a Radon measure, and since A is an arbitrary Radon measure with compact support, the kernel N is a dispersion. It only remains to verify the uniqueness of the kernel N, implied by the following observation: Let N and M be two diffusions such that Nf = Mf for every function f E re.;r: The kernels N and M are then identical. The measures E~ and E~M are, indeed, for every x E E, Radon measures that coincide on re.;r; they are thus equal and hence, for every positive universally measurable function f, Mf~ = <E~M,f> = <E~,f> = Nr·

The proof of the following theorem (which we shall in large part omit) is very similar to that of Theorem 11. It is, in fact, a little simpler, since we suppose that the space Eis LCC. T12 THEOREM Let E be an LCC space, and let ~loc be the vector space of locally bounded universally measurable functions on E. (a) Let N be a dispersion on E. Then the function f .JVV'+ Nf maps re.;r into ~loc' (b) Conversely, let V be a positive linear mapping of re.;r into ~loc; there then exists a dispersion N on E such that Nf = Vffor every function f Ere.;r, and such a kernel is unique.

IX, 13

178

Kernels and Resolvents

Proof We shall give a proof by contradiction for assertion (a). Suppose that there exists a function f E = + 00, or equivalently that = + 00. The measure itN is thus not a Radon measure, and this contradicts the hypothesis according to which N is a dispersion. In order to establish assertion (b), one denotes by N re the Radon measure defined by

and shows that the mapping X.JVV'-)- Nif) is universally measurable whenfis first a bounded Borel function with compact support, then an arbitrary positive Borel function, and finally a universally measurable function, as in the proof of the preceding theorem. 13

Examples of kernels

(a) Let (E,C) be a measurable space. Set I(x,A) = I..ix)

(x

E

E, A

E

C).

We obtain a kernel I, which satisfies the relation If = f for every measurable function f It is called the identity kernel. (b) Let n(x,y) be a positive function on the set E x E, measurable with respect to the a-field C x C, and let A. be a fixed positive measure on E. A kernel on (E,C) is obtained by setting N(x,A) = n(x,y) dJ.(y) (x E E,A E tf).

L

It is evident that N(x,j)

= SE n(x,y)f(y) dJ.(y).

Such a function n(x,y) is called a kernel function. It should be well noted that it is necessary to choose a measure A. in order to associate a kernel with a kernel function. Suppose for example that E is the space R3 (given the Borel a-field, and the Euclidian metric d). The kernel function r(x,y) = Ijd(x,y)

is the Newtonian kernel function in R3. If A. is chosen to be a Lebesgue measure, the function N(x,j) =

f.

RS

n(x,y)f(y) dA.(y)

is the Newtonian potential of the mass distribution with density f [one also considers the potentials ofpositive measures of the form x

-"I/II'+f.

RS

n(x,y) dlt(y),

but we are not interested in these potentials for now]. More generally, the kernel of order (X in Rn (0 < (X < 2; the case (X = 2 is that of the Newtonian kernel) is the kernel function [d(x,y)]cx-n. (c) Convolution kernels. Let E be a locally compact group, written multiplicatively, and let It be a positive measure on E. Define, for every x E E and every positive universally measurable function f, NI"

= tf(X Y ) dp.(y).

179

The Potential Theory of a Single Kernel

IX, 14, D15

We obtain a continuous kernel on E. Let A be a positive measure on E. We have (AN,!)

= (A,Nf) =

fE dA(X) f!(XY) df'(Y) = (A' f',f),

where the asterisk denotes convolution. We thus have AN = A * fl, which explains the name "convolution kernel" given to kernels of this type (note that we do not have Nf = fl *f).

2.

The Potential Theory

of a

Single Kernel

The following problem is of sufficient generality to contain almost all the classical forms of potential theory: given a family (Ni)iE I of dispersions on an LCC space E, to study the measurable functions f on E that satisfy the inequalities Nif
for every

i El

(or the Radon measures fl that satisfy the inequalities flNi < fl). The very special case of this problem, where the family (Ni) is a one-parameter semigroup of sub-Markov kernels, will occupy us for most of this book. As an introduction to this already very complex case, we quickly treat the case where the family (Ni) is reduced to a single kernel N. We do not use probabilistic methods in this section, which is essentially a compilation of results of Deny [(50), (51)] and Doob. The reader can find the probabilistic interpretations in Doob's paper (61). Excessive functions 14 We begin by studying excessive functions, and this study is more interesting for kernels on an abstract space. We thus denote by (E,C) a measurable space, and by N a kernel on (E,C). We set N° = I (the identity kernel), and denote by Nn the nth power of N under composition, for n > 1. The functions we call excessive (respectively, invariant) are called superharmonic (harmonic) by Deny, and superregular (regular) by Doob.

D15 DEFINITION Let f be a positive measurable function; f is said to be excessive with respect to the kernel N if Nf
One says that f is invariant with respect to the kernel N

iff is everywhere finite and if

Nf=f

The words "with respect to the kernel N" will be omitted in the rest of this section. Example The kernel N is sub-Markov (respectively, Markov) if and only if the constant 1 is excessive (respectively, invariant). Remarks (a) It is evident that the inequalities f> Nf> ... > Nnf> Nn+1J> ...

hold iffis excessive. We denote by N°Ofthe limit of N'1asn is not in general a kernel).

~ 00

(the mappingf.AN+ N°Of

IX, T16, D17, T18

180

Kernels and Resolvents

(b) Definition 14 does not limit the size of exCessive functions. The constant

+ 00, for

example, is an often undesirable excessive function. We are thus led to consider certain classes of excessive functions, for example, the class of excessive functions [such that the function N OO f is everywhere finite, or the class of excessive functions f such that the function Nf is everywhere finite. These last functions are called (following Doob) excessive functions in the strict sense. Note that the set {f = oo} is then negligible for every measure Br£N. (c) Classical potential theory also deals with superharmonic functions, which are not necessarily positive. We could introduce here the "signed excessive functions," which are measurable functions f having the following properties: (ex) for every x

E

-

00


< + 00

> O.

((3)

E and every n

for every

x

E

E; Nn(x,f-)

< + 00

Nf
The first condition implies that the integrals Nn(x,f) are all defined. The notion of a (signed) excessive function in the strict sense is obtained by replacing condition (ex) by the condition N(x, If I) < + 00 for every n > o. Signed invariant functions are assumed to be finite. These definitions are borrowed from Doob. We do not use them in this book. T16 THEOREM Let f and g be two excessive functions, and ex, {3 be two positive constants. The functions exf + {3g and fAg are then excessive. Let (fn)neN be a pointwise convergentsequence ofexcessivefunctions. Thefunctionlimn-+oofn is then excessive.

The proof is immediate. Potentials D17

DEFINITION

The potential kernel associated with a kernel N is

Letf be a positive measurablefunction. The function Gf is called the potential of the function f

We shall often use the obvious identity NG

+ 1= GN + 1= G.

Kernels of the form AG, where G is the potential associated with a kernel N, and A is a positive constant, are called elementary kernels by Deny. T18 THEOREM Let f be a positive measurable function. The function g = Gf is then excessive, and at every point x E E NOOgr£

=0

or

NOOgr£

=

00.

Conversely, every excessive function g that has this property is the potential of a positive measurable function. Proof It is clear that

00

NkGjr£ =

!

Nnjr£. n=k

(18.1)

The Potential Theory of a Single Kernel

181

IX, TI9

It then follows at once (for k = I) that the function Gfis excessive. Let x be a point such that N OO Gfx < 00; there then exists an integer k such that the first member of (I8.!) is finite, and it follows that the series in the second member converges. We thus have 00

lim

2 NnplJ = lim N 7JGjX = 0,

7J-+ 00

7J

7J-+ 00

or NOOGfX = 0. Conversely, let g be an excessive function such that the function NOOg takes only the values and 00. Let f be the function defined by

°

f(x)

=(

g(X) - Ngx

at every point x where g(x) is finite.

00

if g(x) =

00.

We show that Gf = g. It is clear that equality holds on the set {g = oo}, since then Gf ~ f = 00. Let x be a point where g is finite. The function g is then integrable for every measure Nk(x,dy) and the same is true off, which is dominated by g. We can thus write from which we deduce that k

2 NnplJ =

g(x) - Nk+1ga:.

n=O

Finally, the relation NOOg X< g(x)

< 00 implies that NOOgx = 0, so that 00

2 Nnfa: =

g(x).

n=O

Thus we have Gf = g.

(Riesz decomposition) Let g be an excessive function such that the function NOOg is finite; g can then be written uniquely as the sum of an invariant function and a potential. T19

COROLLARY

Proof Let x be a point of E, and let k be an integer such that Nkgx < 00. The function Nk-lg is integrable for the measure ea;N and dominates the functions N7Jg (p > k). It thus follows from Lebesgue's theorem that (ea:N,N OO g)

= Hm (ea:N ,N7Jg) = Hm N 7J+1ga: = 7J-+ 00

N OO ga:.

7J-+ 00

The function NOOg is hence invariant. Let f be the excessive function g - NOOg. It is clear that NOOf = 0, and the preceding theorem implies that f is the potential of the function f-Nf. Suppose that we have a decomposition of the form

g=

f' + h,

where f' is excessive and h is invariant. Then N OOg = NOOf'

+ h,

and thus we obtain h ~ NOOg, with equality if and only if NOOf' = 0, i.e., iff' is a potential. This implies the uniqueness of the decomposition of g into an invariant function and a potential, and shows that NOOg is the largest invariant minorant of g.

IX, T2Q-T22

T20 N°Og

Kernels and Resolvents

182

Let g be a finite potential (or more generally a potential such that everywhere). Every excessive function dominated by g is then a potential.

COROLLARY

< 00

T21 THEOREM Suppose that the kernel G is proper (No. 2). Every excessive function f is then the limit of an increasing sequence offinite potentials. Proof Since E is the union of a sequence of sets with finite potentials, the function Gl is the limit of an increasing sequence of finite potentials gn. Let fn = ngn: the potentials fn converge everywhere to + 00. It then suffices to note that the functions fn A fare potentials, which increase to f

The reduite of an excessive function on a set We begin by defining notation. Let A be a measurable set and let A' be its complement. We denote byJA (respectively,JA ,) the kernel defined byJAf = fIA (respectively,JA,f= fIA,) for every measurable function f; by N A (respectively, NA') the kernel NJA (respectively, NJA,); by GA (respectively, GA,) the potential kernel associated with the kernel N A (respectively, N A'). T22 THEOREM Let f be an excessive function. The collection of excessive functions that dominate f on A has a smallest element, equal to

(22.1) This function is called the reduite off on A. Proof Denote by HA the kernel J A HA! = g. The inequality

+ JA,GA,NA .

It is evident that HA = HAJA . Set

k

JAf + 2,JA,N~.4:NAf<1 2l=O

is clear for k = 0; we show by induction that it is true for every k. Apply the kernel N to both sides, obtaining k+l

NAf + 2, N~:NAf 2l=1

< Nf·

Apply now JA', then add JA! to both sides. There comes k+l

JAf + 2JA,N~,NAf< JAf + JA,Nf
The induction is thus possible. This inequality gives the relation g = HAf < f for k -+ 00. It is clear that g = f at every point of A. Let us prove thatg is excessive. We compute Ng, using the identity 1+ NA,GA , = GA,. We obtain Ng = NHA != N A !+ NA,GA,NA != GA,NA! We thus have Ng = g at every point of A'. On the other hand, the relations Ng < N! JAI, and therefore HAh > HA! = g. Now h majorizes HAh; we thus have h > g, and g actually is the smallest excessive function that majorizes f on A.

The Potential Theory of a Single Kernel

183

IX, T24

23 Remarks (a) The potential of the function g - Ng is at most equal to g, and it is equal to g if N°Og = O. This happens at least in the following two cases: (1) if the functionfis a finite potential, since then N°Og < Nj = 0; (2) if the potential of the function JAf is finite, since GJAf > f on A, so that GJAf > g everywhere, and consequently N°Og < N°O(GJAi) = O. Suppose in particular that the kernel N is sub-Markov, and that the potential G(IA) is finite. The reduite of the function 1 on A is then a potential, which is called the equilibrium potential of A. (b) We are going to indicate a characterization of potentials by means of the reduite, analogous to the characterization most commonly used in classical potential theory. We suppose that the kernel G is proper, and we retain the notation of the preceding paragraphs. Denote by g an excessive function such that the function Ng is finite. We can now show that g is a potential if and only iflim n __ oo HAng = 0 for every decreasing sequence (An)neN of measurable sets which has empty intersection. Suppose in fact that g is a potential, and put HAtlg = h n • The functions h n decrease as n increases, and we have seen that Nh n = h n on (An. Let h = lim n __ oo h n. We have Nh = h from Lebesgue's theorem, and this implies the equality h = 0 from T19. Conversely, suppose that g is not a potential. There then exists a nonzero invariant function h, dominated by g. We are going to construct a decreasing sequence of sets An An = 0. such that HAnh = h for every n, and Consider first a set A' such that the function GA,h is finite, and put A = (A'. We have NAh + NA,h = h, and consequently

nn

HAh = JAh

+ JA,GA,NAh =

= JAh + JA,h

JAh

+ JA,GA,(h -

NA,h)

= h.

Since the kernel G is proper. we can choose an increasing sequence of measurable sets B n such that G(IB n ) < 00 for every nand Un B n = E. Then let A~ = B n

n

{h

< n},

and

An = (A~.

We have for every n GA~h < nG(/B) < 00. It follows from the preceding results that HAnh = h for every n, and consequently lim n __ oo HAng > h #: 0, while An = 0.

nn

~

Let h be a positive measurablefunction, equal to zero on the complement of A, and let f be its potential Gh. Every excessive function that dominates f on A dominates it everywhere. T24

THEOREM

Proof Let u be an excessive function that dominates f on A, and let v be the excessive function u A f We denote by j the positive function equal to v - Nv on the set {v < oo}, and to + 00 on the set {v = oo}. The potential of j is equal to + 00 on the set {v = oo}, and to v - N°Ov on {v < oo}; it is thus everywhere less than v (T18). Apply the kernel N to both sides of the inequality v < Gh, yielding Nv

< NGh,

from which, by adding h to both sides, we obtain h

+ Nv < h + NGh =

Gh.

We have Ghx = vex) on A and consequently, at every point x of A hex) ~ vex) - Nv X = j(x).

n {j <

oo},

IX, T25-T27, D28

184

Kernels and Resolvents

This inequality evidently holds also on {j = oo}. It thus holds on all of the set {h and hence everywhere. We therefore have also so that v = f, and consequently f

f= Gh < u.

> O} c

A,

< Gj < v,

T25 COROLLARY (Domination principle) Let g and h be two positive measurable functions, and let A be the set {h > O}. The relation

Ggz implies the inequality Gg

> Gh

Z

for every

x

E

A

> Gh.

The function 1 is excessive if the kernel N is sub-Markov, and the same holds for every function of the form a + Vg, where g is positive and measurable and where a is a positive constant. We thus have the following result. T26 COROLLARY (Complete maximum principle) Suppose that the kernel N is sub-Markov. Let g and h be two positive measurable functions, a a positive constant. The relation a + Ggz > Ghz for every x such that h(x) > 0

implies the inequality a

+ Gg > Gh.

Remark These two "principles" are also satisfied for kernels proportional to G, Le., for the elementary kernels of Deny (No. 17). It is interesting to note that the excessive functions can be characterized without explicit mention of the kernel N. T27 THEOREM Suppose that the kernel G is proper. A positive measurable function f is excessive if and only if the following property is satisfied: For every measurable function h (not necessarily positive) with a potential Gh that is welldefined andfinite, the relation

f(x)

> Gh

f(x)

> Gh

implies

Z

for et'ery x such that

Z

for every

x

E

h(x)

>0

E.

(27.1)

We postpone until later (Nos. 70 and 72) the proof of this theorem. Excessive measures The theory of excessive measures is, in general, easier than that of excessive functions, and we often put the emphasis on this latter theory, here and in all that follows. We suppose now that E is a locally compact, a-compact space and that N is a diffusion-kernel on E. All of the measures we consider will be defined on the a-field !!liE), and will be positive. The results below are borrowed from article [(52)] of Deny, where a more general notion of "kernel" is used, which is more satisfactory for the theory of excessive measures. D28 DEFINITION A Radon measure ft on E is said to be excessive (respectively, invariant) with respect to the kernel N if ftN < N (respectively, ftN = N). The measure ftN is then finite on compact sets, and is hence a Radon measure (No. 10). The following theorem corresponds to Theorem 16. Notice the disappearance of the countability restrictions.

185

The Potential Theory of a Single Kernel

IX, T29, 30, T31-T33

T29 THEOREM (a) Let A and fl be two excessive measures, cx and {J be two positive numbers. The measures CXA + {Jfl, A A fl are then excessive. (b) Let fl be a Radon measure, equal to the weak limit of a family (fli)iel of excessive measures, which is filtering either to the right or to the left. The measure fl then is excessive. Proof Statement (a) is obvious. To establish (b), we need only prove that (fl,Nf) < (fl,f) for every function fE ~}. Now we have (fli,Nf) < (fli ,f), and (fl,f) = limi (fli,f). The point to establish thus is the relation: (fl,Nf)

< lim (fli,Nf).

(29.1)

i

This is obvious if the family is filtering to the left. Assume it is filtering to the right, and denote by hi a density of fli with respect to fl. Let U be any relatively compact open set: The functions h/u increase with i, their integrals remain bounded, and they thus converge in the space Ll(fl). The relation limi fli(g) = fl(g) holds for any function g E~} that has its support in U; the Ll limit of the functions h/u thus is equal to I u , and we get fl(f) = sup fli(f) i

for every universally measurable function f that is positive, bounded, and equal to 0 on the complement of U. This relation now extends, by an increasing passage to the limit, to all universally measurable positive functions, and (29.1) follows. 30 The potential kernel G is not necessarily a diffusion. We shall say that a Radon measure fl belongs to the domain of G if the measure flG is finite on compact sets (it is then a Radon measure). If fl is excessive, we set flNa) = lim n flNn. T31 THEOREM (Riesz decomposition) Let fl be an excessive measure. Then fl can be written uniquely as the sum of an invariant measure and of a potential. To be precise, the measure fl - flN belongs to the domain of G, the measure flNa) = lim flN n is invariant, and n

The proof of this theorem is identical to that of Theorem 19. It can also be verified, as in No. 20, that every excessive measure dominated by a potential flG (where fl belongs to the domain of G) is a potential. The following theorem is obvious. We mention itonly for the sake of its name. T32 THEOREM ("Principle of the uniqueness of masses") Let A and fl be two n:easures belonging to the domain of G. The relation AG = flG then implies A = fl. Proof We have in fact A + AGN = AG = flG = fl + flGN, and the measures AGN and flGN are two equal Radon measures. The following theorem corresponds to Theorem 22 and uses the same notation. We will not give a proof for it.

T33 THEOREM Let fl be an excessive measure. The collection of excessive measures that dominate fl on A has a smallest element fl', equal to

IX, T34, 35

Kernels and Resolvents

186

This measure could be called the "reduite" of p, on A, but in general it isn't. It can be verified, as in No. 22, that the measures p,' and p,'N are equal on A'. Suppose, in particular"that p, is a potential AG. The measure p,', since it is dominated by p" is the potential of a well-determined measure A', which is called the balayee* of A on A. We have A' = p,' - p,' N, so that A' is carried by A. The statement analogous to Theorem 24 is true for excessive measures (the proof carries over without change). Let then A" be a second measure carried by A, whose potential coincides with AG on A. The potentials A'G and A"G have the same restriction to A. They are thus equal, and hence A' = A". We therefore have the following theorem. T34 THEOREM (Principle of balayage) Let A be a measure that belongs to the domain of G, and let A be a universally measurable set. There exists a unique measure A' with the following properties: (1) A' is carried by A. (2) A'G < AG, and these two potentials have the same restriction to A.

Appendix: Connections with Martingale Theory 35 The analogies between potential theory and martingale theory can perhaps be illumi-

nated by the following remarks. Let (.o,§',P) be a complete probability space, and let (§'n)nEN be an increasing family of sub-a-fields of §'. Denote by E the set N x .0, and by C the a-field on E consisting of the subsets of the form

U {n}

X

An'

nEN

where each set An is §'n-measurable. The C-measurable mappings from E into R are then of the form (n,w) ~ Xn(w),

where each partial mapping X n is §'n-measurable. The definition of stochastic processes adapted to the family (§'n) is thus recovered. Introduce on the set of these processes the equivalence relation defined by f"'Ooo./

(Xn)nEN

f"'Ooo./

(Yn)nEN if and only if X n = Y n a.s. for every n EN.

Let X = (Xn)nEN be a process with positive values adapted to the family (§'J. For each n, denote by Yn a version of the generalized conditional expectation of X n+1 with respect to §'n' and define The mapping N is not well defined, due to the indeterminacy in the choice of conditional expectations, but by passing to the quotient by the equivalence relation one can obtain a mapping that formally has all the properties of a sub-Markov kernel-in particular, the behavior under passage to a monotone limit. The "excessive (respectively, invariant) functions" with respect to the "kernel" N are then the equivalence classes of generalized supermartingales (respectively, martingales), and all of f"'Ooo./,

• "Swept out measure."

187

Semigroups and Resolvents

IX, D36, D37, 38

the elementary theory we have developed carries over without difficulty. Naturally, no truly important theorem on supermartingales is obtained by this method. In particular, the fundamental theorems on the behavior of paths have no parallel in potential theory.

3. Semi8roups and Resolvents The results of this section are not deep, but they are very useful technical tools. They come mostly from Hunt's papers (78). Semigroups of kernels D36 DEFINITION Let (E,tff) be a measurable space. A family (Nt)teR+ [respectively (Nt)t> 0] ofkernels on (E,tff) is said to be a semigroup ofkernels (respectively, a semigroup in the broad sense) if the relation

holds for every pair (s,t) of numbers >0 (respectively, >0). The semigroup is said to be sub~Markov (Markov) if all of the kernels N t are sub~Markov (Markov). A semigroup ofdispersions on a locally compact space constitutes a particular case of this definition. A semigroup in the broad sense can always be transformed into a true semigroup. It suffices to set No = I (the identity kernel). We prefer, however, to maintain the distinction between these two types of semigroups. Let (Nt ) be a semigroup of kernels. It is possible to produce from it new semigroups of kernels (N:) (where p denotes a number >0) by setting

Nf = e-ptNt · These semigroups are sometimes better behaved than (Nt ) itself. Supermedian and excessive functions D37 DEFINITION Let (Nt) be a semigroup in the broad sense on (E,tff). A positive function f defined on E is said to be p~supermedian (p > 0) with respect to the semigroup (Nt) if f is

tff~measurable and

e-PtNtf < f

for every

t

> O.

(37.1)

The function f is said to be p~excessive if, moreover, Hm e-PtNtf = f.

(37.2)

t--+O

The function f is said to be p-invariant

iff

e-PtNtf= f

is everywhere finite and if for every

t

> O.

38 Remarks (a) Functions that are O-supermedian (O-excessive) are called simply supermedian (excessive). (b) The functions that are p-supermedian (p-excessive) with respect to the semigroup (Nt ) are identical to the supermedian (excessive) functions with respect to the semigroup (N:).

IX, 39

188

Kernels and Resolvents

(c) A p-supermedian (p-excessive) function is also q-supermedian (q-excessive) for every

q > p. One can thus say that "the larger p is, the more p-supermedian (p-excessive) functions

there are." (d) Relation (37.1) implies that the function t ~ e-ptNt!X is decreasing for every x E E. Moreover, if condition (37.2) is satisfied, this function is right continuous from Lebesgue's theorem. (e) Let! be a p-supermedian (p-excessive) function. The function N t! is then p-supermedian (p-excessive) for every t > 0. (!) Let (!n)nEN be a sequence of p-supermedian functions. It follows immediately from Fatou's lemma that the function! = limninf!n is also supermedian. Suppose that the sequence is increasing, and that the functions!n are p-excessive. We then have I

=

sup In n

=

sup sup e-PtNtln = sup sup e-PtNt!n n

t -PtN! = sup e t,

t

n

t

so that! is p-excessive. (g) Supermedian or excessive measures are defined similarly. The theory of excessive measures cannot be developed satisfactorily under our current hypotheses. We shall see, in turn, that the theory of excessive measures becomes very simple-much simpler than that of excessive functions-when suitable hypotheses are made on the semigroup (Nt ). Resolvents

Let (Nt) be a semigroup in the broad sense. We say that (Nt ) is a measurable semigroup if the function

39

(t,X) ~ Nix,!)

is measurable (with respect to the natural product a-field on ]O,oo[ x E) for every positive C-measurable function! Then, for every number p > 0, define Vp(X,!) =

f.oo e-PtNt(x,A) dt

(A

E

C).

(39.1)

It is clear that the mapping (x,A) ~ Vix,A) is a kernel V p, and that the notation Vix,!) we have used is consistent. The family of kernels (Vp)p>o is called the resolvent of the semigroup (Nt), and the kernel V = Vo is called the potential of the semigroup (Nt). More generally, let p, be a bounded measure on the half-line R+. Define a kernel NI' by

the relation Nil) = f.ooNtfdP,(t).

(39.2)

The kernels N t are of this form (for p, = Ct), just as are the kernels Vp (p, is then the measure with density e- pt on R+, which we denote by e p ). It is easy to verify that N;.Np.

the symbol

=

= Np..;' =

N;.*p.

* denoting convolution. The formula e + (q - p)e * e p = q

q

where q and p are two numbers such that q Vq

Np.N;.,

(39.3)

e p,

> P > 0, then gives us the fundamental formula

+ (q -

p)Vq V2J = V p,

(39.4)

IX, D40, 41, D42, D43, 44

Semigroups and Resolvents

189

which is known as the resolvent equation. Now we are going to forget measurable semigroups for a moment and study the families of kernels which satisfy (39.4) for their own sake. D40 DEFINITION (E,C), such that

A resolvent on a measurable space (E,C) is a family ( Vp) p> 0 of kernels on

(40.1)

and

for every pair of numbers p, q such that q > P > O. A resolvent (Vp) is said to be proper (respectively, sub-Markov, Markov) if the kernels V p are all proper (respectively, if the kernels p V p are all sub-Markov, Markov).

We shall mainly be interested in sub-Markov resolvents in the rest of this book. The reader can simplify his task by supposing, for the rest of this section, that all the resolvents considered are sub-Markov. 41 Let f be a positive measurable function. According to formula (40.1) the function p ~ Vpfis decreasing. We can thus put Vof

=

Vf = sup Vpf = lim Vpf. p

p-+O

Let (fn) be a sequence of positive measurable functions that increases to f We have Vf = sup Vpf = sup sup VIn p p n

=

sup sup VIn n

=

P

sup Vfn n

so that V is a kernel. It is easily verified that VVp = VpV; V = V p + pVVp (p

> 0).

D42 DEFINITION Let f be a positive measurable function. The function Vrf (r the r-potential off The function Vf = Vof is called the potential off D43

DEFINITION

> 0) is called

We say that the resolvent (Vp) is closed if the kernel V is proper.

Suppose that E is a locally compact, a-compact space. We then say, in a slightly more precise sense, that the resolvent (Vp) is closed if all of the kernels Vip > 0) are dispersion kernels. 44 Let (Vp ) be a resolvent and let r be a positive number. The family of kernels (p

> 0)

is a new resolvent. Suppose that the resolvent (Vp ) is proper: The resolvent (V;) is then closed for every r > O. Indeed, V~f = lim p _ o V p+rf < Vrf for every positive measurable functionf(we shall see later that the inequality is, in fact, an equality). This property is the reason for the interest in the resolvents (V;). Suppose that the resolvent (Vp ) is associated with a measurable semigroup (Nt ) by formula (39.1). The resolvent (V;) is then associated with the semigroup (ND = (e-rtNt ).

Supermedian and excessive functions We now define supermedian and excessive functions with respect to a resolvent. The connection between this definition and Definition 37 is examined later (No. 65).

IX, D45, T46

Kernels and Resolvents

190

A positive measurable function f defined on E is said to be r-supermedian (r > 0) with respect to the resolvent (V1J if D45

DEFINITION

pVP+rf
for every

> O.

(45.1)

if in addition

The function f is said to be r-excessive

Hm p V1J+rf =

The function f is said to be r-invariant

p

iff

f.

(45.2)

is everywhere finite and if

PV 1J+rf = f

for every

p

> O.

The words "with respect to the resolvent (V1J )" will usually be omitted. Functions that are O-excessive (O-supermedian, O-invariant) will be called simply excessive (supermedian, invariant). The r-supermedian (r-excessive) functions with respect to the resolvent (V1J) are identical with the supermedian (excessive) functions with respect to the resolvent (V;). First properties

We suppose henceforth that the resolvent (V1J) is proper.

(a) Let f be a positive measurable function, and x a point in E. The function p ~ V1Jfx is then decreasing, right continuous, and continuous on every open interval where it is finite. (b) Let f be an r-supermedian function. The function p ~ pVr+1JfIX is then increasing and continuous for every x E E. T46

THEOREM

Proof The fact that the function p ~ V1JfIX is decreasing follows immediately from the resolvent equation, and has already been used. Let Po, p, e be three numbers such that o < Po < p, 0 < e < p - Po, and V1Jo f lX < 00. We then have V1JfIX = V1J+e f IX + eV1J V1J+ef IX and V1J _ef IX = V1JfIX + eV1J V1J_ef IX, where the quantities eV1JV'lJ-efIX, eV1JV1J+efIX are dominated by(p - po)V1J V 1Jo f'x = V 1Jo f lX V1JfIX < 00. It then follows that the function p ~ V1Jpx is continuous on the interval ]Po,oo[. Consider next an arbitrary number p > 0; since the kernel V1J is proper, f is the limit of an increasing sequence of positive functions fn such that the functions V1Jfn are finite. The function q ~ Vqr is thus equal, on ]p,oo[, to the upper envelope of the continuous functions q ~ Vqfn IX. It is therefore decreasing and lower semicontinuous (l.s.c.), and consequently right continuous. In summary, the function p ~ V1JfIX is decreasing, and has at most one point of discontinuity Po, to the right of which it is finite and continuous, and to the left of which it equals + 00. Suppose next that fis r-supermedian, and let p and q be two numbers such that 0 < P < q. We then have and consequently, applying the kernel Vr+q ,

< (q - p)Vr+qf, p)Vr+qVr+1J f < pVr+qf + (q -

p(q - p)Vr+qVr+1J! and

pVr+qf + p(q -

p)Vr+qf= qVr+qf

Semigroups and Resolvents

191

IX, T47-T49

Since the left side is equal to PVr+pf from the resolvent equation, we see that the function p JW+ PVr+ p! is increasing. Now this function can have, from (a), only a single point of discontinuity Po, to the left of which it equals + 00, and to the right of which it is finite; this cannot happen for an increasing function, and it follows that the function p .A./II'+ Vr + p is continuous. T47 THEOREM Let f be a positive measurable function. (a) The function f is r-supermedian if and only if f is s-supermedian for every s > r. (b) Suppose that f is r-supermedian and that there exists a number s > 0 such that f is s-excessive. The function f is then r-excessive.

Proof The relation p Vr+p! < f implies p Vs+p! < f for every s > r according to T46(a). Conversely, the relation p V s+p! < f for every s > r implies p Vr+pf < f from the right continuity of the function s JW+ Vs+pf. Let rand s be two positive numbers. The equalities lim p ~+pf = I p-+ 00 can be written, respectively,

and

lim PYs+pf = I p-+ 00

lim (p - s)Vpf = f, p-+ 00 and consequently are equivalent, since the ratio (p - r)/(p - s) tends to 1 as p --+ and

lim (p - r) Vpf = I

p -+ 00

00.

T48 THEOREM (a) Let f and g be two r-supermedian (respectively, r-excessive) functions, and let lX, fJ be two positive constants. The function lXf + fJg is then r-supermedian (respectively, r-excessive). The function fAg is r-supermedian. (b) Let (fn)neN be a sequence of r-supermedian functions. The function f = lill1 inffn is n-+oo then r-supermedian. (c) Let (fn)ne N be an increasing sequence of r-excessive functions. The function f = limnfn is then r-excessive.

Proof We have

P Vr+if A g)

< (p Vr+pf) A (p Vr+pg) < fAg.

This function is hence r-supermedian. In order to establish (b), we use Fatou's lemma, which implies that p

~+p (limninfIn) < limninf p ~H!n < limn infIn·

Under the hypothesis of (c), we have, from T36(b), lim p ~+pf = sup p ~+pf = sup sup p ~+pfn p-+ 00 p P n

= so that f is r-excessive.

sup sup p ~+pfn n

=

P

sup In n

= f,

T49 THEOREM Let q and r be two positive numbers, and f an r-supermedian function. The function Vqf is also r-supermedian.

Proof For every p

> 0, we have p Vr+pVq! = Vq(p Vr+p!)

< Vqf·

IX, T5Q-T53

Kernels and Resolvents

192

Resolvent identities

T50

LEMMA

Let f be a positive measurable function. The function Vrf is r-supermedian.

(We shall see later that this function, under very general conditions, is actually r-excessive.) Proof This is an immediate consequence of the resolvent equation, pVr+pVr!

+

Vr+p!= Vrf

(50.1)

T51 LEMMA Let f be a positive measurable function with a finite r-potential Vrf The functions of the form (51.1) are then finite. Proof Let e be a strictly positive number such that r function (51.1) is dominated by (-v,.+e)k-l-v,.!=

+ e < Pb ... , r + e < Pk.

~l (e-v,.+el-l-v,.f <

e

The

Ll -v,.f,

e

from T49 and T50. T52 THEOREM Let f be a positive measurable function such that all the functions Vrf (r > 0) are finite. The function r JVV'+ V r! is then infinitely differentiable on the interval ]0,00[, and we have the relations (52.1)

and

n

d r -v,.! = n! ( -1 )n+l(-v,.t(I - r -v,.)f. dr n

(52.2)

Proof From the resolvent equation and T46(a) we have

lim Vqf - -v,.! = - lim Vq-v,.f = q-+r q - r q-+r

-(-v,.)~.

The two formulas are then established by induction: We leave the details of the proof to the reader. The following lemmas will permit us to establish another important identity. T53 LEMMA Let r be a number >0, and h a finite, positive, measurable function such that all of the functions Vp+,n are finite (p > 0). Suppose that there exists a p > 0 such that p V p+,.h = h;

(53.1)

the function h is then r-invariant. Proof The equality

Vq+,.h = V p+,n

+ (p -

q) Vq+rVp+,.h

holds for every q < P (from the resolvent equation), and also for every q the functions that enter are finite. Replacing V p+,.h by hip we obtain 1 Vq+rh = - h

p

+

and consequently qVq+,.h = h for every q > O.

p-q Vq+rh, p

> p, since all of

193

Semigroups and Resolvents

IX, T54, T55, 56

T54 LEMMA Let f be a positive measurable function with finite r-potential (r function is then the only r-invariant function dominated by VrI

> 0). The zero

Proof We have Vrf = lim£--+o Vr+£f (T46). Since the function Vrf is finite, we can write

lim eVr+£Vrf = lim (~f £-+0

~+£f) =

o.

£-+0

Let then h be an r-invariant function dominated by Vrf; we have h = e~+£h = lim e~+£h £-+0

< lim e~+£~f= o. £-+0

The following theorem is stated only for the kernel V and for a closed resolvent. It extends to the kernels Vr [consider the resolvents (V;) of No. 44] and then, as r ~ 0, to the kernel V even when the resolvent (Vp ) is not closed. T55 THEOREM * Suppose that the resolvent (Vp) is dosed, and let p be a number have the identity:

>0.

We

00

pV = !(pVp)n.

(55.1)

n=l

Proof Both sides being kernels, and the left-hand side being a proper kernel, it suffices to verify the equality 00

pVf= !(pVp)nf n=l

for every positive measurable function f with finite potential VI It follows immediately from the resolvent equation that, for all n

> 0, pVf= pVp[

+ (PVp)'1 + ... + (pVp)n-y + (pVp)npVI

We thus need only show that lim (p Vp)nVf =

o.

n-+oo

But these functions decrease when n increases, and are dominated by Vf(see T49 and T50). The limit h = lim(pVp)nVf n-+oo

thus exists and clearly satisfies the relation p Vph = h (Lebesgue's theorem). It is hence invariant from T53, and zero from T54.

A supplementary hypothesis 56 We suppose from now on that the resolvent (Vp ) satisfies the following hypothesis, which will be studied in more detail in No. 68.

There exists a number s gn such that

> 0, and an increasing sequence offinite s-supermedian functions limg n = n-+oo

• This identity has been used by Deny (see No. 68).

+00.

(56.1)

IX, T57, 058, 059

Kernels and Resolvents

194

We continue to suppose also that the resolvent is proper. Hypothesis (56.1) is satisfied in two very important cases: when the resolvent is sub-Markov (take s = 0 and gn = n); when the kernels V2) are strictly positive (i.e., when all of the measures Ea;V2) are different from 0). Indeed, choose any s > 0; since the kernel Vs is proper, the function 1 is equal to the limit of an increasing sequence (h n ) of positive functions with finite s-potentials. Put gn = nV ~n; the functions gn are s-supermedian (T50) and since the function V sl is everywhere strictly positive we have Hm gn = lim n~l = + 00. n

n

Here is an important consequence of this hypothesis.

T57 THEOREM r-excessive.

Let f be a positive measurable function; the function Vrf (r

~

0) is then

Proof We have the relation pV2J+r Vrf + V2J+rf = Vrf. Suppose first that/is bounded by one of the functions gn of No. 56. We then have, for large enoughp,

V2J+rf < V2J+rgn

=

1

p+r-s

(p

+

r - s)V2J+rgn

1 < p+r-s gm

and thus limj)-+oo V2J+rf = 0, so that the function Vrfis r-excessive. To treat the case wheref is arbitrary it then suffices to note that the functions Vr(f A gn) are r-excessive and to apply T48.

Regularization of supermedian functions

Letfbe an r-supermedian function (r we can put

> 0). Since the function p.A.J"V'+ pV2J+rfis increasing,

J=

lim p V2J+rf· 2)-+ 00

This function is r-supermedian (T49 and T48) and dominated by f. Also for every s > 0, so that depends only on f, and not on r.

J

D58

DEFINITION

D59

DEFINITION

ViIA.)

=

J=

limj)-+oo p V2J+sf

The function 1 is called the regularization of the r-supermedian function f.

Let A be a measurable set; A is said to be a set of potential zero 0 for every p > o.

if

It suffices that V2)(IA.) = 0 for a single value of p. Indeed this implies that Vq(IA.) = 0 for every q > P from T46(a), and for q < P we have

Vq(IA.) = ViIA.)

+ (q -

p)VqViIA.) = O.

We employ in what follows the expression "almost everywhere," when it will not lead to ambiguity, as synonymous with the expression "except for the points of a set of potential zero." Let f and g be two positive measurable functions. equal almost everywhere. The potentials V2)fand V2)g are then equal for every p > O. In particular, iffandg are r-excessive we have f = limj)-+oo p V2J+rf = limj)-+oo p Vr+2)g = g.

IX, T60, 061, T62

Semigroups and Resolvents

195

T60 THEOREM Let f be an r-supermedian function; the function I is then r-excessive, equal to f almost everywhere, and is the largest r-excessive function dominated by f.

Proof The function pVr+pfis r-supermedian (T49) and (r + p)-excessive (T57), and hence r-excessive (T47). It follows from T48(c) that the functionlis r-excessive. Let g be an r-supermedian function dominated almost everywhere by f. We have p Vr+pg < P Vr+pf for every p, and consequently also g < Thus I is, in particular, the largest rexcessive function dominated by f. It remains to show that we actually havel = falmost everywhere. To see this, consider a number t greater than r and the number s of No. 56. The functions gn of No. 56 and the function f are also t-supermedian, and I = limp -+ oo p Vp+tf. The functions fn = fA gn are t-supermedian and finite, so thatf = sUPnfn and

J.

I

= sup pVp+tf = p

sup sup pVpHfn p

n

=

sup sup pVp+tfn = sup In. n

n

p

We thus need only know that In = fn almost everywhere or, since the functions fn are finite and dominate lm that Vtln = Vtfn. But we have, from the relation pVt+pfn
p-- 00

Excessive functions and potentials D61 DEFINITION Let f be an r-excessive function (r > 0). Then f is said to be purely r-excessive, or to be an r-potential, if there exists no r-invariant function dominated by f and distinct from o.

The expression ''! is an r-potential" by no means implies that there exists a positive function g such that f = Vrg; it is borrowed from classical potential theory in the unit disk, where the superharmonic functions that satisfy this condition are effectively Green potentials (of positive measures). We use rather the expression',! is purely r-excessive" in this chapter. T62

THEOREM

(Riesz decomposition)

Let f be an r-excessive function such that the function h = lim p Vp+rf p--O

is finite. The function h is then r-invariant and the function f - h is purely r-excessive. This decomposition off into an r-invariant function and a purely r-excessive function is unique. Proof Since the function p ~ p VP+r f is increasing, and p ~ VP+rf is decreasing, the function h is finite if and only if all of the functions Vp+rf are finite. Suppose then that h is finite. It follows from Lebesgue's theorem that qYa+r h

= limpqVq+rVp+rl= lim p--o

pq (Vp+rf- Ya+rf) p--o q - P

= h.

Thus h is r-invariant. If h' is an r-invariant function dominated by f we have h' = pVp+,h'

< pVp+rf

for every p

> 0,

and consequently h' < h. Thus h is the largest r-invariant minorant off. It then follows that the function f - h is purely r-excessive (If it admitted a nonzero r-invariant minorant k,

196

Kernels and Resolvents

IX, T63, T64

h + k would be a minorant off, and larger than h.) We leave to the reader the uniqueness of the decomposition, which is easily proved.

Let f be an r-excessive function, such that the functions Vp+rf are finite for every p > O. The function f is purely r-excessive if and only if T63

COROLLARY

lim p VfJ+rf = O.

fJ-+O

This applies in particular to a finite functionf of the form Vrg (T54). The following theorem is particularly useful.

-+

(a) Every r-excessive function (r > 0) is the limit of an increasing sequence offinite r-potentials ofpositive functions. (b) If the resolvent (VfJ) is closed, then property (a) holds for r = 0 also. If the resolvent is sub-Markov, the r-potentials considered can moreover be supposed bounded. T64

THEOREM

Proof Property (a) can be deduced immediately from property (b) by replacing the resolvent (VfJ) by the resolvent (V;) of No. 44. We shall thus establish only (b), supposing that the resolvent is closed. The proof will be divided into several parts. (1) Let f be a purely excessive finite function. Define

(p

> 0);

this function is positive. Since all of the functions VfJf are finite for p > 0 (from the inequality pVfJf 0), and we can write

VqDfJf = p(Vqf - pVqVfJf) = p[(Vqf - (p - q)VqVfJf) - qVqVfJf] = p(VfJf - qVqVfJf)

< pVfJf
It then follows that the function VDfJfis finite and equal to pVfJf - Vilimq--+o qVqf). This

limit is zero sincefis purely excessive. We thus have VDfJf= pVfJf (2) Consider the function VI. If this function is zero at a point x, then all of the measures pVfJ(x,dy) are zero. Every excessive functionfis thus zero at x. In other words, if we put j = lim nVl,

n-+oo

we have f = f A j. Since the kernel V is proper the function I is the limit of an increasing sequence of functions in with finite potentials. Define

jn = nVin· The functions jn are purely excessive (T54), and their upper envelope is equal to j. (3) Letfbe an arbitrary excessive function. Putg n = f Ajn (f Ajn A n in the sub-Markov case), and denote by hn the function gm the regularization of the supermedian function gn' Since hn is dominated by jn it is finite and purely excessive. The sequence (h n) is increasing and converges to f, since its limit is an excessive function which is equal almost everywhere to f The function (p,n) -"t/It+ P VfJh n is increasing in p and n. We thus have

f

=

sup sup p VfJh n = lim nVnh n. n

fJ

n

Semigroups and Resolvents

197

IX, T65, T66, D67

Put n Vnh n = fn. The sequence (fn) increases to f and the functions fn are the potentials of positive functions (bounded in the sub-Markov case), from (1). We can now examine the connection between the excessive functions with respect to a measurable semigroup and the excessive functions with respect to the resolvent of this semigroup. We restrict ourselves to the sub-Markov case. T65 THEOREM Let (Nt ) be a measurable semigroup of sub-Markov kernels, and let (V p) be the resolvent associated with this semigroup (see No. 39). (1) Every r-supermedian function (r > 0) with respect to the semigroup (Nt ) is r-supermedian with respect to the resolvent (V p). (2) The r-excessive functions with respect to the semigroup are identical to those with respect to the resolvent.

Proof We begin by studying the case where r = 0, supposing the resolvent (Vp) to be closed. Letfbe a supermedian function with respect to the semigroup (Nt ). For every x E E we have

pVpf~ =

ioope-PtNtf~dt
0

so that f is supermedian with respect to the resolvent. Suppose next that f is supermedian with respect to (Nt). Since the function t .J\./\i'+ Ntf~ is decreasing, we have limpVpf~ = lim Ntf~. p-+ 00 t-+O In particular, iffis excessive with respect to the semigroup (Nt ) we have f(x) for every x, and f is excessive with respect to the resolvent. Let g be a positive measurable function. We have

(65.1) lim:p--oopVpf~ -

NtVg~ = 100 N8g~ ds, so that the function Vg is excessive with respect to the semigroup. Finally, let f be an excessive function with respect to the resolvent. Since the latter is closed there exist positive functions gn with potentials that increase to f (T46). The functions Vg n are then excessive with respect to (Nt ), and f has the same property from No. 38(f). Now stop supposing that the resolvent (Vp ) is closed, and apply the result obtained above to the closed resolvent (V;) (r > 0) and to the semigroup (e-rtNt). We obtain the statement of the theorem for the case where r is strictly positive. The case where r is zero then follows immediately, since a function is supermedian (excessive) if and only if it is r-supermedian (r-excessive) for every r > O. The importance of (65.1) makes it worth being stated explicitly. T66 COROLLARY Keep the notation of No. 65. Let f be an r-supermedian function with respect to the semigroup (Nt ). We have

J= t-+O lim Ntf· The domination theorem D67 DEFINITION A point x E E is said to be nonpermanent [for the resolvent (Vp)] if the measure e~Vp is zero for every p ~ O. Otherwise, the point x is said to be permanent.

Kernels and Resolvents

IX, T68

198

Suppose that there exists a p > 0 such that cxVp = O. We then have cxVa = 0 for every q > P and, for q < p, cxVa = cxV2J + (q - p)cxV2JVa = 0, so that the point x is nonpermanent. We denote by E2J the set of permanent points of E. We have E2J = {x: Vl x > O}, so that E2J is measurable. The following theorem makes clear the significance of the hypothesis of No. 56. Here is an example of a resolvent that does not satisfy the hyothesis of T68. Take a space E composed of two points a and b and put, for every p > 0,

caV2J = 0

and

CbV2J = Ca'

The compositions V2J Va are identically zero, so that the resolvent equation is satisfied. The point a is nonpermanent. and the set {a} is not of potential zero. The difficult part of Theorem 68 [the implication (c) => (d)] is adapted from Deny (52). ~

T6t" THEOREM Let (V2J) be a proper resolvent. The following statements are equivalent: (a) Property (56.1) is satisfiedfor an s > 0; (b) The set of nonpermanent points is ofpotential zero; (c) Property (56.1) is satisfied for every s > 0, and for s = 0 if the resolvent is closed; (d) Let f be a positive measurable function, and u a supermedian function such that

u(x)

> Vfx

f(x)

> o.

f(x)

> O.

for every x such that

Then we have u > Vf; (e) Let f and g be two positive measurable functions such that VgX We then have Vg

> Vfx

at every point x such that

> Vf ("domination principle").

Proof We begin by establishing the equivalence of (a), (b), and (c). The set Eo of nonpermanent points is the set of points where the supermedian function + 00 differs from its regularization, so that (a) => (b) from T60 [which is a consequence of (a)]. With the supposition that (b) is satisfied, equip the set E2J of permanent points with the a-field induced by tt. Since the set Eo is negligible for every measure cX V2J , we can define a resolvent (W2J) on E2J by putting Wix,f) = V 2J(x,f') for every p > 0, every x E E 2J , and every positive measurable function f defined on E 2J , f' denoting any measurable extension off to E. The kernels W 2J are then strictly positive on E 2J , and this implies the existence for every s > 0 of an increasing sequence of functions h n defined on E2J , which are finite, s-supermedian with respect to the resolvent (W2J ), and which tend to + 00 on E2J (this point was established in No. 56). It then suffices to put

gn(x)

=(

hn(X) for x

E

E2J

n

E

Eo

for x

to obtain s-supermedian functions with respect to (V2J ), which satisfy (56.1). Finally, the implication (c) => (a) is clear. The reasoning of No. 56 shows that the functions gn exist also for s = 0 if the resolvent is closed.

IX, T69, T70

Semigroups and Resolvents

199

The rest of the theorem will be established using the scheme (c) => (d) => (e) => (b). It will suffice to prove the implication (c) => (d) in the case where the resolvent is closed. Indeed, this implication will then be established for each of the resolvents (V;) (r > 0). Since the function u is r-supermedian for every r > 0, the relation

>

u(x)

VflX

> Vrr

for every x such that

f(x)

>0

will then imply u

>

Vrf

for every

> 0,

r

and consequently also u > Vof, which is the desired result. Suppose then that the resolvent is closed, and denote by (gn) an increasing sequence of finite supermedian functions that tend to + 00. Put fn = f A gn' We have u(x)

>

Vfn

for every x'such that

IX

fn(x)

> 0,

and it suffices to show that u > Vfn for every n. We shall use, to this end, the elementary domination principle of the preceding section (T24). Let p be a number >0, and let N be the kernelpVp • The potential kernel associated with N(in the sense of No. 17) is equal to / + PV from the identity in No. 55. Every supermedian fuqction with respect to the resolvent (Vp ) is excessive with respect to N (in the sense of No. 14). The relation gn(x)

+ pu(x) ~fn(x) + pVfn(x)

at every point x such that

fix)

>0

hence implies, from T24, the relation gn

+ pu > fn + pVfn'

Since p is arbitrary and the functions gn and fn are finite, this implies u > Vfm and assertion (d) is established. The implication (d) => (e) is clear. Finally, we establish the implication (e) => (b). The function VC/Eo) is zero at every point of Eo. We thus have

o=

Vox

> V(/Eo)1X

at every point x such that

IEo(x)

> O.

Thus V(IE o) = 0 everywhere from (e), and this implies that Eo is a set of potential zero. The following statement is very useful. T69 COROLLARY Properties (d) and (e) of the preceding statement are satisfied by every sub-Markov resolvent and by every proper resolvent (Vp) with strictly positive kernels.

Let (Vp ) be a closed resolvent that satisfies the equivalent conditions of No. 68. It is sometimes of interest to know how to determine if a function g is supermedian with respect to the resolvent (V p), without having to form the functions pVpg. T70 THEOREM A positive measurable function g is supermedian if and only if the following property holds: For every measurable function h (not necessarily positive) with a well-defined and finite potential Vh, the relation (70.1) g(x) > VhlX for every x such that hex) > 0 implies g(x)

> Vh lX

for every x

E

(70.2)

E.

Proof Suppose that g is supermedian. The relation (70.1) can also be written g(x)

+

V(h-y

>

V(h+)1X

for every x such that

h+(x)

> O.

IX, T71

Kernels and Resolvents

200

We then have, from T68(d), which is equivalent to (70.2). Conversely suppose that g satisfies the property in the statement. Denote by f a positive measurable function dominated by g and with finite potential VI The function h = p(f - p V"f) then admits a well-defined and finite potential, equal to p V"f Now we have on the set

{g - pV"f> O}

and a fortiori on the set {f - p V"f > O} = {h > O}. We thus have g > Vh = P V"f everywhere. Since the kernel V is proper, g is equal to the limit of an increasing sequence of functions fn of the preceding type. We thus have g > PV"g, which shows that g is supermedian. The pseudo-reduite of a function

The notion we define now does not coincide with the classical notion of the reduite of f (which would be the lower e:Q.velope of the excessive functions that dominate f on A). This is why we call it the pseudo-reduite of I It is in fact not certain that the following theorem has any usefulness, and we shall only outline its proof.

T7t

Let A be a measurable set and fa supermedian function with respect to the resolvent (V,,). The collection of supermedian functions that dominate f on A has a smallest element, which we shall call the pseudo-reduite off on A. THEOREM

Proof For every p > 0, put N" = pV", and denote by g" the reduite off on A relative to the kernel N" (No. 22). The reader can easily verify the following facts: (a) Every excessive function with respect to the kernel N" is excessive with respect to every kernel N q for q < p. (b) When p increases, there are thus fewer and fewer excessive functions with respect to N", so that the reduite g" increases. Put g = lim~oo g". (c) The function g is supermedian with respect to the resolvent (V,,). It is equal to f on A, and every supermedian function that dominates f on A dominates g everywhere. It then follows that g is the desired pseudo-reduite ofI Remark Suppose that A is a set of potential zero; the pseudo-reduite of f on A is then clearly equal to fIA • There would be a very different result with the classical reduite.

Connections between Sections 2 and 3

Let N be a kernel, and let

be the potential kernel associated with N. We are going to show that a resolvent (V,,) can be constructed so that G = Vo, and that this resolvent is, in turn, associated with a semigroup of kernels. Put, for every number a in ]0,1],

201

Semigroups and Resolvents

Let b be a number such that 0

< b < a. Then

GaGo = GoGa = I

and consequently

+ (a + b)N + (a 2 + ab + b2)N2 + ... ,

(a - b)GoGa + bGo

Put then, for every p

~

IX, T71

= (a

- b)GaGo + bGo

=

aGa.

0, V2J

=

1

p+1

G1 /(21+1)·

The kernels V2J constitute a resolvent such that Vo = G. Put, on the other hand, for every t ~ 0, 2 2 tN t N Pt = e-t(l + +~ + ...).

1!

It is easily verified that the kernels (Pt) constitute a measurable semigroup with resolvent

(V2J)-this last point follows from the possibility of integrating the exponential series term by term. Suppose now, for simplicity, that the kernel G is proper (it could be supposed only that the kernels Ga are proper for every a > 0). Since the kernel G is strictly positive, the resolvent (V2J) satisfies the hypothesis of No. 56. Let A be a set of potential zero. The relation GIA ~ lA shows that A is empty, and it follows (T60) that the supermedian functions with respect to the resolvent (V2J) are excessive. Let/be an excessive function with respect to N. We have p VJ =

(1 + p +1 1 p+ 1 p

Nf +

(p

1

+

N 2f

1)2

+ ...)

~ P ~ / ( 1 + p: 1 + (p ~ 1)' + .. -) = J, so that f is supermedian (and hence excessive) with respect to (V2J). Conversely, suppose that / is excessive with respect to (V2J). Then / is the limit of an increasing sequence of potentials (T63(b», and hence it is excessive with respect to N. The resolvent equation is such a useful analytic tool that one could occasionally think of using it in the elementary situation of Section 2. Theorem 27, for example, reduces immediately to Theorem 70-whose proof by resolvents is very natural. We shall see later on other examples of the use of the resolvents (V21) associated with a kernel G.

X

CHAPTER

Construction of Resolvents and Semigroups

We now study, following Hunt, this problem: Given a kernel V, which satisfies the complete maximum principle, does there exist a sub-Markov resolvent (Vp ) such that Vo = V? Is this resolvent associated with a semigroup? The answer to this question is only partially known, but what is known shows that all "nice" kernels of potential theory fit into Hunt's probabilistic theory. Only Nos. 14 and 16 are indispensable for understanding the following chapters. We consider only proper kernels in this chapter.

The Domination Principle

1.

Dl

Let V be a proper kernel on a measurable space (E,tC); V is said to satisfy the domination principle if for every pair (f,g) ofpositive measurable functions, the relation DEFINITION

VfX

> VgX

for every x

E

E such that

g(x)

>0

implies XEE.

IX.T25 and IX.T69 furnish examples of kernels that satisfy the domination principle. D2 DEFINITION A kernel V is said to satisfy the complete maximum principle if for every constant a > 0 and for every pair (f,g) ofpositive measurable functions the relation

a

+ Vfx > VgX

implies a

+

Vfx

for every x such that

> VgX

for every

g(x)

>0

XEE.

This principle clearly implies the domination principle. We have seen examples of kernels that satisfy the complete maximum principle in IX.26 and in IX.69 (the subMarkov case). 202

X, 3, T4

The Domination Principle

203

3 Here is another, very useful, form of the complete maximum principle. Let f be a measurable function (not necessarily positive) such that Vf makes sense, and suppose that the function Vftakes value> 0 at certain points. Let P = {x:f(x) > O}. If V satisfies the complete maximum principle, we have sup Vfe = sup Vfl:. reEE reEP

(3.1)

(This property is sometimes called the "weak principle of the positive maximum.") To establish (3.1), denote the right side by a; we have on the set

{x:f+(x)

> O} =

P,

and consequently a+ + V(f-) > V(f+) everywhere, so that a+ > Vf. Since the function Vfattains strictly positive values we have finally a+ > 0, and hence a+ = a. This establishes (3.1). Conversely, it is easy to see that property (3.1) implies (for a proper kernel) the complete maximum principle. We shall be particularly interested in the case where E is a locally compact, a-compact space given the a-field of universally measurable sets, and where V is a continuous diffusion-kernel on E. The following theorem then allows us to simplify the verification of the domination principle. The strict positivity hypothesis made on V will be commented on in No. 5.

T4

THEOREM

Suppose that V is a continuous and strictly positive diffuSion-kernel, and that

the relation Vfre

> vgre

for every x such that

g(x)

>0

(4.1)

implies, when f and g belong to ~}(E), Vr

> vgre

for every

xEE.

(4.2)

The kernel V then satisfies the domination principle. Proof Let f and g be two positive universally measurable functions such that

for every x such that

g(x)

> o.

We shall show that Vf > Vg. Let AI be the set of l.s.c. (lower semicontinuous) functions that dominate f, and let B g be the set of bounded, positive, u.s.c. (upper semicontinuous) functions dominated by g. We have the following relations, which are immediate consequences of classical results from the theory of Radon measures:

Vj= inf Vj~ f'EA,

Vg = sup Vg'. g'EB g

We have, on the other hand, V(X,g')

= f

J{g'>O}

V(x,dy)g'(y)

=

sup

f

Kcompact JK Kc{g'>O}

V(x,dy)g'(y).

X,5

204

Construction of Resolvents and Semigroups

It thus suffices to show that

Vf' > V(g'IK)

for every functionf' E AI' every functiong' E B g , and every compact K contained in {g' Let then ep be a positive continuous function with compact support, such that Vepa;

>0

for every

x

E

> O}.

K.

The existence of such a function is an immediate consequence of the Borel-Lebesgue theorem, since the kernel V is strictly positive. For every e > 0 we have V(f'

+

eepY

> V(g'IKY

for every

x

E

K.

Denote by C the set of functions h' E ce~(E) dominated by f' + eep. The family of functions Vh'(h' E C) is filtering to the right, and admits f' + eep as its upper envelope; these functions are, on the other hand, continuous, whereas the function V(g'IK ) is U.S.c. (cf. IX.IO). Theorem 6 then implies the existence of a function h' E C such that Vh'a;

> V(g'IK)a;

for every

x

E

K.

Since the function Vh' is continuous, and the function V(g'IK) is u.s.c., there exists a compact neighborhood L of K such that Vh'a;

> V(g'IK)a;

for every

x

E

L.

Denote by D the set of functions j' E ce}(E) with support in L, which dominate g'I K' Another application of Theorem 6, analogous to that above, shows that there exists a function j' E D such that Vh'a; > Vj'a; for every x E L. This inequality then holds for every x such that j'(x) x, from (4.2). We thus have, V(f'

+

eep)

> 0, and consequently also for every

> Vh' > Vj' > V(g'IK).

which concludes the proof, since e was arbitrary. S Remarks (a) Let E1) be the set of permanent points for the kernel V (see IX.D67). We have E1) = {x: VIa; > O}. Since the function VI is l.s.c., E1) is open. Instead of supposing that V is strictly positive, suppose that the set (E1) is of potential zero. For every positive Borel function f defined on E1) and every x E E1) set W(x,j) = V(x,f'), where f' denotes any Borel extension off to E. The kernel W defined in this way on E1) is continuous and strictly positive, and thus satisfies the domination principle if V has the property of the statement. It then follows that the kernel V itself satisfies the domination principle. (b) Suppose that the continuous kernel V satisfies the complete maximum principle for the elements of ce}(E); it can then be shown, exactly as above, that V satisfies the complete maximum principle. It is not necessary to suppose that V is strictly positive: the function Vf + eVep of the foregoing proof can be replaced by Vf + e. Here now is the topological lemma we have used in the course of this proof, and which we shall have occasion to use again. It is a very easy generalization of the classical Dini's lemma.

205

Construction of Resolvents

X, T6, T7, T8, 9

T6 THEOREM Let K be a compact space, fan l.s.c. function on K, and g a u.s.c. function on K such that f(x) > g(x) for every x E K. Let :Yt' be a set of continuous functions, filtering to the right, with upper envelope f; there then exists a function h E :Yt' such that on K.

f>h>g

Proof For each x E K, choose a function ha: E :Yt' such that ha:Cx) > g(x). Since g is a u.s.c. function, we have hiy) > g(y) for all points y in a neighborhood Va: of x. There then exist a finite number of neighborhoods Va:'1 ... , Va: n , which cover K, and it suffices to take for h an element of :Yt' that dominates the functions ha: 1 , ... , h~""n .

Some consequences of the domination principle The lemmas we give now are borrowed from Deny (53). T7 THEOREM Let V be a kernel that satisfies the domination principle, and let f and g be two finite, positive, measurable functions such that the functions Vf and Vg are finite. Then, if p denotes a constant > 0, the equality f+pVf=g+pVg implies the equality f = g. Proof Putf' = f - (f A g) and g' = g - (f A g); we havef' A g' = 0 andf' + pVf' = g' + p Vg'. Then Vg'a: > Vf'a: at every point x such that g'(x) = 0 and consequently at every point x such that f'(x) > O. The domination principle then implies the inequality Vg' > Vf', but we can show in the same way Vf' > Vg', so that Vf' = Vg', hencef' = g' and finally f = g. ~

T8 THEOREM Let V be a proper kernel which satisfies the domination principle; there then exists at most one resolvent (Vp) such that Vo = v.

Proof Let (Vp) and (Wp) be two resolvents such that Vo = Wo = V, and let f be a positive, finite, measurable function such that the function Vf is finite. From the resolvent equation we have (I + pV)Vpf= (I + pV)Wpf= Vf for every p > O. It then follows from the preceding theorem that Vpf = Wpf. Since the kernel V is proper, every positive measurable function g is the limit of an increasing sequence of functions of the same type as! We thus have also Vpg = Wpg, and the theorem is established. The following result will be pointed out without proof: if the kernel V satisfies the domination principle, so do all of the kernels I + PV (p > 0). This result is clear when there exists a resolvent (Vp ) such that Vo = V, from IX.T55 and IX.T25.

2. Construction

if Resolvents

9 Let V be a kernel that satisfies the complete maximum principle. We give in this section sufficient conditions for the existence of a resolvent (Vp) such that Vo = V. Such a resolvent,

X, TIO

206

Construction of Resolvents and Semigroups

if it exists, is necessarily sub-Markov since the constant I is supermedian from the complete maximum principle and IX.T70. Here first is an example which shows that such a resolvent does not always exist. Let (Up) be a closed Markov resolvent on a measurable space (E,C); the kernel U = Uo satisfies the complete maximum principle from IX.T69. Let F be the set obtained by adjoining a point (X to E and let ff be the (J- field generated by C and {(X}. For every positive ff-measurable functionf defined on F set Vfx

=

Uf'x + f«(X)

for every

x

and

E,

E

vpt = !«(X),

where f' denotes the restriction off to E. It can easily be verified that V is a kernel and satisfies the complete maximum principle. We show that there cannot exist a sub-Markov resolvent (V p) such that Vo = V. If there were one, in fact, we would have (1

+ pV)Vpf=

Vf

for every positive measurable function f Suppose that f is zero at (X and denote by g the function equal to U pt' on E, and to zero at (X; then (1

+ pV)g =

VI,

and consequently Vpf = g from T7. In particular, we would have pVilE) = lE·

This equality and the inequality p VpI


ViI{a}Y = 0

would imply for every

x

E

E,

which would in turn imply the same property when p ~ 0, contradicting the definition of V. We begin by establishing, following Hunt (78), the existence of the resolvent (Vp ) associated with a uniform kernel.

TIO THEOREM Let V be a kernel on the measurable space (E,C), such that the function VI is bounded, and which satisfies the complete maximum principle. There then exists a subMarkov resolvent (Vp) such that Vo = V. * Proof Let r'§ be the space of bounded measurable functions, a Banach space under the uniform convergence norm (written 11 • 11). The restriction to r§ of the mapping f Jo.N+ Vf is a bounded operator, which we again denote by V. Let P be a number> O. There exists at most one bounded operator Vp on r§ such that pVVp = pVpV = V - Vp,

since this relation implies (1 + P V) Vp = V, and the kernel of the operator (I + PV) is zero from T8. With the supposition that such an operator V p exists, we point out several of its properties. (a) V p is positive. Let g E r'§ be a function < O. We show that a contradiction is obtained by supposing that the function Vpg attains a strictly positive value. We have in fact

* This resolvent is unique from T8.

207

X, TI0

Construction of Resolvents

and consequently, if the maximum of the function V1Jg is > 0, sup V1Jgll: =

Il:EE

sup V1Jg ll: > 0 Il:E{(g-1JVPg) > O}

from (3.1). This is absurd, since V1Jgx < 0 at every point x such that g(x) > p V1Jgx. (b) IlpV1J 11 < 1. Since V1J is positive, it suffices to verify the relation p V1Jl < 1. Now we have

1 >pV1Jpl for every x such that 1 - pV1Jl x > O. In other words,

1 > V[p(1 - pV1Jl)]X for every x such that p(1 - pV1Jl)X > 0, and consequently, from the complete maximum principle, 1 > V[P(1 - P V1Jl)] = P V1J1.

(c) The existence of V1J for some p

1/11 VII for p

=

> 0 implies that of V1H-e for 0 < e < p (or 0 < e <

0).

Consider, indeed, the series of operators

V1J(I - eV1J + e 2V;+ ...

+ (_l)ne nV:+

which converges for lel < 1/11 V1J II, and consequently for sum of this series by A, we have

.. '),

IcI < P

from (b). Denoting the

eV1JA = eAV1J = V1J - A. The operator V commutes with V1J , and hence with A. It then follows that

pV(V1J - A) = epVV1JA = e(V - V1J)A, or

pVV1J

+ eV1JA =

(p

+ e)VA,

or finally

V-A=(p+e)VA. We can thus set V1J+ e = A. Now consider the existence of the operators V1J: set Vo = V, and use the series to define V1J for pE [0,1/11 VII[. A second extension gives us V1J for pE [0,2/11 VII], a third extension doubles this interval, etc. The operator V1J can thus be defined for p > O. It is clear, moreover, that V1J is a continuous function of p on [0,00[. Let p and q be two numbers> O. We verify the relation

V1J - Vq = (q - p)V1JVq (which implies in particular, by interchanging p and q, that the operators V1J and Vq commute). It will suffice for us to verify the relation

since the kernel of the operator (I + PV) is zero. The computation is then immediate. Let then (fn)nEN be a decreasing sequence of bounded positive functions, which converges to zero. For every p we have lim VJn n ..... <Xl

<

lim Vfn = 0, n ..... <Xl

X, Tll

Construction of Resolvents and Semigroups

208

from the fact that V is a uniform kernel. The operators V 1l are hence the restrictions to t§ of kernels on (E,C), for which we use the same symbols. The relations: V1l Vqf = VqV1l f, VJ= Vqf + (q - p)V1l~!

(0

< p < q),

Vi = Hm V1l! (= sup V1l!) (continuity at p = 0), 1l.. . O 1l are satisfied for bounded, positive, measurable functions f, and hence also when f is positive measurable. The kernels V1l thus constitute the desired sub-Markov resolvent. Remark Suppose that E is a locally compact space, and that V is a continuous kernel on E which tends to 0 at infinity. All of the arguments we have made on the Banach space t§ can then be made on the Banach space f'Co(E), and it then follows that the kernels V1l are continuous and tend to 0 at infinity. The following theorem is in essence due to Hunt [(78), pp. 351-358]. It has been refined by Lion [see (87) and (88)], from whom we borrow the following proof. It should be recalled that the complete maximum principle is a necessary condition for the existence of a resolvent of the type considered (IX.T69), and that such a resolvent is unique (T8). .....

Ttt THEOREM Let E be a locally compact, a-compact* space, and let V be a continuous kernel on E which tends to 0 at infinity and satisfies the complete maximum principle. There then exists a sub-M arkov resolvent (V1l) on E such that Vo = V, consisting of continuous kernels which tend to 0 at infinity.

Proof We limit ourselves to constructing the restrictions of kernels V 1l to the space f'Co(E), and leave to the reader the task of extending them to the universally measurable functions. Let (Kn)neN be a sequence of compact subsets of E such that E = Un K n. For each n, let fn be a continuous function with compact support, with values in [0,1], and equal to 1 on K n. The function Vfn belongs to f'Co: it is thus bounded, and coefficients An > 0 can be chosen so that the series !n An and !n An 11 Vfn 11 converge. We then set

an = (na) A 1.

The function a belongs to f'Co, is strictly positive everywhere, and the function Va belongs to f'co. The functions an are continuous and increase to 1 as n ---+ 00: the set {an = I} ends up containing every compact subset K of E, provided n is large enough. The functions Van, finally, are bounded. For every n and every positive universally measurable function/we now set

It can easily be verified that

vn is a continuous kernel,

which tends to 0 at infinity and satisfies the complete maximum principle. Since the function Vnl = V(a n) is bounded, there exists a sub-Markov resolvent (V;:) consisting of continuous kernels, which tend to 0 at infinity, and such that Vf: = vn. • It is supposed that E is given the a-field f!ju(E), as in all problems that concern dispersion-kernels.

209

Construction of Resolvents

Let m and n be two integers such that n

X, TU

< m, and let fbe an element of ~}. We show that

Denote these functions by kn,k m, respectively. From the resolvent equation we obtain

kn =

vn(:' -

Pk n)

= VU -

pankn),

and an analogous formula for km. Consequently we have

k n - km = V(pamkm - pankn). A contradiction is then obtained by supposing that the function km - k n takes on a strictly positive value; in fact it then takes on strictly positive values on the set {p(ank n - amkm) > O} which is absurd since an < am. But the function f has compact support; the function V;(f/a n) is thus equal to V;f whenever n is large enough. It follows that the limit

VJ=

lim

V;f

(11.1)

n-+oo

exists (and is u.s.c.) for every function f E ~}. This property extends by uniform convergence to the elements of ~t, from the relation lip 1 < 1, then to ~o by linearity. It is clear that IlpVpfll < Ilf 11· Let g be an element of ~}; the function Vg belongs to ~t, and we thus have

V;

VpVg = lim V;Vg.

(11.2)

n-+oo

But we have then

pV:Vg = Pv;vn(.K.) = v n(-!.) - V;(-!.) = Vg - V:(-!.).

(11.3)

an an an an The left side thus increases with n, and the function p V pVg is thus l.s.c. We saw above that it was u.s.c., and hence it is continuous. Finally, a passage to the limit shows immediately that this function is dominated by Vg. It thus belongs to ~t. Let then f be an element of ~}; choose the function g E~} so that f ~ Vg, and set h = Vg - f The functions Vpf and Vph are u.s.c., and their sum is the function VpVg, which belongs to
(q - p)V:V;f= V;f - V;f (p> then shows that the VIJ form a resolvent.

0, q > 0)

X,12

To establish the second point, note that for every

functionfE~}

Hm pV;Vnj = lim (Vnj - V;f) = 1>-+0

Since every function g also have

210

Construction of Resolvents and Semigroups

E~}

and every n we have

o.

1>-+0

is dominated by a function of the form vnf (fE ~}), we Hm pV;g =

o.

1>-+0

Now we have seen that V1>g < V;g for everyp, whenever n is large enough so that an is equal to 1 on the support of g. We thus have a fortiori Hm p V1>g =

,

o.

jl->-O

\

This extends to functions g E ~t by uniform convergence, in view of the inequality lip V 1> I1 < 1. Let then f be an element of~}; the function g = Vf belongs to ~t and we thus have Hm pV1>Vj = o. 1> ..... 0

But we have p V1> Vf = Vf - V1>! [an obvious passage to the limit starting with (11.3)]. We therefore have also lim V1>f = VI, 1>-+0

which concludes the proof.

3. Construction

cif Semigroups

12 The construction of the previous section allows us to associate a sub-Markov resolvent with every "nice" kernel, which satisfies the complete maximum principle. We are now going to give sufficient conditions for such a resolvent to be associated with a sub-Markov semigroup. The essential tool for the construction of the semigroup is the Hille-Yosida theorem, which we state here without mentioning infinitesimal generators, a subject which the reader can find treated in the following works: Dunford and Schwartz (67), Hille and Phillips (77), Yosida (121), and also Loeve (89), which has the advantage of giving an introduction to the Russian work on infinitesimal generators of Markov semigroups. We begin by recalling several results on semigroups and resolvents in Banach spaces. Let @J be a Banach space, ordered by a closed convex proper* cone @J+ (we take @J+ = {O} if 81 doesn't have a natural order structure). The only topology we shall consider on 81 will be the strong topology defined by the norm (written 1I • 11). An operator A on 81 is said to be sub-Markov if IIA 11 < 1 and if A is positive (Ax E 81+ for every x E 81+). A sub-Markov semigroup on 81 is a family (Tt)t>o of sub-Markov operators on @J, such that TsTt = T s+ t for every s > 0, t > O.

We always complete this definition by putting To = I, but this convention is not necessary. The semigroup is said to be strongly continuous if Hm 1;x = x

for every

X E

81.

t-+O

• We understand a proper cone (in French: cone sail/ant) to be a cone P such that P

(1

(-P) =

o.

Construction of Semigroups

211

X, T13

It can then be easily shown that the function t .A.N+ Ttx is continuous on the interval [0,00[. A sub-Markov resolvent on fJB is a family (V2»2»o of operators on fJB, such that the

operators pV2> are sub-Markov and the resolvent equation holds: for every

p

> 0, q > 0.

(12.1)

Although this expression is not classical, we say that the resolvent (V2» is strongly continuous if for every X E fJB. (12.2) This definition can be put in another, very useful, form: It follows immediately from (12.1> that the image VifJB) does not depend on p; denote it by!:». The resolvent is then strongly continuous if and only if!:» is dense in fJB. Condition (12.2), in fact, implies immediately that !:» is dense. Conversely, the relation x = Vay implies limpV2>x

=

2> .... 00

lim (VaY - V2>Y

+ qV2>Vqy) =

VqY

=

x.

2> .... 00

Relation (12.2) thus holds for every x E!:». Since the operators p V2> are sub-Markov, it holds for every x E !:», and hence for every x if !:» is dense. Let (Tt) be a strongly continuous sub-Markov semigroup on fJB; a sub-Markov resolvent on fJB can then be defined by setting for every

X E

fJB.

Let x' be a continuous linear functional on fJB, orthogonal to !:». It can easily be verified that (x,x' ) = Hm (p V2>x,x') = 0. 2> .... 00

We thus have x' = 0, and!:» is dense in fJB from the Hahn-Banach theorem. The resolvent (V2» is hence strongly continuous. We call it the resolvent of the semigroup (Tt). Here then is the Hille-Yosida theorem. The proof we give is borrowed in large part from Yosida (121) and Neveu. * We shall only indicate the steps, leaving the verification of details to the reader. ...

Tt3 THEOREM Let (V2» be a strongly continuous sub-Markov resolvent on fJB. There then exists a strongly continuous sub-Markov semigroup (Tt) with (V2» as its resolvent, and this semigroup is unique. Proof We begin by supposing that there exists an operator V such that, for every p

> 0, (13.1)

It can then be easily verified that V(fJB) = !:». We also set !:»2 = V2(fJB); we then have !:»2 = V;(fJB) for every p 0, and the relation

>

Hm (p V2»2x

=

X

for every

X E

fJB

2> .... 00

• See "Theory of Markov Semigroups," University of Calif. Publications in Statistics, 2 (1958), 319-394.

X, TB shows that

Construction of Resolvents and Semigroups !?)2

212

is dense in f16. Note also the relation

(a) For every p

(I

> 0 set

+ pV)(I -

(13.2)

pV1J) = I.

A1J = p(pV1J - I), T:1J)

= exp (tA1J) =

e-1Jt exp (tp . pV1J).

It can easily be verified that the operators T~1J) constitute a strongly continuous sub-Markov

semigroup (we even have lim 1 Tt(1J) - III = 0). t--+O

We are going to show that these semigroups converge to the desired semigroup (Tt) whenp-+ 00. (b) With this in mind, note that the formula

~ A 1J

dp

= - (p v:

1J

is a consequence of the formula: V1J It then follows that rp

=-

-

1.. A2 p2

1)2 = -

1J

Y; (which comes from the resolvent equation).

~ 7:(1J) = _ ..!..- T(1J)A 2 t 2 t 1J. dp p If x belongs to !?)2 we have, since x is of the form y 2y,

.E:..- T(1J) t X dP

.!...2 T(p)(p v:p)2y. t

2 2 .!....2 T(lI) t A 1J V y = -

= -

P

P

The norm of the latter is at most equal to t Ily 1 jp 2, and we thus obtain, by integrating,

1I T~1J) x

T~q) xii < t .! _.! . Ilyll.

-

p

q

We can hence set Ttx = liIllp--+oo T~p)x for every x E !?)2. Since the operators T~p) are sub-Markov, this limit also exists for every x E !!)2 = f1l, and defines a sub-Markov operator Tt on f1l. The function t JV'.I+ Ttx is the uniform limit of the functions t JV'.I+ T~p)x on every compact interval of R+, when x belongs to !?)2. It is therefore continuous on !?)2' and hence on !?)2 = f1l by passage to the limit. The relation T~p)T~p) = Ts~~ then passes to the limit, and it follows that (Tt) is a strongly continuous semigroup, for which we still must find the resolvent. (c) We have d - T(p)

dt

t

= -d

dt

exp (tA )

=

T(p) A

1J

t

1J

and consequently, all of the operators T~1J), A 1J , V1J commuting, d .. - T(1J) t

dt

vi = T(p) A t

1J

Vx

=-

T(1J)(pV: )x t 1J

=

-pV: T(1J) x. 1J t

213

X,014

Construction of Semigroups

This derivative thus converges to - Ttx when p ~ 00, the convergence being uniform on every compact interval of [0,00[, and we obtain the formula d

-~Vx= -~x.

dt

Denote the resolvent of (Tt) by (Wp ). Integrating the above formula by parts, it follows that Wpx =

J:

00

e-Pt~x

dt = -

o

f.oo e- pt -d (~Vx) dt dt

0

= Vx - p f.ooe-Pt~vx dt,

or Wp(I + P V) = V. Now V p satisfies an analogous formula, and the operator I + P V is invertible, from (13.2). We thus have Wp = Vp as desired. (d) Let (TD be a second strongly continuous semigroup which has (Vp ) as its resolvent, and let x' be an element of the dual f!J' of f!J. The two continuous functions t AN+ (Ttx,x' > and t AN+ (T;,x' > have the same Laplace transform and are hence equal. The equality of the two semigroups can now be deduced. It remains for us to free ourselves of the auxiliary hypothesis concerning the existence of V. The proof of uniqueness given above clearly is independent of this hypothesis. Take then an arbitrary strongly continuous sub-Markov resolvent (Vp)p>o, and consider for every s > 0 the resolvent (Vs,p)p>o defined by

These resolvents satisfy the auxiliary hypothesis. There thus exists for each of them a strongly continuous sub-Markov semigroup (Ts,t) such that

Now the semigroup (e-(S-r)t Tr,t) also satisfies this relation for every number r thus have, from the uniqueness established above, T s,t = e-(s-r) t T r,t

(0

E

[O,s]. We

< r < s).

It then follows that the semigroup (est Ts,t) does not depend on s, and is strongly continuous and sub-Markov: We denote it by (Tt). The resolvent of (Tt) clearly is (Vp).

The Hille-Yosida theorem will allow us to complete the construction of the semigroup associated with a kernel that satisfies the complete maximum principle. We begin with a definition.

f:

D14 DEFINITION Let E be a locally compact, a-compact space and let (Pt)teR+ be a semigroup of sub-Markov dispersion-kernels on E. We say that (Pt) is a Feller semigroup if: (1) Each kernel Pt is continuous and tends to 0 at infinity. ~ (2) Po = I, andfor every functionfE ~o(E)Jimt-+o Pt! = f niformly on E.

A

Such a semigroup is not necessarily measurdte (in the sens f IX.39) when E is given the a-field f!Ju(E). A resolvent can, however, be associated with it in the following manner.

X, TI5

Construction of Resolvents and Semigroups

214

(a) Let f be an element of CC%; the mapping t ~ Ptf of R+ into CCo is bounded and continuous, which allows us to set

(integrating in CCo).

Vflf= f.ooe-fltptfdt

We thus define a positive linear mapping of CC% into ~o (which extends moreover to an operator on CCo, of norm at most equal to lip). It then follows from IX.T11 that the mapping f ~ Vflfis the restriction to ~% of a dispersion-kernel Vfl on E (continuous, and tending to 0 at infinity). (b) Let £ be the collection of bounded Bore! functions f such that the function t -A.J\t+ (p"Ptf) is Borel for every bounded Radon measure p, on E, and such that the following relations are satisfied.

<,.,V.f) =

fe-"(P,Pd) dt

Vflf = Vqf + (q - P)VflVqf

(p

(p

> 0)

> 0, q > 0).

(14.1)

(14.2)

£ contains the bounded, positive, l.s.c. functions (an obvious passage to the limit starting with the continuous functions). It then follows from I.T20 that £ contains all bounded Borel functions. (c) Let f be a bounded universally measurable function, and let f' and f" be two bounded Borel functions such thatf' < f < f", and (p,v'p,f') = (p,Vfl,f"). The function t -A.J\t+ (p"Ptf) is then bounded between two Borel functions that are equal a.e.; it is thus Lebesgue measurable, and relation (14.1) is still true. It can be verified in the same manner that (14.2) is true. Here then is Hunt's theorem.

-+

T15 THEOREM Let V be a continuous dispersion-kernel on E, which tends to 0 at infinity and satisfies the complete maximum principle. * Suppose that the image of ~%(E) under V is dense in ~o(E). There then exists a Feller semigroup (Pt) on E such that Vf =

f.oo Ptf dt

(15.1)

for every positive universally measurable function f This semigroup is unique. Proof We saw in No. 11 that there exists a sub-Markov resolvent (Vfl) on the Banach space ~o(E) such that Vf = limfl~o V flf for every function f E CC%(E). The relation Vf= ViI

+ pV)f

(fE~%)

shows that the image of ~0 under V fl contains the image of ~% under V. The resolvent (Vfl) is therefore strongly continuous, and from T13 there exists a strongly continuous sub-Markov semigroup (Pt) on ~o such that for every p > 0 and every x E E,

v"r =

fe-.tpJ"dt

Relation (15.1) is then deduced, whenfbelongs to

(fE 'C}). ~},

by lettingp tend to O.

• In other words, from IX.Tll and X.T4, a positive linear mapping V of r;% into '6'0 such that the relation + VpJ > Vg3: on {g > O} (a a positive constant, f e ~}-, g e ~~) implies the same inequality for every x.

a

Construction of Semigroups

215

The mappingf JW+ Ptf is, from IX.Ttt, the restriction to Pt (continuous, tending to 0 at infinity). The relations

X,16 ~o

of a sub-Markov kernel

Vf= f.\f dt and can then be extended to universally measurable functions, as in No. t4. The existence of a Feller semigroup satisfying (15.1) is thus established. Let (P;) be a second Feller semigroup having the same property, and let (V;) be its resolvent. It follows from (15.1) that V~ = V, and from T8 that V; = V p for every p. Let then f be an element of ~J("; the continuous functions t J\III'+ Pt/x and t ~ P;/x have the same Laplace transforms and are hence identical; so that the kernels Pt and P; are themselves equal. Remark It can be shown that (Pt) is the only semigroup of sub-Markov kernels that satisfies (15.1) and such that, for every function f E ~J(", the function (t,x) J\III'+ Ptfx is measurable with respect to the a-field 88(R+) x 88iE). We shall not prove this result. Passage from the sub-Markov case to the Markov case t6 Suppose that the semigroup (Pt) we have constructed is Markov. We shall see later that, using probabilistic methods, the potential theory relative to the kernel V can be studied in a very detailed manner. These methods are not directly applicable in the subMarkov case, where we have to reduce to the Markov case by the following method. Let (Pt) be a Feller semigroup of sub-Markov kernels on a measurable space (E,C). Adjoin to E an additional element 0, put E U {o} = E', and denote by C' the a-field generated by C and the set {o}. Define then kernels P; on (E',C') by setting P't(x,A) = Pt(x,A) for x P~(x,{

oD =

P~(o,A) =

E

E, AcE, A

1 - Pt(x,E) for x

IA(o)

(A

E

E

E;

E

C; (16.1)

C').

It is trivial to verify that we thus obtain Markov kernels on (E',C'), which again constitute

a semlgroup. We adopt the following very important convention: We identify every function defined on E with its extension to E' which vanishes at the point o. It is clear with this convention that Pt/ = P;f for every function f defined on E and, iff is defined on E', Pd = !(o)

+ Pt(! -

j(o)).

(16.2)

A sub-Markov resolvent (VJI) on (E,tC) can be extended in the same way to a Markov resolvent (V;) on (E',C'), by putting

pV;! =

j(o)

+ PYv(! -

j(o)).

(16.3)

It will be noted that if (VJI) is the resolvent of (Pt), then (V;) is the resolvent of (P;). Let us next consider the case where E is a locally compact, a-compact space, and where the sub-Markov semigroup (Pt) is a Feller semigroup on E. E' can then be considered to be the Alexandrov (one-point) compactification of E, 0 being the point at infinity (an

X, 17, D18, TI9

Construction of Resolvents and Semigroups

216

isolated point if E is compact). In this case the semigroup (P;) is a (Markov) Feller semigroup on E'. Analogous considerations apply to resolvents that take elements of CCo(E) into CCo(E). Ray resolvents

17 Let E be a locally compact, a-compact space, and let (Vp ) be a sub-Markov resolvent on the Banach space CCo(E). The hypothesis of strong continuity on this resolvent plays an essential role in the construction of the semigroup (Pt) associated with (Vp ) through the Hille-Yosida theorem. We seek now to replace strong continuity by a less restrictive condition, following Ray (106). The results that follow will not be used in later chapters. From No. 16 above, we lose no generality in limiting ourselves to the study of Markov resolvents (V p) on a compact space E with kernels that leave the space CC(E) invariant. We note first that for each p > 0, if f/ p is the convex cone of continuous p-supermedian functions (IX.T45), the vector space f/ p - f/ p is independent of p. To see this, let p and q be such that 0 < P < q. Every p-supermedian function is then q-supermedian (IX.T47), and it suffices to show that every continuous q-supermedian function f is equal to the difference of two continuous p-supermedian functions. Now we have f = [f + (q - p) Vpf] - (q - p) Vpf; these two functions are continuous, and the second is psupermedian from IX.T50. It will thus suffice to show that the function h + (q - p)Vph is p-supermedian for every q-supermedian function h. This property holds when h is of the form Vqg (g > 0), since then h + (q - p) Vph = Vpg; it thus holds for every excessive function h from IX.T64. We conclude finally by noting that, for every p, a function is p-supermedian if and only if it is equal almost everywhere to a p-excessive function, from IX.T60. We can now pose the following definition.

DI8

Let (Vp) be a Markov resolvent consisting of continuous diffusion-kernels on a compact space E. We say that (Vp) is a Ray resolvent if the cone f/ q of continuous q-supermedian functions separates the points of E for some q > O. DEFINITION

The condition then holds for every q > O. The cone f/ q is closed under the operation A; the space f/ q - f/ q is thus closed under the operations V and A , contains the constants, and separates the points of E. It is then dense in CC(E) by the Stone-Weierstrass theorem. TI9 THEOREM (Ray) Let (Vp) be a Ray resolvent. There then exists a unique measurable semigroup (Pt) on the measurable space (E,gjo(E» with (Vp) as its resolvent, which has the following property: The function t ~ Ptf X is right continuous for every function f E CC(E) and every x E E. Proof Let q be a number> 0, and let J be the closure in CC(E) of the image space Vp(CC(E». J is a Banach space, invariant under the operators V p, on which the resolvent (Vp ) is strongly continuous. There thus exists a strongly continuous sub-Markov semigroup (Pt) on J such that

(p> O,!E J).

(19.1)

The function I belongs to J, and Vpl = lip. It follows from the uniqueness of Laplace transforms that Ptl = 1 for every t ~ O.

Construction 01 Semigroups

217

X, T19

LetIbe an element of !7q' The functions p V'P+ql increase to the q-excessive regularization lof/whenp-+ 00 (IX.T46 and T60). We set (19.2) Pt! = Hm Pt(pV'P+qf), 'P-+ 00 and in particular Pol = J. We next extend the mapping I .J\I\t+ P tlto!7q - !7q by linearity. To show that the mapping t.J\l\t+ Ptlx is right continuous (and free of oscillatory discontinuities), we begin with the case where I is of the form Vqg, g E ~+(E). We then have oo

e-qtPtf = e-qtpt(f. e-qSPsf dS)

=

f.oo e-qSPsg ds.

The function t.J\l\t+ e-qtPtlx is thus continuous and decreasing. Suppose next that/belongs to !7q; the functions V'P+ql are of the preceding type, taking for g the positive function (I - qV'P+ql)· It follows from (19.2) that the function t.J\l\t+ e-qtPtlx is decreasing and l.s.c.-i.e. right continuous. The stated result is then clear by linearity when I belongs to !7q - !7q. Suppose next that I belongs to !7q - !7q and is positive; I is then the difference of two continuous q-supermedian functions g and h such that g > h. This inequality implies that pV'P+qg ~ pV'P+~ for every p, and hence Ptg > Pth for every t from (19.2). The relation I ~ 0 thus implies Ptl > O. We note finally that the function Ptf, when I belongs to !7q' is the upper envelope of an increasing sequence of continuous functions. Ptl is thus a Baire function, and this result extends to !7q - !7q by linearity. We have seen in No. 18 that the space !7q - !7q is dense in ~(E). The positivity of Pt implies on the other hand the relation IIPtll1 < IIIII (in the uniform norm). The mapping 1.J\I\t+ Ptl thus extends by continuity to a linear mapping, of norm 1, from ~(E) into ~(E), the space of bounded, 86o(E)-measurable functions on E [where 86o(E) is the Baire a-field on El. It can then be shown, as in IX.II-12, that the mappings Pt so defined are the restrictions to ~(E) of Markov kernels on the measurable space (E,86o(E)), which we denote by the same symbols. We show that the following three properties hold for every bounded Baire function I: (s ~ 0, t

> 0);

(19.3)

(b) The function (t,x).J\I\t+ Ptlx is measurable with respect to the a-field 86(R+)

(c)

V.I =

X

86o(E);

f.oo e-"'P,f dt

(p

> 0).

These three properties are indeed true when I belongs to J: (a) and (c) from the HilleYosida theorem, and (b) from the strong continuity of the semigroup, which implies that the function (t,x) .J\I\t+ Pt/x is continuous. They extend then to the case where I belongs to !7q from (19.2) and Lebesgue's monotone convergence theorem, then to ~(E) by linearity and continuity. The space :Ye of bounded Baire functions, which satisfy (a), (b) and (c), contains ~(E), is closed under passage to monotone limits, and therefore includes all bounded Baire functions from I.T20. The existence of the desired semigroup is thus established. Let (P;) be a second semigroup which has the same properties, and let I be an element of CC(E); the functions t.J\l\t+ ptr and t.J\l\t+ P;!X are right continuous and

X, 020, T21

Construction of Resolvents and Semigroups

218

have the same Laplace transforms. They are hence equal, and we see then that the kernels Pt and P~ are equal. In Chapter XI we study concepts analogous to those which we introduce now, following Ray. We keep the notation of the preceding numbers. D20 DEFINITION We say that the point x E E is a branching point for the Ray resolvent (VI}) if there exists a number p > 0 and a positive measure # of mass 1, distinct from ex, such that (20.1) (#,f> < f(x) for every function fE //1.1. TIt THEOREM The following properties are equivalent: (a) x is not a branching point for (VlJ); (b) exPo = ex (c) lim Ptfx = f(x) for every function f E ~(E) t-+O

(d) lim qVq / x = f(x) for every function f E ~(E). q-+oo

Proof We have from formula (19.2) for every functionfE

Po/x = limqVlJ+q flll

//1.1'

< f(x).

q-+oo

The measure exPo thus satisfies inequality (20.1), so that (a) implies (b). The implication (b) => (c) follows from the right continuity of the function t ~ Ptf X for t = O. The implication (c) =>(d) is a well-known property of Laplace transforms, and it only remains to show that (d) implies (a). Let # be a positive measure of mass 1 such that #(f) < f(x) for every function f E // 1.1. Let g be a continuous function with values in [0,1]. Apply # to both sides of the equality 1 = pVlJg

+ pVi 1 -

g).

The two potentials belong to // 1.1' so that (#, VlJg> = V lJtJ:. We thus have (#,!> = lex) for every functionfEJ", from the relation! = limq -+ oo qVlJ+q! Now wehave!(x) =f(x) from (d); the relation! < f then gives us (#,f> > f(x) for every function f E // 1.1. Since the measure # satisfies (20.1), this inequality can be replaced by an equality. Since the space // 1.1 - / / 1.1 is dense in ~(E), we have # = ex and property (a) follows.

CHAPTER

XI

Convex Cones and Extremal Elements

The main object of this chapter is to prove Choquet's fundamental theorem on integral representations in compact convex sets. This theorem could be considered now as a particular case of a theory of "balayage" defined by a convex cone of continuous functions on a compact set. Since this general theory has as yet no other important applications, we preferred to present it after Choquet's theorem, in Section 3, in order not to impose it on the reader interested only in convex cones. In Section 1 we have grouped a few auxiliary results on compact sets, which are not all indispensable for our purpose, but which are sometimes hard to find in the literature. All vector spaces considered in this chapter will be supposed real.

1.

Compact Convex Sets

Sublinear functions

Dl DEFINITION Let E be a vector space. A real-valued function p defined on E is said to be a sublinear function if it is subadditive: p(x

+ y) < p(x) + p(y)

(x, Y E E)

and positive homogeneous:

p(AX) = Ap(X)

(x

E

E, A > 0).

A linear functional f on E is said to be dominated by p if f(x) < p(x) for every x We shall need the following form of the Hahn-Banach theorem.

E

E.

T2 THEOREM Let E be a vector space. p a sublinear function on E, Fa subspace of E, and f a linear functional on F such that

f(x)

< p(x)

for every x

E

F.

There then exists a linear functional g on F, which is dominated by p and extends f. 219

XI, T3, 4

220

Convex Cones and Extremal Elements

Proof* Let fJJ be the set of all ordered pairs (h,H), where H is a subspace of E containing F, and h a linear functional on H, dominated by p on H and extending f: fJJ is clearly nonempty, and inductive for the order relation < defined by:

< (h',H»~ (H c

«h,H)

H' and h' extends h).

Let then (g,G) be a maximal element of fJJ (from Zorn's lemma); the theorem will be established if we show that G = E, which follows immediately from the next lemma. Let (h,H) be an element of fJJ, a an element of E that does not belong to H, and H' the vector space H E8 Ra. Let h' be the linear functional on H' defined by

LEMMA

+

h'(x

=

ra)

Then h' will be dominated by p on H'

h(x)

+

(x E H, rE R).

rA

if and only if

sup [h(x) - p(x - a)] a;EH

< A < inf [p(y + a) -

h(y)].

'YEH

These two conditions are always consistent. Proof of Lemma Since the two functions h' and p are positive-homogeneous, h' will be dominated by p if and, only if h'(x

+

a) = h(x)

+

A
+

a)

h'(x - a) = h(x) - A
for every x

E

H. The consistency of the two conditions follows from the inequality:

h(x)

+

h(y) = h(x

which gives us, for every x

E

+ y)


+ y)


+ p(y +

a)

H and every y E H,

. h(x) - p(x - a)

< p(y + a) -

h(y).

We illustrate the preceding theorem by proving a result on the extension of positive linear functionals, which Choquet (40) has used to study the moments problem. This theorem can be omitted with no inconvenience. T3 THEOREM Let E be an ordered vector space, H a subspace of E, and H the vector space consisting of the elements of E that are dominated by and dominate elements of H. Every positive linear functional h defined on H can be extended to a positive linear functional on H.

Proof The function p defined on

H by

p(x) = inf h(u)

(x E 11)

uEH u2::a;

is sublinear and equal to h on H. There thus exists a linear functional h' on H, which is dominated by p and extends h. We have p(x) < 0 for every x < 0 belonging to H, and hence h'(x) < O. The linear functional h' is thus positive. 4 Here now are several remarks on the subject of Theorem 2 (we keep the notation of that theorem). • Borrowed from Dunford-Schwartz (67) p. 62. See also No. 4(b) below.

221

XI, 5, T6

Compact Convex Sets

(a) Let x be a point of E; denote by F the vector space Rx, and by f the unique linear functional on F such that f(x) = p(x). The sublinearity of p then implies the relation o
fact which implies the following result: Any sublinear function p is the upper envelope of the linear functionals dominated by p.

Here is a very important direct consequence: There exists a unique linear functional dominated by p if and only if p itself is a linear functional. (b) Suppose now that E is a locally convex topological vector space (LCS) and that the sublinear function p is continuous. Denote by E' the product LCS R x E, by A' the convex open set {(t,x) E E': t > p(x)} (the set of points of E' situated above the graph of p), and by M ' the linear manifold {(t,x): x E F, t = f(x)} (the graph of f). There then exists, from Theorem 1, p. 69 of Bourbaki (16), a closed hyperplane H', which contains M ' and does not meet A'. It can be verified directly that H' is the graph ofa continuous linear functional on E which is dominated by p and extends f

Theorem 2 can be proved by this reasoning, since any vector space can be given the finest locally convex topology (under which all finite convex functions are continuous). Compact convex sets 5 Let E be a LCS, and K a compact convex subset of E. We denote by d the vector space consisting of the continuous affine functions on K, i.e., of the continuous functions f from K into R such that tf(x)

+ (1

- t)f(y) = f(tx

+ (1

- t)y)

(x,y E K, t E [0,1]).

The subspace of d consisting of the restrictions to K of the continuous affine functionals on E separates the points of K; hence this is also true, even more so, of d. We denote by [/ the convex cone in CC(K) consisting of the concave continuous functions. This cone is closed under the operation A.; the vector space [/ - [/ is thus closed under the operations A. and v, contains the constants, and separates the points of K. [/ - [/ is therefore dense in the Banach space CC(K), from the Stone-Weierstrass theorem. * Let E' be the dual of E. The topology induced on K by the weak topology a(E,E ' ) on E is Hausdorff and coarser than the initial topology on K. These two topologies are thus the same. We now establish two results borrowed from Mokobodzki (103). Note that T6 can be deduced from T7 and Dini's lemma. T6

Every continuous affinefunction f on K is the uniform limit on K ofcontinuous affine functions on E. THEOREM

Denote by F the LCS R x E. Set a = inf:l:EKf(x), and consider the following two subsets of F: u= {(t,x)ER x K: a - e < t f(x)},

Prooft

• See Bourbaki (16).

t This proof was communicated to us by R. Phelps.

XI, T7

Convex Cones and Extremal Elements

222

where e is a number >0. These two sets are disjoint, the first is compact convex, and the second is convex closed. There hence exists a closed affine hyperplane H* which separates U and V; H cannot be parallel to the line R X {O} since every hyperplane of this type that intersects U also intersects V. H is hence the graph of a continuous affine function g, which is within e off on K. Let B be a Banach space and let B' be the dual of B, with the weak topology. Take for K the unit ball of B', and consider a linear functional f on B', which is weakly continuous when restricted to K. f is the uniform limit on K of a sequence (fn)neN of weakly continuous linear functionals on B'. These functionals arise from elements of B, and ~ a Cauchy sequence in B; it then follows that f itself arises from an element of B. We have thus established an important theorem due to Banach [see, e.g., Bourbaki (17), p. 74; Dunford and Schwartz (67), p. 428]. Several analogous results follow in the same way from Theorem 6. The expression "g is strictly dominated by f" in the following statement means that g(x) < f(x) for every x E K. Example of an application

"

prt.n1.-

T7 THEOREM (a) Let f be a finite convex l.s.c. function defined on K. Denote by d, the set of restrictions to K of continuous affine functions on E, which are strictly dominated by f on K; we then have (7.1) f= sup g. ge.9l1f

(b) Suppose in addition that f is affine. The set d, is then filtering to the right. (c) Let f be a finite convex u.s.c. function defined on K; the set re, of continuous convex functions on K which strictly dominate f is then filtering to the left, and

f= inf g.

(7.2)

ge~,

Proof Using the notation of the proof of T6 we set, under hypothesis (a), W

= {(t,x) E R

X K:

t

> f(x)}.

W is then a closed convex subset of F = R X E; for every point z = (s,y) of R x K such that there exists a closed hyperplane H, which strictly separates z from W. This hyper-

plane cannot be parallel to R x {O}; it is hence the graph of a function g E d" and relation (7.1) follows. Suppose next thatfis affine; to establish (b), we recall that the convex hull of the union of two convex compact subsets Band B' of F is compact [being the image of [0, I] x B x B' under the mapping (t,x,y) ~ tx + (1 - t)y]. Let then hand h' be two elements of d, and a a constant dominated by hand h' on K. Denote by B the compact convex set {(t,x) ER x K: a < t < h(x)}, and by B' the analogous set with h replaced by h'. The reader can verify directly from the fact that f is affine that the convex hull C of B U B' is disjoint from W. Since C is compact there exists a closed hyperplane, which strictly separates C and W; this hyperplane is the graph of an affine function g that dominates hand h' and is strictly dominated by f on K. Statement (b) is thus established. Suppose finally thatfis convex u.s.c. Let g and g' be two convex bounded l.s.c. functions on K that strictly dominate f We shall show that there exists a function h E re, dominated

* Bourbaki (16), Chapter 1I, Section 3, Prop. 4, p. 73; Dunford and Schwartz (67), V. 2.7, Theorem 10, p.417.

Compact Convex Sets

223

XI, 8

by g and g'-this will imply in particular that re! is filtering to the left. Let a be a constant that dominates g and g'; denote by B the compact convex set {(t,x) E R X K: g(z)

< t < a},

and by B' the analogous set with g replaced by g'. Let C be the convex hull of the union B U B', which is compact. Set k(x) = inf {t: (t,x) E Cl. This function is convex l.s.c. and strictly dominates f It hence is equal, from (a), to the upper envelope of the set ..#k' Denote by :Ye the set of functions of the form ho V hI V ..• V hn

(n EN; ho, ... , hn E ..#k) ;

:Ye is a family of continuous functions, which is filtering to the right. There then exists from X.T6 a function h E :Ye, which strictly dominates f

It only remains to show that the lower envelope of re! is equal to f It suffices from the above to construct for each point x E K and each t > f(x) a convex l.s.c. function g t.aJ on K, which strictly dominates f and is such that g t.ix) = t. Let us thus choose a number b, which dominates f on K (such a number exists since fis u.s.c. and finite), and denote byGt,aJ the convex hull of the point (t,x) and of {b} x K: Gt.aJ is a "stalactite" hanging over the graph off It then suffices to put

gt.iy) = inf {s ER: (s,y) E G t.aJ }.

Extreme points of a compact convex set 8 We denote by Jt+ (respectively, Jtt) the collection of positive (respectively, positive with unit mass) Radon measures on K. The barycenter of a measure ft E.Lt will be written b(p,). We begin by recalling, or proving quickly, some elementary properties of barycenters. (a) Let ft be a measure in .Lt with barycenter x, and let f be a convex (respectively, afJine),jinite, u.s.c. or l.s.c.function on K. We then have

f(x)


f(y) dp,(y)

[respectivelY,f(x)

=

L

l

f(y) dp,(y)

Equality of the two sides in the case where f is continuous affine is, in fact, the very definition of b(ft); one passes from there to the inequality for a continuous or l.s.c. convex function by means of T7(a) and II.T36. The case of a convex u.s.c. function then follows from TI(c) and II.T36. Equality for a l.s.c. or u.s.c. affine function is obtained finally by considering the convex functions f and - f (b) Let L be a compact subset of K. The closed convex hull of L is equal to the set of barycenters of the measures p, E .Lt, which are carried by L. This is a well-known property [see, e.g., Bourbaki (18), Chap. Ill, Section 4, Prop. 7, p.87]. (c) Every measure A. E.Lt is the weak limit of discrete measures belonging to .Lt and having the same barycenter as A.. Here (following Choquet) is the principle of the construction of such measures. Since the topology of E is locally convex and K is compact, there exist arbitrarily fine open coverings of K consisting of a finite number of convex open sets COl' CO2, ••• , COn' Put

XI, 9

224

Convex Cones and Extremal Elements

Each point Xi belongs to wi , the measure () is discrete and has the same barycenter as A, and we have IA(/) - ()(/)/ < 'YJ for every continuous function I whose oscillation on each W i is less than 'YJ. It then follows that () is very near to A in the weak sense whenever the covering is fine enough. 9 Recall that a point x

E

R is said to be extreme if it admits no representation of the form

x = ty

+ (1

- t)z

(t

E

[0,1], yE K, z

E

K)

(9.1)

other than the trivial ones for which t = 0 or 1, or y = z = x. Here are several properties of extreme points. (a) A point x is extreme if and only if the set of measures p, E.Lt such that b(p,) = x consists of Ere alone.

Suppose that x admits a nontrivial representation of the form (9.1). The measure tE1I + (1 - t)E z is then distinct from Ere and has x as its barycenter. Conversely, suppose that there exists a measure p, E .Lt distinct from Ere' and such that b{p,) = x. Let y #: x be a point in the support of p, and let V be a closed convex neighborhood of y, which does not contain x; we then have 0 < p,(V) < 1. Put V' = K",-V, A = Iv . p,1p,(V), A' = Iv' . p,1p,(V'), y = b(A), z =b(x'), and t = p,(V). The point x then admits the nontrivial representation x = ty + (1 - t)z, and is hence not extreme. (b) Let Y be a compact subset of K, and Z the closed convex hull of Y. Every extreme point of Z then belongs to Y. Every point x of Z is, in fact, the barycenter of a measure p, E .Lt carried by Y [No. 8(b)]. If x belongs to Z"'- Y this measure is distinct from Ere' and x cannot be extreme in Z from No.9(a). (c) We call (following Choquet) every intersection of K with an open half-space of E a slice of K. Choquet has established the following property: A point x E K is extreme if and only if the slices of K containing x form a fundamental system of neighborhoods of x. Suppose first that x is not extreme, and consider a nontrivial representation of the form x = ty + (1 - t)z. Every slice of K containing x contains at least one of the points y,z. Let then V, W be neighborhoods of x, which do not contain y,z, respectively; the neighbor-

hood V n W contains no slice of K containing x. Conversely, let x be an extreme point of K. Since the topology of K is induced by the weak topology a(E,E'), every neighborhood of x contains a finite intersection TnT.n···nT I 2 n

of slices of K, which contain x. Let Cb C2 , ••• , Cn be the (compact and convex) complements of Tb ... , T n , respectively. Since the point x is extreme, and does not belong to the union Cl U C 2 U ... U Cn' it cannot belong to the convex hull C of this union. Now C is compact; there thus exists a closed hyperplane H, which strictly separates C from x, and H determines a slice T of K, containing x and contained in T I n T 2 n ... n Tn •

225

XI, DI0, Tll, T12

Compact Convex Sets

The Krein-Milman theorem

When the compact set K is metrizable, Choquet's theorem can easily be proved without relying on the Krein-Milman theorem (we shall do this in Section 2). However, we will need the Krein-Milman theorem to prove the Choquet theorem in the nonmetrizable case. We follow the development of Bauer (2). DI0 DEFINITION A face* of the compact convex set K is any nonempty convex subset F of K, which has the property Every measure p, E.At such that b(p,)

E

F is carried by F.

(10.1)

Let x be a point of K; {x} is then a face if and only if x is extreme. Tll THEOREM (a) Let (Fi)iEI be a family of faces of K, with nonempty intersection F. F is then a face of K. (b) Let F be a face of K and fa finite l.s.c. concave function on K. The set G ={x E F: f(x)

= inff(Y)} 1JEF

is then a face of K. (c) Every face of K contains at least one extreme point. Proof (a) The intersection of two faces of K is clearly either empty or a face of K. We can thus assume that the family (Fi)iEI is filtering to the left. Let then p, be a measure in .At such that b(p,) E F; we have fl(Fi ) = 1 for every i E I, and thus fl(F) = 1 from II.T35. F is hence a face of K. (b) The set G is nonempty and compact since f is l.s.c. Put a = inf1JEF f(Y); we have p,(f) > a for every measure p, E .At carried by F, with equality if and only if fl is carried by G. Consider then a measure A E.At such that b(A) E G; since F is a face, A is carried by F. Since the functionfis concave we have also from No. 8(a) that a = j(b(A»

> A(f).

The measure A is thus carried by G, and it follows that G is a face. The family of faces of K, ordered by inclusion, is inductive downwards by (a). Every face thus contains a minimal face (Zorn's lemma). Assertion (c) will thus be established if we show that a face F, which contains at least two distinct points y and z, is not minimal. Indeed, denote by f a continuous affine function that separates y and z; the set G of points of Fwherefattains its minimum on Fis then a face of K [from (b)], which is contained in F and distinct from F. Here are several important consequences of this theorem. The first is known as Bauer's "minimum principle." T12 THEOREM Let f be a finite l.s.c. concave function on K, which is positive at every extreme point,. f is then positive on K.

Proof The set of points of K where f attains its minimum is a face of K, which contains an extreme point from Tll(c). This minimum is thus positive. • One usually calls faces of a compact convex polyhedron KeRn the compact convex subsets F of K, which satisfy property (10.1).

XI, T13, TI4

226

Convex Cones and Extremal Elements

This property also holds when f is a finite, u.s.c., concave function: f is then, in fact, the lower envelope of the set of continuous affine functions, which dominate f, and the latter are positive from the preceding theorem. * T13

Let K and K' be two compact convex sets, u a continuous affine mapping of K onto K', and x'anextreme point of K'. There then exists at least one extreme point x of K such that x' = u(x). THEOREM

Proof It can be verified directly that the set u-1 [{x'}] is a face of K.

T14

THEOREM

(Krein-Milman)

The closed convex hull of the set of extreme points of K

is equal to K. Proof Let L be the closed convex hull of the set of extreme points. If there exists an x E ~L, there exists a continuous affine function h on E such that f(x) < and infllEL f(Y) ~ 0, which contradicts T12.

°

Convex cones with compact base It frequently happens that the compact convex set K appears as the base of a closed

convex cone C with apex 0, i.e., as the intersection of C and a closed hyperplane that misses 0. Moreover, one can always reduce to this situation by identifying K with the subset K' = {I} x K of the product vector space R x E, and then putting C = {tx: t E R+, X E K'}. It is then appropriate to modify the notation of this chapter in the following manner. (a) Every real-valued function f defined on K can be considered as the restriction to K of a unique positive-homogeneous function defined on C, which lve still denote by f We are thus led to denote by re (respectively, .91,.:7) the collection of continuous (respectively, continuous affine, continuous concave) positive homogeneous functions defined on C. (b) Every positive linear functional on the space re ("conical measure" on C) can be written f.A.N+ p(f) where p denotes a positive measure on K. We denote by Ba; (X E C) the conical measure f .A.N+ f(x); it is clear that Bta; = tBa; for every t > 0. Let p be a measure in Jt+; the point x E C characterized by the relation: f(x) = p(f) for every function fEd is called the resultant of p, written rCfl).

(15.1)

In other words we have reO) = 0, rep) = p(l) . b(plp(1» for every measure p ¥= 0, and in particular rep) = b(p) if p belongs to Jtt. The inequality (8.1) then gives us f(rCfl»

< p(f)

(15.2)

for every measure p E Jt+ and every positive-homogeneous convex function f on C, which is l.s.c. or U.S.C. (c) A point x E K is extreme if and only if the relation x = y + Z (Y,z E C) implies the existence of a number t E [0, I] such that y = tx and z = (1 - t)x, or again, if every element y of C which is dominated by x in the intrinsic ordert of C, is proportional to x.

* These results are improved in Bauer (7).

t The intrinsic order of C is defined by x ~ Y <=> for some Z E C, x

+Z = Y

(x, Y E

C).

The Choquet Theorem

227

2.

XI, D16, 17, DI8

The Choquet Theorem

We keep henceforth the notation of No. 15. The idea of proving the Choquet theorem by means of an order relation on 1+ is due to Bishop and DeLeeuw (8), but the order relation below was used for the first time by Choquet (33). D16

DEFINITION

We denote by (A.

-< the order relation on vI/+ defined by

-< ft) <=> (A.(f) > ft(f) for every function f

E

!7).

(16.1)

Remarks (a) This is indeed an order relation, since the vector space !7 - !7 is dense in CC (for uniform convergence on K). (b) The relation A. -< ft implies A.(f) = ft(f) for every functionf E d, and thus r(A.) = r(ft). (c) We have Cx -< ft for every x E C and every measure ft E..4+ such that r(ft) = x. The measures Cx (x E C) are thus the minimal elements of J(+ under the relation -< . (d) The relation A. -< ft says that ft is "nearer to the boundary of K" than A.. We seek to study the measures on K which are "as near the boundary of K as possible," i.e., the maximal measures under the order -<. These measures will be called simply maximal in what follows. (e) The positive measures ft such that A. -< ft are also called balayages of ,1.* This expression will be justified in No. 50. The following definition, which we borrow from Mokobodzki (102) (but which has been used before, in different forms, by several other authors), will play a fundamental role. A lower envelope! [J - (- j)] can be similarly defined. 17

D18 DEFINITION Let f be a function in CC. The function

inf g

(18.1)

ge!/ g?f

is called the upper envelope off, and denoted by

J.

This function is clearly positive homogeneous, concave, and u.s.c. We have f <J and 1(Y) < sUPxeKf(x) for every yE K, so thatJis bounded. For every measure A. E..4+ we can thus set p;.(f) = ,1(/) = inf A.(g), (18.2) ge!/ g?f

the second equality following from II.T36 since !7 is closed under the operation A. The following properties are immediate:

f=J if fE!7;

(if) =

if

for every function f ~

(f + g)

<J + g

E

CC and every

(f,g E CC).

It then follows that the functionf ~ p;.(f) is sublinear on CC.

* That is, "swept out" measures.

t

> 0;

XI, T19-T22

Convex Cones and Extremal Elements

228

T19 THEOREM Let A. be a positive measure. The linear functionals on~, dominated by the sublinear function P;.' are identical with the positive measures on K, which are balayages of A.. Proof Let fl be a balayage of A.; for every function f E ~ and every function g E f/ such that g > f we then have fl(f) < fl(g) < 1(g), which yields the inequality fl(f) < pif) < p;.(f) by passage to the limit inferior on g. Conversely, let fl be a linear functional on ~, which is dominated by the sublinear function P;.. The relation f < 0 implies pif) < 0, and thus fl(f) < 0; fl is thus a positive measure on. K. Let f be a function in f/; then f = J, hence p;.(f) = 1(f)andfl(f) < 1(f). It then follows that fl is a balayage of 1. T20

COROLLARY

For every measure 1

E

Jt+ and every function f

E ~

p;.(f) = sup fl(f).

we have (20.1)

p.evlt+ p.>-;'

Proof The sublinear function P;. is the upper envelope of the family of linear functions which it dominates (No. 4), and the latter are the balayages of 1 from the preceding result. The existence and characterization of maximal measures

-.

T21

THEOREM

Every measure 1

E

Jt+ admits a maximal balayage.

Proof Let Jt;. be the family of balayages of 1, ordered by the relation -< ; it will suffice to show that Jt;. is inductive (Zorn's lemma). Now let i J\,f\,f+ fli be an increasing mapping of a totally ordered set I into Jt; since the measures fli are positive and have the same total mass, they admit a weak cluster point fl. We have fl(f) = limi fllf) for every function f E f/, since the mapping i J\,f\,f+ fllf) is decreasing. Since the set f/ - f/ is dense in ~, fl is the weak limit of the fli' and also the least upper bound of the fli under the order -< . This establishes the theorem. Here now is the most important result concerning maximal measures; it is due to Mokobodzki (102). -.

T22 (a) (b) (c) (d)

Let 1 be a positive measure on K. The following statements are equivalent The measure 1 is maximal; 1(f) = 1(j) for every function f E ~; 1(f) = 1(1) for every function f E - f/; The measure 1 is carried by each of the sets B t = {x E K:f(x) = lex)} (fE -!/). THEOREM

Proof We know that 1 is maximal if and only if the set of balayages of 1 consists of 1 alone, or again (T19 and No. 4) if 1 = P;.' Statements (a) and (b) are thus equivalent, and (b) clearly implies (c). We show conversely that (c) implies (a): Let fl be a balayage of 1; we have fl{f) > 1{f) for every functionfE -f/, and also fl{f) < pif) = 1{f), from (c). Since the space f/ - f/ is dense in ~ under uniform convergence on K, we have fl = 1, and it follows that 1 is maximal. Finally, the relation f < implies immediately the equivalence of (c) and (d).

1

229

The Choquet Theorem

XI, T23-T25

Maximal measures and extreme points (The metrizable case) We denote by OK the set of extreme points of K (it will be noted that this set has not yet entered the discussion). The first result does not require that K be metrizable. T23 THEOREM (a) f(x) = J(x) for every point x (b) OK = Bf

n

E

OK and every functionfE~;

fE-!/'

(c) Let A be a positive measure on K, such that every compact subset disjoint from OK is A-negligible,. A is then maximal. Proof If x is extreme every balayage of Ca; is equal to Ca; [9(a)]; (a) then follows from formula (20.1). We thus have OK c nfE-!/' Bf • Conversely, let x be a point of this intersection; the measure Ca; is then maximal from T22. Every measure p, E 1+ such that r(p,) = x is thus equal to Ca; [17(c)], and x is extreme from 9(a). Suppose finally that A satisfies the hypothesis of (c), and let f be an element of~. The function f is u.s.c., so that K"""Bf is the union of the sequence of compact sets {J - f > Iln }(n EN). These sets are disjoint from OK' their union is thus A-negligible, and so A is carried by Bf • It then follows from T22 that A is maximal. The idea of using a strictly convex function in the proof of the following theorem is borrowed from Bonsall (12) (this article contains a very short and elegant proof of the Choquet theorem in the metrizable case, which is the origin of the proof we give here).

J-

T24 THEOREM Suppose that K is metrizable. The set OK is then the intersection of a sequence of open sets. A measure A E 1+ is maximal if and only if it is carried by OK'

Proof Since the set K is metrizable, the space ~ (given the norm of uniform convergence on K) admits a countable dense subset. Since the space f/ - f/ is dense in ~, there exists a sequence Cfn)nEN of elements of -f/, which separates the points of K. We can suppose that all of these functions lie between - I and 1 on K; put then

f= nEN !

21nf~.

This function is also convex continuous and is linear on no open segment contained in K. Since the function J is concave, we thus have J(x) > f(x) at every nonextreme point x E K. It follows then from T23(a) that B f = OK; OK is thus the intersection of the sequence of open sets f < Iln} (n EN), and every maximal measure A is carried by OK from T22(d). Conversely, if A is carried by OK' A is carried by every set B g (g E -f/) from T23(a), and thus is maximal from T22(d). Here then is Choquet's existence theorem for the metrizable case.

{l-

...

T25 THEOREM Suppose that K is metrizable. Every point x measure fl carried by the set of extreme points of K.

E

K is then the resultant of a

Proof Let p, be a maximal balayage of Ca; (T21); we have rep,) = x, and p, is carried by OK'

The uniqueness theorem The version of the uniqueness theorem that we give is borrowed from the article by Loomis (91) [see Cartier, Fell, Meyer (24)]. We begin by establishing, following Cartier, the identity between the order -< and the "strong" order introduced by Loomis.

Convex Cones and Extremal Elements

XI, T26, 27 T26

THEOREM

230

Let A and fl be two positive measures on K; the following three properties

are equivalent; (a) A -< fl; (b) For every finite family (A i)i=l n ofpositive measures on K such that A = ~~=1 Ai, there exists a finite family (fli)i=l n ofpositive measures on K, such that Ai -< fli for

and

i = 1, . . . , n;

(c) The same statement as (b), replacing Ai -< fli by r(Ai)

= r(fli)'

Proof* Suppose that (a) holds. Let E be the product vector space ~n, and let F be the subspace of E consisting of the elements of E of the form (1,1, ... ,f) (n times, f E ~. Consider the sublinear function p on E defined by ,fn) = A1(h) + A2(!2) + ... + An(!n)' The linear functional (I, I, ,f) -A/II'+ fl(f) on F is dominated by p on F. It can thus be extended to all of E, from T2, by a linear functional dominated by p. This functional can be written (f1,h, ... ,fn) -A/II'+ #1(f1) + fl2(f2) + ... + flnCfn)' P(f1,f2'

flh ... , fln denoting linear functionals on ~. We have fllf) < Ai(!) for every i and every function f E ~; consequently fli is a balayage of Ai from T19. Finally, we have 1i fli = fl,

and property (b) is established. Property (b) clearly implies (c). To show that (c) implies (a), consider a functionfE [/, and a number 8 > O. Cover K with a finite number of closed convex sets Wb W 2 , ••• , W m on each of which the oscillation of/is less than 8. Put ei = W i "'" Uj
A(f) =

1 Ai(f) > 1 [f(r{A i» i

- 8A i {I)]

i

from the condition on the oscillation of the (positive-homogeneous) function This last expression is equal to ~f{r(fli»

- 8A{I)

> ~ fli{f)

- 8A{I)

=

f

on

Wi•

fl(f) - 8A{I),

i

i

from (15.2). We thus have A -< fl, and the theorem is established. Let x be an element of C. We define a subdivision of x to be any finite family (X i)i=1.2. ... .n of elements of C such that x = ~:=1 Xi' The subdivisions of a measure fl E 1+ are defined similarly. The set of all subdivisions of x is partially orderedt by the relation "s is less fine than t," which we write s -l t, and which is stated, if s = (X i)i=l. ... .n and t = (yj)j=l . ... .k' as

27

(s -l t}<::>(there exists a partition of {I, 2, ... , Xi

k} into n sets J1 ,

•••

,In such that

= ~jEJiYj for all i).

* One may note that the first part of the proof can be generalized as follows: let E be a vector space, PI' P2' ... ,pn be sublinear functions on E, x' be a linear functional dominated by PI + P2 + ... + pn. One may then find linear functionals x~, x~, ... , x~ on E, dominated, respectively, by Ph P2' ... ,pn, such that x' = x~ + x~ + ... + x~. We shall not give details here, since No. 51 will yield an extension of this result to "continuous sums" of sublinear functions. t This translates the French "preordonne"; note that (s -I t and t -I s) doesn't imply s = t.

231

XI, T28, T29

The Choquet Theorem

We associate with every subdivision s = (X i )i=1 . ... .n of x the measure e s = ~:=1 erei • The relation s 1 t clearly implies e s -< et. The relation e s -< ft, where ft is an element of Jt+, is equivalent from T26 to the existence of a subdivision (fti)i=1 . ... .n of ft such that Xi = r(fti) for every i. Let S be a collection of subdivisions of x, which is filtering to the right for the relation 1 ; we say then simply that S is afiltering set. Such sets are natural objects of study in certain applications of Choquet's theory, in particular, in the theory of group representations, which was the origin of Loomis's work. Let ft be a positive measure on K, and let (fti)i=1.2 . ... .n and (ft;)i=1.2 . ... .k be two subdivisions of p; there then exist positive measures Ai; (i < i < n, 1 <j < k) such that Pi = ~i Aii for every i, and P; = ~i Aii for every j. * Denote then by Sp. the collection of subdivisions of X = rep) of the form (r(Pi»i=l . ... ,n' where (Pi)i=l . ... .n denotes a subdivision of p. It is clear from above that this set is filtering. The following theorem then summarizes the main results of Loomis's theory. T28 THEOREM (a) Let A and P be two positive measures. The relations A -< p and Sl c Sp. are equivalent. (b) Let R be a filtering set of subdivisions of x. The set of measures e s (s ER) then admits an upper bound for the order -<, which we denote by eR' We have R c S(ER); if R is of the form Sp. (p E Jt+) we have eR = p. (c) Every filtering set R is contained in a maximal filtering set (for the relation c). (d) A filtering set R is maximal if and only if there exists a maximal measure pE Jt+ such that R = S p.' This measure is then unique.

Proof Statement (a) is a simple translation of T26. Since the relation -< is a true order, we see in particular that the relation Sl = Sp. (p, A E Jt+) implies A = p. To establish (b), note that the relations s 1 t, f E!/ imply (e s ,!) > (et,f). Since the set !/ - !/ is dense in rc, it follows that the mapping s ~ es from R into JI+ admits a single cluster point in the weak topology, along the filter of sections of the partially ordered set R. This mapping thus admits a weak limit eR' and it can be verified easily that eR is the supremum of the measures e s (s ER). The relation e s -< eR for every subdivision SE R implies R c S(ER) from T26. Finally, if R is of the form Sp.' we have e s -< p for every SE R, hence eR -< p; on the other hand R = Sp. c S(ER)' so that p -< eR and lastly p = eR. The collection of filtering sets of subdivisions of x, ordered by inclusion, is clearly inductive; (c) is thus an immediate consequence of Zorn's lemma. Suppose finally that R is a maximal filtering set, and let p be a measure such that ReSp." We have R c S(ER)' R eSp" thus R = Sp. = S(ER) and finally p = eR. In other words, eR is the only measure p such that R esp.. It then follows in particular that eR is maximal. Conversely, let p be a maximal measure, and let R be a maximal filtering set containing Sp. [one exists, from (c)]. The measure eR is then maximal, we have p -< eR and thus p = eR. Consequently Sp. = R, and Sp. is indeed a maximal filtering set. Theorem 28 implies Choquet's uniqueness theorem:

-+

T29 THEOREM The following two statements are equivalent: (a) The cone C is a lattice under its intrinsic order. (b) For every point x E C there exists a unique maximal measure ft E JI+ such that rep) = x.

* Let /i (respectively, I;) be Borel functions on K into [0,1], such that Pi We then set Ail = fl;p.

= ~p

(respectively,

pi = f;p).

Convex Cones and Extremal Elements

XI, T30

232

Proof Suppose that the cone C is a lattice. Let x be an element of C, and let (X i )i=l, ... . n and (X;)i=l . ... ,k be two subdivisions of x. The "decomposition lemma" [see Bourbaki, (18) Chapter. 11, Section 1, No. 1, p. 19] implies the existence of elements Yii (1 < i < n, 1 <j < k) of C such that Xi = ~i Yii for every i, and = ~i Yii for every j. In other words the set of all subdivisions of x is filtering; it is therefore the unique maximal filtering set and (b) follows from T28. To establish the converse, we note first that the set 1~ of maximal measures is always a convex cone, which is a lattice for its intrinsic order. To see this let A and ft be two maximal measures; A and ft then charge no set B, (fE -f/; cr. No. 22). The same is then true of the measures A + ft, A " ft, and A V ft (these last symbols with respect to the natural order on 1+), measures which are thus maximal from T22. It follows that 1~ is a cone and that A " ft and A V ft are, respectively, the inf and the sup of A and ft in 1~. Now the function ft Jo..N+ r(ft) maps 1~ onto C (T21), and it is clearly additive and increasing when ..,/I~ and C are given their intrinsic orderings. Suppose that property (b) is satisfied: this mapping is then an isomorphism, and it follows that C is a lattice. We say (following Choquet) that K is a simplex if the cone C is a lattice. The reader will find other characterizations of the case of uniqueness in the article by Choquet-Meyer

x;

(44).

The following theorem will be generalized later (No. 36).

Suppose that K is a simplex. For every x E K, let ft:» be the unique maximal measure with barycenter x. The mapping x .J\I\/'+ ft:»(f) is then Borel for every function f E cc. Let A be a positive measure; the unique maximal measure ft such that A -< p is given by the formula T30

THEOREM

(f E rc).

(30.1)

Proof For every measure A E 1+ and every functionfE -f/ we have from (20.1) Pl(f)

= ACj)=

sup ()(f)

=

ft(/),

;'-<6

ft denoting the unique maximal measure such that r(ft) = reA). It follows, taking A = e:» (x E K), that

lex) = p:»(f) (fE -[/).

The function x Jo..N+ ftif) is thus u.s.c. when f belongs to - f/; it is hence Borel when f belongs to f/ - f/, and this property extends to every f E CC by passage to the uniform limit on K. Keeping the same meaning as above for A and p we have, for every function f E - f/,

ft(f)

=

A(])

=

fKl(X) dA(X)

=

fK fti/ ) dA(X).

The two sides of (30.1) therefore coincide for f E - f/, and consequently also (by linearity and continuity) for every f E CC. Maximal measures and extreme points (general case) We resume the discussion of Nos. 23-24 in order to show that the maximal measures are still, in a weak sense, "carried" by OK when K is not metrizable. The situation is, however, much less satisfactory than in the metrizable case. We follow a proof of Choquet's.

233

XI, T31, T32

The Choquet Theorem

T31 THEOREM Let fl be a maximal measure, and (!n)n?1 a decreasing sequence o/positive continuous functions on K, which converges to 0 at every point of OK; we then have limn-+oop(fn) = O. Proof We have fl( -c) = fl( -c) for every function c E

~

(T22); i.e.

fl(c) = sup fl(g). g6-!/ gS;C

Let us use this relation to construct inductively continuous functions gl' g2' ... , belonging to -/7 and satisfying the following properties (where e denotes a number> 0):

> fl(fJ - e/2; ft(gJ > ft(gl AfJ - e/4;

fl(gl)

gl
< gl Af2'

gn

< gn-l Afm

We then have the inequalities: fl(fn - gn) = fl(fn - fn A gn-l) < fl(fn-l - gn-l)

+ (fn A gn-l -

gn)

+ e/2n

It can be easily verified by induction that this quantity is less than e(1 - 1/2n ) < e. Denote by g the convex u.s.c. function infn gn; it is < 0 at the points of OK and hence everywhere (T12), and we thus have fl(infnfn) < fl(infnfn - g) < e. The theorem is established. The following theorem, borrowed from the article by Choquet-Meyer (44), is an improvement of a result of Bishop and DeLeeuw (8). We denote by r the paving consisting of the compact subsets of K.

T32

THEOREM

Every maximal measure fl is carried by every r-analytic set containing OK'

Proof We are going to begin by showing that fl is carried by every element of ra, which contains OK' Let A be such a set and B the complement of A; it will suffice for us to show that fl(C) = 0 for every compact Cc B. Now B is the intersection of a decreasing sequence (Gn)neN of open sets. Choose then a decreasing sequence (fn)neNof positive continuous functions such that fn equals 1 on C, and 0 off of Gn' The preceding theorem implies the relation fl(limnfn) = 0, and thus fl( C) = O. Set then, for every subset A of K, fl*(A) = inf fl(B). Be.Jra

B:JA

This function is a Choquet capacity with respect to the paving r , from III.T23. Every r -analytic set A containing OK is thus capacitable. Now the outer capacity of such a set equals fl(l) from above, the inner capacity is hence also equal to ft(1), and it follows that ft is carried by A. Let B be a compact Baire set disjoint from OK; the complement of B is r-analytic from II.T29: fl therefore does not charge B. The set K"".0K is thus inner negligible for the restriction of fl to the Baire a-jield of K.

XI, 33, 34, T35, T36

Convex Cones and Extremal Elements

234

Theorem 32 implies the identity of the maximal measures and the measures carried by OK when OK is a .?r-analytic set (and in particular when OK is compact). It has been possible, on the other hand, to construct examples: of nonmaximal measures carried by every .?r-analytic set containing OK [Mokobodzki, in Choquet-Meyer (44)]. of maximal measures, which charge compact subsets disjoint from OK [Bishop and DeLeeuw(8)].

33

The existence of dilations 34 The problem whose definitive solution is given below has been studied by several authors. We mention in particular Blackwell (9) (who established Theorem 36 for an interval on the line) and Sherman (108) (who proved it for discrete measures). An unpublished paper by Fell [who considered the "strong" order of Loomis (91) in place of the order -<] has been an essential stage in the solution of the problem, due to Cartier. Theorem 36 has, on the other hand, been established by Mokobodzki (unpublished). We denote by vi{ the space of measures on K with the weak topology. The mapping x ~ ea; permits the identification of K with a compact subset of vII+. We denote by D the closed convex cone of the product topological vector space vi{ x vii consisting of pairs (A,,u) such that A -< ,u. The set ~ of pairs (A,,u) E D such that 1.(1) = 1(= ,u(1)) is a compact base for D. We denote finally by Do the compact set of elements of D of the form (ea;,1J) (x E K). Suppose that K is metrizable. We say that a Markov kernel T on the Borel O'-field of K (see the beginning of Chapter IX) is a dilation on Kif r(ea;T) = x for every x E K. T35 THEOREM (Cartier) Let (A,,u) be an element of D; there exists a positive measure () on Do whose resultant (in the space vi{ X 1) is equal to (A,,u). Proof Since the theorem is clear for the pair (0,0), we reduce to the case where (A,,u) belongs to ~. All that then remains to be proved, from 8(b) is that ~ is the closed convex hull of Do, or (T14) that every extreme point of ~ belongs to Do. Now let (rx.,{J) be an element of ~, which does not belong to Do; the measure rx. is the sum of two nonproportional positive measures rx.' and rx.". There then exist from T26, two measures {J' and {J" such that rx.' -< {J', rx." -< {J", and {J = {J' + (Ju; we thus have (rx.,{J) = (rx.' ,(J') + (rx." ,(J").

The pair (rx.,{J), the sum of two nonproportional elements of Do, is hence not extreme in Do, and the theorem is established. It will be noticed that this statement did not suppose that K was metrizable. This is not true of the following theorem. T36

(Cartier) Suppose that K is metrizable; let A and,u be two positive measures on K. The relation A -< ,u is equivalent to the existence of a dilation Ton K such that AT = ,u. THEOREM

Proof Suppose first that there exists such a dilation. Let f be an element of f/; we have

p.(f)

=

t

T(x,f) dA(X)

< tf(x) dA(X) = AU),

from (15.2), and therefore A -< ,u. Conversely, suppose that A -< ,u; consider the measure () on Do associated with the pair (A,,u) by the preceding theorem, and denote by h the continuous mapping (sa;,fj) ~ x

235

The Choquet Theorem

XI, 37

of Do onto K. The compact space K is metrizable, and hence so are the image of () under h equals A, () admits a disintegration

o=

.Lt and

Do. Since

IK O. dA(X),

where «()re}lJEK is a measurable family of probability laws on Do (in the sense of I1.D13; the compact metric spaces K and Do are naturally given their Borel a-fields) such that ()re is carried by h-I({x}) for every x E K. * We denote by T re the resultant of ()1lJ (in 1); TIlJ is an element of Jlt, and the mapping X.A.f\l+ TIlJ(A) is Borel on K for every Borel subset A of K. There hence exists a Markov kernel T on K such that 8 re T = T re for every x. Since

J:

TIlJ dA(X) the resultant of () in .L X .L is (A,ft), we have K established. Theorem 36 will be generalized later (No. 52).

= AT = ft,

and the theorem is

Extension to certain cones without a compact base 37 Let E be a locally convex space, and let C be a closed convex cone contained in E, containing no line, and with apex o. We say that a point x E C, distinct from 0, belongs to an extremal ray of C if the relations x = y + Z, Y E C, Z E C imply the existence of a number t E [0,1] such that y = tx, Z = (I - t)x. One can ask under what conditions C has the following properties (which are satisfied for cones with compact base): (1) The closed convex hull of the union of the extremal rays of C is equal to C (a generalization of the Krein-Milman theorem); (2) Every point x E C is the resultant of a positive measure ft on a compact subset of C, carried in a more or less precise sense by the union of the extremal rays of C (a generalization of Choquet's theorem). The reader can consult Choquet's notes on this subject, cited in the bibliography. We limit ourselves here to indicating a procedure that sometimes allows a reduction to the theory of compact convex sets. Following Choquet we call every compact subset of C of the form {h < I} a cap of C, where h is a mapping from C into R+ U { + oo} which is linear in the following sense: h(O) = 0 h(x

+ y) = h(x) + h(y)

h(tx) = th(x)

C)

(x,y

E

(x

C, t

E

E

R+).

Since the set {h < I} is closed, the function h is l.s.c. on C. Let X o be a point of C different from 0; suppose that X o belongs to a cap {h We can suppose that h(xo) = 1. Set H

= {x: h(x) ~ I}

HI

= {x: h(x) = I}.

< I} of C.

and

• Bourbaki, (20): Chapter VI, Section 3, No. 1, Theorem 1, p. 58. The generalization of T36, which will be given in No. 52, will be proved without making use of this theorem.

XI, 37

Convex Cones and Extremal Elements

236

We shall see that the set H b although it is not, in general, compact, can play the role of a base of C for the integral representation of elements of H. We want to find the extreme elements of H. It is clear first of all that 0 is extreme, since C contains no line. We have hey) # 0 for every yE H different from 0, since if not H would contain the half-line R+y, and would not be compact. Let then y be a point such that 0 # hey) # I, and let YI be the point yjh(y) E HI; y admits the representation y = [I - hey)] . 0 + hey) . YI and thus cannot be extreme. In other words, we have hex) = 1 for every extreme point x # O. Let then x = y + z be a decomposition of x (y E C, Z E C) such that y # 0, Z # 0; set YI = yjh(y), ZI = zjh(z). We have

= h(y) . YI + h(z) . ZI; since x is extreme, we have YI = ZI = x: Y and Z are thus proportional to x, x

and it follows that x belongs to an extremal ray of C. Conversely, the reader can verify directly that every point of HI that belongs to an extremal ray is an extreme point of H. In other words, the extreme points of Hare: the point 0 and the points of HI located on the extremal rays of C. We denote by 8}] the set of these latter points. Now apply Choquet's theorem to the point X o (recall that h(xo) = I) and the compact convex set H: there exists a maximal probability measure p, whose barycenter is equal to X o' Since the function h is affine l.s.c. on H, we have (No. 8) h(xo)

= 1=

fHh(y) dp,(y).

The measure p, thus does not charge the set {h < I}. It then follows, in particular, that p, has no mass at O. It is known (T32) that the maximal measure p, is carried by the closure of 8H ; since it does not charge {O}, ft is carried by the closure of 81: X o hence belongs to the closed convex hull of 8}]. We have thus obtained a natural extension of the Krein-Milman theorem. Suppose that His metrizable. The maximal measure p, is then carried by 8H , and does not charge {O}; it is thus carried by 8}], and so we have a generalization of Choquet's theorem. Let s = (Xi )i=I.2 • . . . . n be a subdivision of Xo (No. 27); since the function h is increasing, all of the points Xi belong to H and Theorems 26 and 28 apply without modification to any two measures A and p, on H with barycenter xo. Suppose, in particular, that the cone C is a lattice; the set S of all the subdivisions of X o is then filtering, and X o is the barycenter of a unique maximal measure ft carried by H, defined by p,(f) = Hm (es,f> sES

for every positive-homogeneous and continuous function f defined on C, the limit being taken along the directed set S. It will be noted that the cap H does not enter into the formula, a fact which implies the following consequence: Let H = {h < I} and H' = {h' < I} be two caps for C such that h(xo) = h'(xo) = 1. Let p, and ft' be two measures with barycenter x o, carried, respectively, by Hand H', and maximal in these caps. We then have p,(f) = p,'(f) for every continuous and positive-homogeneous function f on C. In other words, p, and p,' define the same "conical measure" on C. Here now is the most important case to which the preceding theory applies.

The Choquet Theorem

237

XI, T38, 039, T40

T38 THEOREM Let X be a locally compact, a-compact space, and let C be a cone ofpositive measures on X, which is closed in the weak topology. Every point of C is then contained in a cap ofC.

Proof Let flo be an element of C, and let (Xn)neN be a sequence of open, relatively compact subsets of X whose union is X. We have fl(Xn) < 00 for every positive measure fl and every n EN; we can thus find a sequence (an)neN of strictly positive numbers such that

We then set, for every positive measure fl, h(fl) =

Z anfl(X n)· n

This function is clearly linear and l.s.c. on .L+(X), and the set of positive measures fl such that h(fl) < I is weakly compact. The set H = {fl E C: h(fl) < I} is therefore a cap of C which contains flo. An application of the Krein-Milman theorem Choquet has given a simple proof of the famous theorem of S. Bernstein on completely monotone functions. We reproduce this proof here, with a view to later applications. We denote by R: the open half-line R+"'{O}. Let f be a real-valued function on R:, and h be a positive number; we shall denote by tJ.hfthe function x ~ f(x + h) - f(x). The iterated difference operators tJ.h1 tJ. h2 ... tJ.hn can be defined in the same way. One can show easily that these operators commute. D39 DEFINITION Let f be a real-valuedfunction defined on R:. We say that f is completely monotone if f is infinitely differentiable and if we have for every integer p

> O.

(39.1)

The function f then is positive, decreasing, and convex.

T40

THEOREM

Let f be a real-valuedfunction defined on R:; the following statements then

are equivalent: (1) f is completely monotone. (2) The function f is positive; for every integer p (h b h2 ••• hp) ofpositive numbers, we have:

>

I, and every finite sequence (40.1)

(3) There exists a positive measure f-t on

f(x) for every x

> O.

R+ such that

= f.oo e-~t dfl(t)

(40.2)

Moreover, this measure is unique.

Proof The implication (3) => (1) and the uniqueness of fl are elementary and well-known results on Laplace transforms. To show that (1) implies (2), we start with the remark that, if g is completely monotone, then the same is true for -tJ.hK. Indeed, the operators

XI, T40 ~h,

238

Convex Cones and Extremal Elements

and DP obviously commute, and we have (-l)P DPLlh8'(x) = (-l)P(DPg(x

+ h) -

DPg(x)) = (-l)P DPHg(X

+ u),

where u belongs to the interval [O,h]; hence the first member is negative. One then easily deduces that the first member of (40.1) is a completely monotone function, and therefore a positive function. This establishes (2). We now prove that (2) implies (3), beginning with the case of bounded functions. Let us denote by C the set of all bounded functions that verify (40.1); C is a convex cone, the elements of which are decreasing and convex functions. The limit lim t -+o+ I(t) thus exists for every function lE C; we shall denote it by 1(0), and call Cl the set of all lE C such that 1(0) < 1. We provide C with the topology of pointwise convergence on R:; it is clear that Cl then is compact. On the other hand, all elements of Cl are convex, and therefore continuous, functions. The topology induced on Cl thus is equal to the topology of pointwise convergence on a countable dense set which is metrizable. One finally sees that Cl is a metrizable cap of C associated with the l.s.c. linear function I ~ 1(0). We now prove that every nonzero extremal point of Cl is an exponential t ~ e-~t (perhaps a constant). * Let indeed I be an extremal point; 1(0) is equal to 1, and I belongs to some extremal ray of C (No. 37). Let us write: I(x) = I(x

+ h) + (/(x)

- I(x

+ h)).

The functions x ~ I(x + h) and x ~ I(x) - I(x + h) belong to C; since lis extremal, they must be proportional to f, and a constant k exists, such that I(x + h) = kl(x). Taking x = 0, we find k = I(h). The relation I(x + h) = l(x)/(h) shows that I is a decreasing exponential function, or the constant 1. Let us denote by E the subset of Cl whose elements are the decreasing exponential functions, the constants 1 and 0; E is a closed set, which contains the extremal points of E. Every point of Cl thus is the barycenter of a positive measure of mass 1 carried by E (theorem of Krein-Milman); if we observe that C = U t~O tCl , we get the representation (3) for bounded completely monotone functions. Let then I be an unbounded function which satisfies (2), and let h be a strictly positive number; the function x ~ !(x + h) belongs to C, and therefore has a representation: f(x

Then we have for every k f(x

+

h

+ h) = iOOe-~t dftit)

(x

> 0).

(40.3)

> 0: +

k) =

iOOe-~te-ktdft",(t) = iOOe-~tdfth+k(t)

which implies dfth+k(t) = e-ktdft",(t) (uniqueness of Laplace transforms) It follows that the measure ft defined by dft(t) = ehtdft",(t) does not depend on h, and (40.3) then becomes f(x

We replace x

+ h) =

iooe-l,,*hltdp(/).

+ h by x and get the representation (40.2).

• In fact, one can easily prove that the decreasing exponential functions and the constant 1 are extremal elements of Cb but we don't need this here.

239

Balayage Defined by a Convex Cone of Functions

3. BalayaBe Defined by a Convex Cone

XI, D41

cif Functions

In all of the classical forbears of Newtonian potential theory certain convex cones of functions, which play a fundamental role, are seen to appear: superharmonic functions, plurisuperharmonic functions, concave functions, and excessive functions. These cones are always closed under the operation A, and the functions of which they consist are generally lower semicontinuous. One can thus imagine the "general potential theory" as the study of convex cones of functions that have these two properties. This ambitious "general theory" so far has certainly not reached its definitive form. Its main interest at this time comes from the better insight it gives into older results (Choquet's theorem, Shilov boundaries) and from having simplified their proofs. This certainly is enough to justify the study here. Given the incomplete state of the theory, it appeared sufficient for us to give the general ideas in their simplest form. We limit ourselves in particular to the study of convex cones of continuous functions. The results of this section are borrowed from Bauer (3) and Mokobodzki. D41 DEFINITION Let X be a compact space and [/ a subset ofCC(X). We denote by partial ordering on .L+(X) defined by for every

fE f/).

-<

the

(41.1)

Let A and ft be two positive measures such that A -< ft; we say then that ft is a balayage of A (relative to f/). Remarks (a) Let [/' be the closed convex cone generated by [/ in CC(X). The partial

orderings defined by [/ and [/' are the same. Two closed convex cones contained in CC(X) are the same if and only if they define the same partial ordering on .L+(X) (HahnBanach theorem). (b) Let [/ be a convex cone contained in CC(X), and let [/1 be the collection of functions of the formfl A f2 A •.• A fn (n EN, fb ... , fn E f/); the identity

shows that !/1 is a convex cone, which is closed under the operation A. Let -< and -< 1, respectively, be the partial orderings associated with !/ and [/1; the relation A -< 1 ft clearly implies A -< ft, and the relations ea; -< A and ea; -< 1 A are equivalent for every x E X. The inequalities ft(fl) < fl(X), ... , ft(fn) < fn(x) indeed imply ft(fl A f2 A ••• A fn)

< fl(X) A ••• A fn(X)

= ea;(fl A ••• A fn)·

(c) We assume henceforth that !/ contains the constant 1. The relation A -< ft then implies A(1) > ft(1); the collection of balayages of a measure A is thus weakly compact.

Suppose that X is a compact convex subset of a locally convex space, and that [/ is the collection of continuous concave functions on X. The relation -< then coincides with that which we have used in the preceding section.

Example

XI , D42--D44

240

Convex Cones and Extremal Elements

D42 DEFINITION A point x E f/ is said to belong to the boundary (of X relative to 9) there exists no balayage of ere distinct from ere'

if

The boundary will be denoted by a!/X. Remarks 41(a) and (b) imply that the boundary is not changed if f/ is replaced by the closed convex cone, closed under A, which is generated by f/.

Example Suppose that there exists afunction f E f/, which attains a strict negative minimum at a point X.~ f(x) < 0; f(y) > f(x) for every yE X""{x}. The point x then belongs to the boundary. Indeed let p be a balayage of p(f) < f(x) and p,(f) = p({x})f(x)

+f, x

ere;

we have

fey) dp,(y)

,{re}

> p,({ x })f(x) + p,(X",,{x })f(x) =

p,(1)f(x),

X""

where the inequality is strict if p charges {x}. The relation 1 E f/ implies that p,(l) < 1, hence p(1)f(x) > f(x), with strict inequality if p,(1) < 1. The comparison of these inequalities then gives p(X""{x}) = 0, p,(1) = 1, and hence p = ere'

D43 DEFINITION Let A be a subset of x; we say that A is a Shilov set (relative to 9) if the relations: f E f/; inff(x) ~ -1 (43.1) re eA

imply the inequality infreex f(x)

> -1.

Remarks (a) The set X is always a Shilov set; the empty set is a Shilov set if and only if every function f E f/ is positive. (b) Let A be a compact subset of X; definition 43 then takes the following form: A is a compact Shilov set if and only if every function f E f/ which takes on a value < 0 attains its minimum at a point of A. (c) The Shilov sets remain the same if f/ is replaced by the closed convex cone generated by f/ and closed under the operation A. D44 DEFINITION Suppose that f/ is a convex cone closed under the operation A and containing the positive constants. Let A be a Shilov set. For every function f E ~(X) we set (44.1) = inf g.

lA

1

ge!/ g?! onA

Ix·

We write in place of

lA

lA

Remarks (a) The relation f < 1 implies < 1; the relation f > -1 implies :;=: -1, from the fact that A is a Shilov set. The function is thus bounded for every function f E ~(X); since it is upper semicontinuous, it is integrable for every measure A E .L+(X). Since the cone f/ is closed under A we have (AlA)

=

lA

inf (A,g).

ge!/ g?! onA.

(b) We set

PA,A (f)

=

(A,lA)'

(44.2)

241

Balayage Defined by a Convex Cone of Functions

XI, T45-T48

This function is finite and sublinear on ~(X). We then have the following theorem, valid under the hypotheses of D44, which generalizes T19. T45 THEOREM Let A be a compact Shilov set. The linear functionals on ~(X) dominated by the sublinear function pA,A are identical with the balayages of A carried by A. Proof Let ft be a balayage of A carried by A; we have ft(f) < ft(g) < A(g) for every function g E f/ which dominates f on A, and hence ft(f) < PA.if). Conversely, let 4> be a linear functional on ~(x) dominated by PA.A; the relation f < implies PA.if) < 0, and hence 4>(f) < 0: 4> is thus a positive measure. The relations f > 0, f = on A, imply PA.if) < 0, and hence 4>(f) < 0. Thus 4> is carried by A. Finally we have PA,if) < A(f) if fbelongs to f/, hence 4>(f) < A(f), and 4> is a balayage of A.

°

T46

COROLLARY

°

We have*

sup

PA ;.(f) =

ft(f).

A-<,p JI.

carried by A

(An immediate consequence of Nos. 4 and 45). Here is another consequence of Theorem 45. It was proved in Choquet-Deny (41) (by another method). T47 THEOREM Let f/ be a closed convex cone contained in ~(X), and containing the positive constants. The following two conditions are equivalent: (a) f/ is closed under the operation A; (b) There exists a family of pairs (Ba:i,fti)iEb where Xi denotes a point of X, and fli a positive measure on X of mass at most equal to 1, such that f/

= {f E ~(X):

f(x i)

< flif)

for every i El}.

(47.1)

Proof The collection of functions defined by the right side of (47.1) is clearly a closed convex cone, closed under the operation A, and containing the positive constants. Conversely, let f/ be a cone having these properties; denote by -< the partial ordering associated with f/, and by f/' the collection of functions f, which satisfy the inequality ft(f) < f(x) for every x E X and every measure ft, which is a balayage of Ba:' Since the cone f/' satisfies condition (b), it suffices to show that f/ = f/'. Now let -<' be the partial ordering associated with f/'; the relations Ba: -< ft and Ba: -<' ft are clearly equivalent. The formula !(x) = sup ft(f) t:c-<JI.

and the analogous formula relative to f/' show that the functions J are the same for the two cones f/ and f/'. It then follows from Theorem 45 that every positive measure A admits the same balayages relative to f/ and f/'. The Hahn-Banach theorem shows finally that the two cones are identical. T48 THEOREM (Bauer) Suppose that the convex cone f/ is closed under the operation A, contains the positive constants, and separates the points of X. (a) Every function fE f/ which attains values < attains its minimum at a point of the boundary. (b) A compact set A is a Shilov set if and only if it contains the boundary.t

°

• Mokobodzki has proved that this extends to U.S.c. functions. t The closure of the boundary 0.9'X is called the Shi/ov boundary of X (with respect to 9').

XI, 49

Convex Cones and Extremal Elements

242

Proof We limit ourselves here to reasoning in the case where [/ contains all of the constants; the general case can then be deduced by the procedure described in No. 49. If [/ contains all of the constants the restriction on fbecomes unnecessary in statement (a). We call a face of X any nonempty compact subset F of X which has the following property: For every x E F, every balayage of Ex is carried by F. (For example, X is a face.) The following assertions can then be established by reasoning identical to that of Theorem 11. Every face contains a minimal face. Let F be a face, and f an element of [/; the collection of points of F where f attains its minimum on F is a face. Every minimal face is a point of the boundary. These three assertions clearly imply (a). To establish (b), we note first that the boundary is a Shilov set from (a); the same thus holds true for any compact set which contains the boundary. Conversely, let A be a compact Shilov set, and let x be a point not in A; there exists at least one linear functional on
One can then reduce to the case of a compact space and of a cone of continuous functions which contains all of the constants, by means of the following construction, which is very similar to the construction used in No. X.16. Denote by 2 = X U {o} the Alexandrov compactification of X (the point at infinity 0 is isolated if X is compact). Identify
+ A') = A + [f A (f' + (A.' -

A»],

and this last function is equal, from (49.1), to the upper envelope of a family, which is filtering to the right, of elements of [/; it thus belongs to [/ from Dini's lemma (applied on 2). Let A be a bounded positive measure on X, and let # be a balayage of A (relative to [/); we have #(1) < A(1) from (49.1). The following measure is then a balayage of A on 2, relative to !?: p, = # + (A(1) - #(1»Ea. Conversely, it is easy to show that every balayage of A relative to !? is of this form. The relation then follows easily.

243

XI, 50

Balayage Defined by a Convex Cone of Functions

Here again are several remarks whose verification is almost trivial. Let A be a Shilov set in X relative to f/; the set A u {a} is then a Shilov set relative to !/. Conversely, let B be a compact Shilov set relative to !/; the boundary point then belongs to B from T48, and B",,{a} is a Shilov set for f/. It then follows that Theorem 48 remains true under the hypotheses of 1:his number: the Shilov sets relative to f/ which are closed in X are the closed sets which contain the boundary f/'X.

a

a

Example Take for X the open unit ball of Rn (n > 2), and for f/ the convex cone consisting of the positive superharmonic functions continuous in the unit ball and zero at the boundary (in other words, vanishing at infinity in X). Every element of f/ is of the form Uv, where v is a positive measure on X and U denotes the Green's kernel. f/ contains all functions of the form Uf, where f is positive, continuous, and has compact support in X; it is well known that f/ satisfies the hypotheses of No. 49. Let A and fl be two bounded positive measures on X; the relation A -< fl implies 50

(AU,f)

=

(A,Uf)

> (fl,Uf) =

(50.1)

(flU,f)

for every functionfE ~}(X), and consequently AU

> flU.

(50.2)

Conversely, this last relation implies A -< fl, since for every function g = Uv E f/ [v E 1+(X)] we have (A,UV)

=

(AU,V)

> (flU,v) = (fl,Uv).

In particular, let fl be the balayage of A (in the classical sense) on an open set w of X; it is known that AU > flU on X, and AU = flU on w. We thus have A -< fl, so that fl is a balayage of Ain the sense of this chapter. This justifies, in part, the use of this terminology. Since the functions of f/ are positive, we have Ba; -< 0 for every x E X, and the boundary is empty. Every compact set A c X is thus a compact Shilov set. The function is hence defined for every function f E f/; this function is well known in classical potential theory, where it is called the reduite off on A.

lA

lA

Theorems on the existence of dispersions We are now going to extend Theorem 36 of the preceding section (on the existence of dilations) to the balayage defined by a convex cone f/. We begin by establishing a very general result, which implies a large number of existence theorems for dispersions; it is due to V. Strassen (117). Here first are several points of vocabulary. Denote by E a Banach space (with norm written 11 • ID and by S the collection of sublinear functions on E; consider a mapping q: w ~ qw from a measurable space (n,§) into S. We shall say that q is bounded if there exists a positive constant K such that Iqw(x) I < K IIx 11 for every x E E and every wE n*; we shall say that q is weakly measurable if the real-valued function w JV\I'+ qw(x) is measurable for every x E E. • The reader can easily verify the relation Iqw(x) - qw(y) I ~ K the functions qw on E.

Ilx - yll,

which implies the continuity of

XI, T51

244

Convex Cones and Extremal Elements

T51 THEOREM (Strassen) Let E be a separable Banach space, and (n,~ a measurable space with a complete bounded positive measure A. Let p: w .A./II'+ p w be a bounded, weakly measurable mapping from n into the set S. Denote by s the sublinear function

s(x)

=

L

PcO<x) dA(W)

(x

E

E).

(51.1)

Let x' be an element of the dual E' of E. The following properties are equivalent: (a) x' is dominated by s [(x',x) < s(x)for every x E E]; (b) There exists a bounded, weakly measurable mapping W.A./II'+ x~ from n into E' such that x' is dominated by p w for A-almost all w, and such that (x',x) =

fn (x~,x) dA.(w)

for every

x

E

E.

(51.2)

Proof It is clear that (b) implies (a). To establish the reverse implication, we first recall several facts from measure theory. A function f defined on n, with values in E, is said be to measurable if it is the uniform limit of measurable elementary functions. A measurable function f is said to be integrable if the quantity

IIfl11

=

fnl'f(w) 1 dA( w)

is finite. One denotes by 2}; the vector space of integrable functions with values in E, by L}; the quotient space of 2}; by the subspace of a.s. zero functions. It can be shown that L};, with the norm 11 • 111 defined above, is a Banach space. Since the measure A is bounded, the constant mappings from n into E are integrable. We can clearly assume, without loss of generality, that A. is a probability law. Denote by () the mapping which associates with each x E E the equivalence class of the constant function equal to x; () is an isomorphism of E onto a subspace of L};, which we identify with E in what follows. Let f be a real-valued integrable function, and x an element of E. We denote by fx the integrable function W.A./II'+ f(w)x with values -in E; it can be shown without difficulty that the vector space generated by the classes of these functions is dense in L1. Let G be a continuous linear functional, of norm K, on the normed space 21 (or, by passing to the quotient, on L1). It can be shown * that there exists a weakly measurable mapping W.A./ll'+ g'(w), with values in the ball of radius K in the Banach space E' such that for every integrable function fwith values in E we have G(f)

=

L

(g'(w),j(w» dJ.(w).

(51.3)

Now let us return to the proof of Theorem 51. Let f be an integrable function with values in E; since the real-valued function W.A./ll'+ Pw(f(w)) is obviously integrable, we can extend the sublinear function s from X to L1 by setting s(f) =

fn PCJlf(w)) dA.(w)

(fE L1:).

• Here is a quick proof of this result. For every x E E, consider the linear functional/ .A./II'+ G(/x) on the space £1, with norm at most equal to K Ilxll. The dual of V being LOO, there exists a unique element hz of LOO, with norm at most equal to K Ilxll, such that G(Jx) = E[fh z ] for every function/ E V; it is c1earthat hz +" = hz + h" and h t% = th z • Let p be a linear and isometric lifting of Loo into.flJ oo (see VII.TU); denote by H z the function p E.flJ oo , and by g' the linear functional x .A./II'+ Hz ( w) on E. The desired properties can be easily verified.

("f

Balayage Defined by a Convex Cone of Functions

245

XI, T52

The linear functional x' on E, dominated by s on E, can be extended from T2 to a linear functional ~' on L};, which is dominated by s. This linear functional can be written

~'(f) = fn (x~,f(w)

dA(W)

(f E L1J),

(51.4)

where W.J\.f\t+ x~ denotes a bounded, weakly measurable mapping from n into E'. Formula (51.2) says that ~' extends x'; the theorem will hence be established if we show that x~ is dominated by pw for almost every w. Let (Xn)nEN be a dense sequence in E; it clearly suffices to verify that a.s. (x~,xn) < Pw(xn) for every nE N. To see this, we write the relation ~'(IAXn) < s(IAxn) for every A E:F as

fA

(X~,Xn) dA(W)


Pw(Xn) dA(W),

and apply remark IL9(a). Here is a consequence of Theorem 51. Let X and Y be two compact metrizable spaces, and let A be a positive measure on X. Denote by Kthe set consisting of the positive measures on Y of mass at most equal to 1; K is compact and metrizable in the weak topology. Let X.A.f\l+ M z be a mapping from X into the set of nonempty compact convex subsets of K. Suppose that the set of pairs (x,O) E X x K such that 0 E M z is closed. For every function f E
It can easily be verified that the function

I

is bounded and upper semicontinuous on X;

we can thus set

p;.(f) = (A,l), and we obtain the following theorem.

Let fl be a positive measure on Y. The following two properties are equivalent: (a) fl(f) < pif) for every function f E
THEOREM

i

Proof It is clear that (b) implies (a). To establish the reverse implication apply the preceding theorem, taking for (Q,:F) the space X with the a-field of A-measurable sets, for E the space
=

Ix

(t.,j) dA(X)

(52.1)

for every functionfE
(trx,f)

< pi!)

for every function f E
(52.2)

Let (!n)nEN be a dense sequence in
XI, T53, 54

Convex Cones and Extremal Elements

246

Remark It frequently happens that Ell E M Il for every x E X. Then the family (T:Jxex defined by TIl if TIl belongs to M Il T'= { Il Ell otherwise

is also a Borel family, and T; E M Il for every x EX. It can, in fact, be shown that one can always find a Borel family equal a.e. to (TIl)IlEX and having this property. Here is the result which directly generalizes Theorem 36. Let X be a compact metrizable space, [/ a convex cone of continuous functions on X, containing the positive constants and closed under the operation A. We denote by -< the partial ordering associated with [/, and we call an [/-dilation any kernel T on X (with the Borel a-field) such that Ell -< EIlT for every x E X. We have the following statement. T53 THEOREM Let A and fl be two positive measures on X. The following properties are equivalent: (a) A -< fl,· (b) There exists an [/-dilation T on X such that AT = fl. Proof We apply the preceding theorem, taking Y = X and denoting by M Il the set of balayages of Ell; conditions 52(a) and 53(a) are equivalent from T45.

The theory of maximal measures 54 Let us return to the situation of No. 48: X is compact (not necessarily metrizable), [/ is a convex cone of continuous functions which is closed under A, contains the positive constants, and separates points. The partial ordering associated with [/ is then an ordering [since [/ - [/ is dense in ~(X) from the Stone-Weierstrass theorem]. It is easy to develop a theory of maximal measures that generalizes that of the preceding section. We point out a few results. Every positive measure admits a maximal balayage. A positive measure A is maximal if and only if A(f) = A(j) for every functionf E ~(X) (a consequence of Nos. 4 and T45). Suppose that X is metrizable. The boundary is then the intersection of a sequence of open sets. The measure A is maximal if and only if it is carried by the boundary. * Theorems 31 and 32, relative to the nonmetrizable case, also extend without difficulty. However, a criterion for the uniqueness of maximal balayages in the general case is not known. This extension of the theory of maximal measures seems as a matter of fact to present rather little of interest for analysis; only the theory of the boundary associated with a vector space of continuous functions, developed by Bishop and DeLeeuw (8) and Bauer (3), has received important applications. But one can show [see Section 6 of ChoquetMeyer (44)] that this theory reduces directly to that of convex cones with compact base. We refer the reader to the article cited for more details.

* The proof of this theorem is slightly different from that of the corresponding theorem T24. One notes first (from T46, for example) that the mapping f.1l/ll+! is continuous under the topology of uniform convergence. Then let (fn)nEN be a dense sequence in ~(X); A will be maximal if and only if A(fn) = M!n) for every n, hence if A is carried by the sets {x: fn(x) = !n(X)} whose intersection is equal to the boundary.

APPENDIX

Chapters I and IT The results of these two chapters should be considered classical. Those given without proof figure in works on measure theory, e.g., Dunford and Schwartz (67), Halmos (72), Loeve (89), and Bourbaki (18) for that which concerns Radon measures. Bourbaki is alone in establishing Theorems 11.35 and 11.36, which are indispensable for potential theory. The notion of the Baire a-field has been popularized by Halmos (72), from which come Theorems 29 and 30. The ecart that defines convergence in probability in No. 22 was communicated by P. Cartier. The theorems on uniform integrability are due to Vitali (1907) with the exception of T22 (La Vallee Poussin, 1915) and of the Dunford-Pettis criterion [established in Dunford (65), and generalized in Dunford-Pettis (66)]. The treatment of conditional independence follows Loeve (89). Chapter ITI The theorems on compact pavings have an obvious topological interpretation [Theorem 4, due to Alexander, is used by Kelley to prove Tykhonov's theorem; see Kelley (82) pp. 139-143]. The notion of a semicompact paving does not admit an analogous interpretation, but has no particular interest either. The presentation of the theory of analytic sets is new. The theorem on separating Suslin sets is adapted from Kuratowski (86). The theorems on uniformization of Borel or analytic sets would be out of our subject. They are found in Kuratowski (86), Bourbaki (14), and especially Sion (109). In a general manner, Sion's articles are recommended for readers who desire to go deeper into the theory of analytic sets. Article (10) of Blackwell shows the natural character of the theory of analytic sets in probability theory: they are seen to appear whenever one wishes to eliminate certain pathological characteristics of general probability spaces. The "topological" form of Choquet's theorem (as opposed to the "abstract" form) is given in Choquet (25)-a work the richness of which we are far from exhausting. It is also the object of two excellent treatments in Brelot (21) and (22), where one will find as well its main applications to potential theory. Articles (28)-(31) of Choquet treat related theorems: Definitions of capacities, applications, theorems of capacitability under weaker 247

Appendix

248

hypotheses. Strassen indicates in (116) certain connections between information theory and that of capacities. The theory of regular measures is essentially due to Alexandrov. It has also been the object of publications by Marczewski and by Ryll-Nardzewski in Fundamenta Mathematica (1953). This theory recently aroused a new mterest. It was first discovered that regular measures on quasicompact (i.e. non-Hausdorff compact) spaces are a natural tool for harmonic analysis on non-Abe1ian groups. On the other hand, we had communication of unpublished papers by L. Schwartz and N. Bourbaki showing that locally bounded regular measures on the Borel field of a Hausdorff space share practically all the "nice" properties of Radon measures. Probabilists are well acquainted with regular measures on Polish spaces, because of a famous paper by Prokhorov. * Chapter IV

The proof of Theorem 9, familiar to readers of Bourbaki, has been popularized in the United States by a paper by Nelson (Ann. Math., 69, 1959). Example 11 is taken from Doob, as is everything having to do with separability, with numbers of upcrossings and downcrossings, etc. [see the classical presentation of the subject in Doob (56)]. The proof of Theorem 22 is very close to the reasoning of Bourbaki concerning "regulated" functions (Functions of Real Variables, Chap. 2, 1, Theorem 3). The appendix to paragraph 2 comes from Meyer (94), and certain points there have been since developed by Feldman (69); there is unfortunately no known application for these theorems. It should be pointed out also that Chung has defined an interesting notion of right-separability for processes, stronger than ordinary separability. See (47). Stopping times have been used, without formal definition, since the beginning of the theory of processes. The idea appeared entirely clearly for the first time with Doob in 1936. Doob has employed them systematically in martingale theory, together with increasing families of a-fields. These ideas have passed from there to the theory of Markov processes, where they play an essential role. Most of the results about measurable processes and stopping times are of a very elementary nature. Many of them will be found in Dynkin (68) where they appear as lemmas in the work on Markov processes. The systematic study of these ideas, undertaken independently by Chung and Doob, and the author, has led to the discovery of a certain number of new and useful results. The work of Chung and Doob (47) goes a great deal farther than ours. On this subject of stopping times, see also the papers by Courrege and Priouret cited in "Additions to the Bibliography." Theorem 53 on hitting times is adapted from a work of Blackwell and Freedman in the probability seminar at Berkeley. It will be noted that the capacitability theorem is not used in its full strength: it is only necessary to know that an analytic set is measurable for every measure [Saks (107), p. 50]. Chapter V

With the exception of several results pointed out in the text, all the theorems of this chapter are due to Doob [(56) or (62)]; we owe some of them, unpublished, to a personal communication. • "Convergence of random processes and limit theorems in probability theory," Theor. Prob. Appl., 1 (1956).

249

Appendix

The general convergence theorems (arbitrary filtering index set) have been studied by Helms (73), Chow (45) and (46), Krickeberg (84), and Krickeberg and Pauc (85). A detailed bibliography will be found at the end of the last article. We have cited here only a result of Helms. Recent study is based on the introduction of "Vitali conditions" implying a kind of almost sure convergence. The theory is unfortunately far from having the simplicity of the convergence theorems which we present here. Chapter VI Again this is entirely due to Doob, with the exception of Theorems 15 and 16, published here for the first time [but 15 is part of the folklore of martingales, and 16 is very close to a result of Ray (106) on Markov processes]. The class (D) was invented by Doob in (56), for certain martingales. Theorems 19-20 are taken from Meyer (96). Applications of martingale theory to potential theory will be found in Doob (57)-(60). Chapter VII The theorem on the existence of the Doob decomposition was first worked out for martingales for the form (/0 Xt), where (Xt ) denotes a Markov process, and / an excessive function. The desired increasing process is then a Markov additive functional. In this form, after a first fundamental study of Volkonskii (120), the problem was solved indev pendently by Sur (118) and Meyer (95); in the particular case where (Xt ) is Brownian motion, other solutions are due to Ventzel, * and McKean and Tanaka.t The uniqueness theorem (in a slightly different form, better adapted to the theory of Markov processes) is established in Meyer (95). The extension to martingales is made in Meyer (96) and (97), to the theory of energy in Meyer (100). We have known of the work of S. Watanabe on Markov processes, in which several results on energy are found and used. Finally, we point out that a more general problem than that of the Doob decomposition (the decomposition of a process into a sum of a martingale and a process whose paths are of bounded variation) has been studied, and in part resolved, by Fisk (71). Fisk's statement is unfortunately not simple. K. Ito and S. Watanabe have introduced an extremely interesting decomposition of a positive supermartingale into a product of a martingale and a natural decreasing process. This may be found in their paper cited in "Additions to the Bibliography." Chapter VIII Some bibliographic comments on Section 1 are found in the text. The applications to the general theory of processes constitute a new draft of Meyer (99), freed of a certain number of mistakes and obscurities. Section 3 is a new draft of the end of Meyer (97). Courrege (48), (49) can be consulted for the applications to stochastic integrals. Chapters IX and X The results of Section 1 are classical. The origin of the results of Section 2 is explained in the text. Theorem 27 (or 70) is probably new. The Appendix to Section 2 makes clear • Dokl. Akad. Nauk. SSSR, 137 (1961), 17-20. t Mem. Coil. Sci. Univ. Kyoto 33 (1960), 479-506.

Appendix

250

the connections between potential theory and Snell's paper (115); the relations between the theory of elementary kernels and the theory of discrete martingales can in fact be pursued farther, so as to give the Doob decomposition in the discrete case (VII, Section 1). The results of Section 3 are due to Hunt in the case of a sub-Markov semigroup; we use systematically resolvents instead of semigroups. Theorem 68 is due to Hunt in the subMarkov case, and to Deny in the general case. The origin of the theorems in Chapter X is explained in the text. The idea of passing from the sub-Markov case to the Markov case in No. 16, by the adjoining of a point at infinity (an idea which will be found again later) is ~orrowed from Doob (1957). The results of these two chapters can be completed on several points. We do not study here the connections between the diverse "principles" of potential theory: in particular, we accord little attention to the aspect of measures and balayage, and we leave aside everything having to do with energy. The reader interested in these questions should consult the article (42) of Choquet and Deny (potential theory on a finite set), and the volumes of the Seminar Brelot-Choquet-Deny. The results of Chapter X, on the other hand, are extended to convolution kernels in Deny (54) and (50). Chapter XI We have profited, in writing this chapter, from a seminar on the Choquet theorem directed by R. Phelps at the University of Washington. The form of the Hahn-Banach theorem given in No. 2 is evidently the "good" one; it is due to Banach himself. We owe Nos. 8(e) and 9(e) to a personal communication from Choquet. The sources of Section 2 are indicated in the text. Herve (74) gives another very simple proof of Choquet's existence theorem in the metrizable case. The first simplified proof of the uniqueness theorem (in the form: uniqueness of the maximal measure representing a point, for a cone which is a lattice) was given by Meyer, but has never been published, since it was immediately improved very considerably by Choquet. In this form, it figures in Choquet and Meyer (44). The connections between the theory of Loomis and that of Choquet (including the existence of dilations) have been studied by Fell and Meyer, but their articles have not been published because of the discovery by Cartier of the identity of the orders of Loomis and Choquet [see Meyer (101)]. The theory of Section 3 can be extended to cones of l.s.c. functions. This extension has been done by Edwards in a paper to appear soon. Mokobodzki (104) knows a nice way of reducing the l.s.c. case to the continuous one, and has some very interesting results which generalize to these cones his theorem on convex functions (T7).

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69. 70. 71. 72. 73.

254

. "A Probability Approach to the Heat Equation," Trans. Am. Math. Soc., 80 (1955), 216-280. . "Probability Methods Applied to the First Boundary Value Problem," Proceedings of the Third Berkeley Symposium on Math. State and Probability, 1954-55, 2, 49-80 (Los Angeles: University of California Press, Berkeley, 1956). . "Probability Theory and the First Boundary Value Problem," IllinoisJ. Math., 2 (1958), 19-36. . "Discrete Potential Theory and Boundaries," J. of Math. Mech., 8 (1959), 433-458. . "Notes on Martingale Theory," Proceedings ofthe Fourth Berkeley Symposium on Math. State andProbe (Los Angeles: University of California Press, Berkeley, 1961), Vol. 2, 95-103. Doob, J. L., J. L. Snell, and R. E. Williamson. Application of Boundary Theory to Sums ofIndependent Random Variables, "Contributions to Probability and Statistics," 182-197 (Stanford: Stanford Univ. Press, 1960). Dubins, L. E. "Rises and Upcrossings of Nonnegative Martingales," Illinois J. Math., 6 (1962), 226-241. Dunford, N. "A Mean Ergodic Theorem," Duke Math. J., 5 (1939), 635-646. Dunford, N., and B. J. Pettis. "Linear Operations on Summable Functions," Trans. Am. Math. Soc., 47 (1940), 323-392. Dunford, N., and J. T. Schwartz. Linear Operators. Part I: General Theory (New York: Interscience Publishers, 1963). Dynkin, E. B. Fundamentals ofthe Theory ofMarkov Processes (in Russian) (Moscow: Fizmatgiz, 1959). The references in the text are to the English translation: Theory of Markov Processes, translated by D. E. Brown (London: Pergamon Press; Englewood Cliffs, N.J.: Prentice-Hall, 1961). Feldman, J., "On Measurability of Stochastic Processes in Product Space," Pacific J. of M., 12 (1962), 113-120. Feller, W. "A Simple Proof for Renewal Theorems," Comm. Pure Appl. Math., 14 (1961), 285-293. Fisk, D. L. Quasimartingales and Stochastic Integrals (Department of Mathematics Research Monographs, Kent State University, Kent, Ohio, 1964). Halmos, P. R. Measure Theory (Princeton, N.J.: D. Van Nostrand Co., 1950). Helms, L. L. (see also Johnson and Helms) "Mean Convergence of Martingales," Trans. Am. Math. Soc., 87 (1958), 439-446.

74. Herve, M. "Sur les representations integrales a l'aide des points extremaux dans un ensemble compact convexe metrisable," Compt. Rend., 253 (1961), 366-368. 75. Hewitt, E. "Integration by Parts for Stieltjes Integrals," Am. Math. Monthly, 67 (1960), 419-423. 76. Hewitt, E., and L. J. Savage. "Symmetric Measures on Cartesian Products," Trans. Am. Math. Soc., 80 (1955), 470-501. 77. Hille, E., and R. S. Phillips. Functional Analysis and Semigroups (Providence, R.I.: American Mathematical Society colloquium publications, 1957). 78. Hunt, G. A. "Markoff Processes and Potentials I, 11," Illinois J. Math., 1 (1957), 44-93, 316-362. 79. Ionescu Tulcea, C. A. "On the Lifting Property 1," J. Math. Anal. Appl., 3 (1961) 537-546.

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80. Johnson, G., and L. L. Helms. "Class (D) Supermartingales," Bull. Am. Math. Soc., 69 (1963), 59-62. 81. Kendall, D. G. "Extreme-Point Methods in Stochastic Analysis," Z. Wahrscheinlichkeitstheorie, 1 (1963), 295-300. 82. Kelley, J. L. General Topology (Princeton, N.J.: D. Van Nostrand Co., 1955). 83. Kelley, J. L., I. Namioka et al. Linear Topological Spaces (Princeton, N.J.: D. Van Nostrand Co., 1964). 84. Krickeberg, K. "Convergence of Martingales with a Directed Index Set," Trans. Am. Math. Soc., 83 (1956), 313-337. 85. Krickeberg, K., and C. Pauc. "Martingales et Derivation," Bull. Soc. Math. 91 0963), 455-544. 86. Kuratowski, C. Topologie 1, 3rd ed., Monografje Matematyczne Tom XX (Warzawa: Polskie Towarzystwo Matematyczne, 1952). 87. Lion, G. "Construction du semi-groupe associe a un noyau de Hunt," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny. Institut Henri Poincare, Paris, 5e annee 1960-61, 9 pages. 88. . "Theoreme de representation d'un noyau par l'integrale d'un semi-groupe," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny, Institut Henri Poincare, Paris, 6e annee, 1961-62, Vo!. 1, 9 pages. 89. Loeve, M. Probability Theory, 2nd ed. (Princeton, N.J.: D. Van Nostrand Co., 1960). 90. Loomis, L. H. An Introduction to Abstract Harmonic Analysis (Princeton, N.J.: D. Van Nostrand Co., 1953). 91. . "Unique Direct Integral Decompositions on Convex Sets," Am. J. Math., 84 (1962), 509-526. 92. Maharam, D. "On a Theorem of Von Neumann," Proc. Am. Math. Soc., 9 (1958), 987-994. 93. . "On Two Theorems of Jessen," Proc. Am. Math. Soc., 9 (1958), 995-999. 94. Meyer, P. A. "Separabilite des processus stochastiques," Compt. Rend., 248 (1959), 3106-3107. 95. . "Fonctionnelles multiplicatives et additives de Markov," Ann. Inst. Fourier, Grenoble, 12 (1962), 125-230. 96. . "A Decomposition Theorem for Supermartingales," Illinois J. Math., 6 (1962), 193-205. 97. . "Decomposition of Supermartingales : The Uniqueness Theorem," Illinois J. Math., 7 (1963),1-17. 98. . "Sur les demonstrations nouvelles du theoreme de Choquet," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny. Institut Henri Poincare, Paris 6e annee 1961-62, Vo!. 2, 9 pages. 99. . "Une presentation de la theoriedesensembles sousliniens. Applicationaux processus stochastiques," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny. Institut Henri Poincare, Paris, 7e annee, 17 pages. 100. . "Interpretation probabiliste de la notion d'energie," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny. Institut Henri Poincare, Paris 7e annee, 1962-63. 101. - - - . "Progres recents dans le theorie des cones convexes a base compacte," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny, Institut Henri Poincare, Paris, 8e annee, 1963-64, 10 pages.

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102. Mokobodzki, G. "Balayzge defini par un cone convexe de fonctions numeriques sur un espace compact," Compt. Rend., 254 (1962), 803-805. 103. . "Quelques proprietes des fonctions numeriques (s.c.i. ou s.c.s.) sur un ensemble convexe compact," presented at Seminaire de Theorie du Potentiel, directed by M. Brelot, G. Choquet, and J. Deny. Institut Henri Poincare, Paris, 6e annee, 1961-62, 3 pages. 104. "Generic unpublished paper." 105. Neveu, J. Bases mathematiques du calcul des probabilites (Paris: Masson et Cie, 1964). 106. Ray, D. "Resolvents, Transition Functions, and Strongly Markovian Processes," Annals Math., 70 (1959), 43-72. 107. Saks, S. Theory of the Integral. 2nd ed. Translated by L. C. Young. (Warszawa: Monografie Matematyczne nO 7, 1937). 108. Sherman, S. "On a Theorem of Hardy, Littlewood, Polya, and Blackwell," Proc. Nat. Acad. Sciences, 37 (1951), 826-831. 109. Sion, M. "On Uniformization of Sets in Topological Spaces," Trans. Am. Math. Soc., 96 (1960), 237-246. 110. . "Topological and Measure Theoretic Properties of Analytic Sets," Proc. Am. Math. Soc., 11 (1960), 769-776. 111. . "On Analytic Sets in Topological Spaces," Trans. Am. Math. Soc., 96 (1960), 341-354. 112. . "Continuous Images of Borel Sets," Proc. Am. Math. Soc., 12 (1961), 385-391. 113. . "On Capacitability and Measurability," Ann. Inst. Fourier., Grenoble, 13 (1963), 83-98. 114. Sion, M., and D. W. Bressler. "The Current Theory of Analytic Sets," Can. J. Math., 16 (1964), 207-230. 115. Snell, J. L. "On a Theorem of Hardy, Littlewood, Polya, and Blackwell," Proc. Nat. A cad. Sciences, 37 (1951), 826-831. 116. Strassen, V. "Messfehler und Information," Z. Wahrscheinlichkeitstheorie, 2 (1964), 273-305. 117. . "The Existence of Probability Measures with Given Marginals," Ann. M. Stat., 36 (1965), 423-439. 118. Sur, M. G. "Continuous Additive Functionals of a Markov Process," Dokl. Akad. Nauk. SSSR, 137 (1961), 800-803 [Soviet Math. 2 (1961), 365-368]. 119. Suslin, M. "Sur une definition des ensembles measurables (B) sans nombres transfinis," Compt. Rend., 164 (1971), 88-91. 120. Volkonskii, V. A. "Additive Functionals of Markov Processes," Tr. Mosk. Mat. Oble. 9 (1960), 143-189. 121. Yosida, K. Lectures on Semi-Group Theory and Its Application to Cauchy's Problem in Partial Differential Equations (Bombay: Tata Institute of Fundamental Research, 1957). Additions to the bibliography The following papers appeared while the book was in press, and they could not be inserted in the bibliography because of the numbering system adopted in this series.

257

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1. Volume 15 (1965) of the Ann. Inst. Fourier contains the Proceedings of the Colloquium on Potential Theory (Orsay, June 1964). Among the papers that might interest the reader, because of their relationship with this book, we may cite: Deny, J. "Principe complet du maximum et contractions" (259-274). Fuglede, B. "Le theoreme du minimax et la theorie fine du potentiel" (65-88). Ho, K., and S. Watanabe. "Transformation of Markov Processes by Multiplicative Functionals" (13-30). Neveu, J. "Relations entre la theorie des martingales et la theorie ergodique" (31-42). 2. The following papers contain results on stopping times, which are very useful complements to the theory given in Chapter IV: Courrege, Ph., and P. Priouret. "Temps d'arret d'une fonction aleatoire, theoremes de decomposition," Compt. Rend., 259(1965), 3933-3935. - - - . (Same title, to appear in 1965 in Publ. Inst. Stat. Univers. de Paris.)

INDEX OF NOTATIONS

(Numbers refer to Section in which notation is first used)

IA (indicator function of A), 1.4

T(A) (O'-field generated by A), 1.6 T(h, i El) (O'-field generated by the ft.), 1.7 B(E) (Borel O'-field), 1.10 B o(E) (Baire 0'- field), 1.10 a.s. (almost surely), 11.2 Ea; (unit mass), 11.4 E[f] (expectation), 11.5 LP, LP, 11/ll p , Loo, LOO, 11/1100,11.8 O'(Ll, LOO), 0'(L2, L2), 11.10 P ® Q (product law), 11.14

I:-

dQ(x) (integral oflaws), IU5

(p,,/), p,(f), 11.34 ... Bp (,u-measurable sets), 11.37 _ B u (universally measurable sets), 11.37 - E[X F], E[X ft., i E I] (conditional expectation), 11.46 -P[A F], P[A ft., i E I] (conditional probability), 11.46 -E[X FIG], 11.46 - A(F) (F-analytic sets), 111.7 1* (outer capacity), 111.23 X H , X t _ (right and left limits), IV.25

I I

IV.30 _Fp{Ta stopping time), IV.35 t(n), H(n), IV.43 X H (X a process, H a random variable), IV.48 DB (hitting time), IV.53 « (stronger order), VII.8 TA (T a stopping time, A prior to T), VII.38 ... V Fs" (Sn an increasing sequence of stopping times), VIII.38 ..(Ft)teR , F oo , F H , +

n

e[(Xt )] (energy), VII.57 259

Index of Notations [8, T[, [8, T] (stochastic intervals), VIII.l3 N(x, A) (kernel), IX.1 Nf(extension of N to functions), IX.1 N(x,f), Nfx, IX.5 ftN (ft a measure), IX.6 MN (composed kernels), IX.8 I (identity kernel), IX.13 Nn,IX.l4 (regularization of a supermedian function), IX.58 b(ft) (barycenter), XI.8 r(ft) (resultant), XI.15 -<, XI.l6 (also XI.41) f (upper envelope), XI.18 OK (extreme points), XI.23 osX (boundary), XI.42

1

260

INDEX

.~~,

A""-B, 2

A /::; B, 2 +Abelain group, 151 additive function, 41 additivity, countable, 12 additivity theorems, martingales and, 78-79 )< A(~)-analytic set, 34 affine function, 29, 222-223 affine mapping, 226 Alexander, J. W., 247 Alexandrov, A. D., 47 n., 248 Alexandrov compactification, 215, 242 analytic set, disjoint, 38; of R, 37 a.s. ("almost surely"), 12 £i(E),8

Baire function, 217 Baire set, 23, 233; compact, 46 Baire a-field, 8, 21, 23, 49, 51, 54, 233, 247 balayage, 1, 227, 229, 242-243; defined by convex cone of functions, 239; theory of, 219 balayee, 186 Banach, S., 250 Banach space, 154, 174,206,208,210,216,222, 243-244,250 barycenter, of measure, 223-224, 236, 238 base space, 52 Bauer, H., 225, 226 n., 241, 246 Bishop, E., 227, 233-234 Blackwell, D., 30, 37-38, 71, 88, 234, 247 Blackwell spaces, 37 Borel, Emile, 247 Borel family, 245-246 Borel function, 108, 147, 214, 231-232, 245

Borel-Lebesgue property, 46 Borel-Lebesgue theorem, 204 Borel mapping, 35, 38 Borel set, 36, 69 Borel a-fields, v, 7-10, 12 n., 13, 21, 23, 25, 35, 54, 176-177, 235, 246, 248; see also a-field Bourbaki, Nicholai (pseud.), 2, 13, 21, 24, 30, 32 n., 33 n., 35 n., 37, 39, 59 n., 175, 203 n., 221-223, 235 n., 247-248 Brelot, M., v, 39,247,250 Brownian motion, 103, 147, 149 canonical process, 53-55 capacities, 39-47; application to measure theory, 44-45; construction of, 40; right-continuous, 45 Caratheodory theorem, 22, 48-49 Cartier, P., 229, 234, 247, 250 Cauchy criterion, 86 Cauchysequence, 86,222
Choquet, G., v, 32, 34 n., 40, 175, 220, 223-224, 227,232-235,241,246-247,250 Choquet-Deny theorem, 151-153 Choquet ~-capacity, 39,42,44,46, 163 Choquettheorem, 32,35,40,71,219,225, 227238,250 Chow, Y. S., 84, 249 Chung, K. L., 68 n., 73, 248 compactification, 35, 215, 242 compact convex set, 219-225; extreme points of, 223-224 compact interval, 60 compactness criterion, 20 compact pavings, 32-33 261

262 compact set, 46-47, 55; metrizable, 225 )l£.rJ~Eber1in-Shmuliantheorem, 20 compact space, 25 energy, computation of, 141; monotone concomplete maximum principle, 1, 184 conditional expectation, 27, 120; fundamental vergence and, 142-146; results on, 140-148 properties of, 28-30; generalized, 28 equilibrium potential, 183 equivalences, 52-53, 58 conditional probability, 28 events, 7 conditioning, 27-31 excessive functions, 179-180, 182, 187-190, 195conical measure, 226 197; reduite of, 182-184 conjugate exponents, 88 excessive measures, 184-186 continuous increasing processes, 123-128 convergence, ~8e'8'- tMOt~<>,-Mti, 87-88; existence theorem, 119 Lebesgue's theorem of, 144 (see also Lebes- expectation, conditional, 27; mathematical, 12gue's theorem); monotone, 45; of random 31 extreme points, 229 variables, 14-15 convergence theorem, applications of, 149-156 convex cones, with compact base, 226; theory !F, 22: 65, 68 faces, family of, 225 of, 1, 219, 239 convexity theorem, martingales and, 78-79 family, countable, 33; of faces, 225; left- and right-continuous, 70-71, 139; of a-fields, 65, convex sets, compact, 219-225 72,85, 104, 110, 114, 129, 132, 152, 186,225; convolution kernel, 179 subdivision of, 230 countable set, 55-56, 62 !F-analytic sets, 34-36, 40 Courrege, P., 113 n., 248 Fatou's lemma, 13, 18, 84, 88, 96, 153, 169, 191 Fell, J. M. G., vi, 229, 234, 250 Daniell extension theorem, 21, 24, 44-45 decomposition, Reisz, see Reisz decomposition Feller, W., 151 Feller semigroup, 213-214, 216 DeLeeuw, K., 227, 233-234 filtering set, 231 denseness property, 46 Deny,J., v, 151, 171, 179, 184,198,205,241,250 finite intersection property, 32 first canonical process, 53 Denzel, J. M. G., vi diffusion-kernel, 216 Fisk, D. L., 249 dilations, existence of, 234-235 !F-measurable elements, 47 Dini's lemma, 204, 209, 221, 242 !F-measurable sets, 7 Dirichlet integral, 146 !F-Suslin set, 35 n. discontinuity, times of, 129 Fubini's theorem, 15, 109, 114 function, additive and subtractive, 44; affine, dispersion (dispersion-kernel), 175-176, 214 29, 222-223; bounded, 238; excessive, 179domination principle or theorem, 13, 184, 197180, 182, 187-190, 195-197; lower semi198,202-205 continuous (l.s.c.), 25, 51, 60 n., 176, 203, 205, Doob, J. L., 1, 10-11, 30, 55-56, 63, 68 n., 77, 209, 223; pseudo-reduite of, 200; reduite of, 84,87,90, 149, 151, 179-180,248-249 Doob decomposition, existence of, 122-123; 182-184; step (elementary random variable), 9,20; sublinear, 219, 230; supermedian, 187uniqueness of, 111-113 190; unbounded, 238; upper semicontinuous Doob's inequality, 82 Doob's theorem, 87-88 (u.s.c), ~04-~5, 222-223, ').27, 232-233 Co."...~~ h1.D·n,-o.t'OlA...<:..,. Dubins, L. E., 30, 79, 83-84, 88 Dunford, N., 14-15,20,23 n., 30,210,220,222, Gabriel, P., vi, 60 n. Getoor, R., vi 247 Dunford-Pettis compactness criterion, 20 l-t J(/, 11, 16-17 Dynkin, E. B., 11 Hahn-Banach theorem, 2 n., 14, 211, 239, 241, 250 Halmos, P. R., 247

n..

Index

263 Hausdorff topological space, 33, 45-46, 51, 59, 70,248 Helms, L. L., 86, 96 n., 101, 102 n., 103, 249 Herve, M., 250 Hewitt, E., 114, 149-150 Hille, E., 210 Hille-Yosida theorem, 210-211, 213, 217 Hirsch, W. M., vi hitting times, measurability of, 71 HOlder's inequality, 88 Hunt, G. A., 1, 71, 82, 202, 208, 250 Hunt's theorem, 214 hyperplane, 222-224

w

Ii,--

kernel function, 178 Kolmogorov's inequality, 82 Krein-Milman theorem, 225-226, 235, 237 Krickeberg, K., 249 Laplace transform, 213, 216, 218, 237 lattice, 3, 236 law, induced, 150 LCC (locally compact) spaces, 8-9, 37, 68, 70, 160, 177, 179 Lebesgue measure, 8, 13, 55 Lebesgue's theorem, 13, 18-19, 21, 45, 89, 100, 106, 108, 111, 113, 115, 123, 126, 144, 174, 181, 214, 217 Levy, Paul, 85 lifting theorem, 154-156 Lion, G., v locally compact space 8-9; see also LCC spaces, Loeve, M., 150 n., 210, 217 Loomis, L. H., 21, 229, 231, 234, 250 l.s.c. (lower semicontinuous) function, 25, 51, 60 n., 176, 203, 205, 209, 223 Lusin spaces, 37

image probability laws, 15, 53 increasing process, 105-111; continuous and discontinuous parts of, 107; integrable, 111113; integration with respect to, 109-111; "natural," 111-113; stochastic process and, 108-109 independence, 26 index set, 48 indicator function, 28 induction hypothesis, 43 inequalities, fundamental, in martingale theory, It 1, 22 1(E),3 . 80-84 ~nf (f, g)~ ~ mapping, 57, 59, 74, 222, 235, 245; affine, 226; mtegrabIlIty, 19-20 i 1_ Borel 35' coordinate 62' identity 7 23' . 'D "".. t,~\ ..u.... et ~l'.'U ..\ \,.,2 . '.' " '" Hlt~8U1t'9" vOW~r ~ r mclusIon 23' kernel as 173' measurable 35 . . h ' . . c . f 12 31 ,Uf' " " , , ~ntegratlOn t ~o~y, resume 0, 62; notation for, 3; positive linear, 176-177; mternally neglIgIble sets, 22-24 universally measurable 178 intersection, nonempty, 37 " Maraham D. 154 ' intersection property, finite, 32 I" Markov additive functionals 124 invariant function, 181 Markov process, 30, 61, 77, 108, 248-249; ~onescu-T~lcea theorem, 154 theory of, 127 Isometry, lInear, 155 Markov semigroup 1 210 215 Ito, K., 123,249 martingale, 1; ac~ssible ~topping times and, Jensen's inequality, 29, 79, 85-86, 163 133-134: applications of theory, 149-164; Johnson, G., 96, 101, 102 n., 103 continuous parameter, 93-103; countable case, 84c-90; defined, 77; examples of, 78; Kelley, J. L., 247 fundamental inequalities in, 80-84; generkernel(s), 1, 173-186; composition of, 175; alized, 77; monotone convergence and, 146continuous, 176, 179, 203; convolution, 178; 148; quasi-left-continuous, 164c-165, 169; defined, 173; diffusion, 203; dispersion, 175right-continuous, 94c-l00, 110, 115, 124, 135, 163-164; square-integrable, 163-165, 167; 176,214; domination principle and, 202-205; elementary, 180; on locally compact space, uniformly integrable, 84c-86, 90-92, 101-103; 175; Markov, 217, 234-235; Newtonian, 103, see also submartingale; supermartingale 140, 178; positive, 199; potential theory of, martingale theory~ v, 77-164, 186-187, 249; 179-187; semigroups of, 187; sub-Markov, applications of, i49-164 173, 179, 183, 186-187, 197, 215; uniform, mathematical eXPectations, and probability laws, 12-31 206,208

"1 "

Index

264

maximal measures, 228-229, 246 paving of, 158-163 measurable processes, 34-74 Pettis, B. J., 20, 241, 247 measurable space, 7, 65 p-excessive function, 187 measure, barycenter of, 223-224, 236, 238; Phelps, R., 250 construction of, 21-24; excessive, 184-186; p-invariant function, 187 maximal, 228-229, 246-247; as reduite of P-negligible set, 22 function, 186; regular, 248; application ,of polar set, 1 capacities to, 44-45; complements to, 32-51; Polish space, 35, 38 convergence theorems and, 153-154 ~polyhedron, faces of, 225 n. metrizable space, 23,50,52,55,61-62; compact, potential, bounded, 123, 128; class D, 123; 59 defined, 180; equilibrium, 183; generation of, 106; Newtonian, 147, 178, 195, 239; regular, Meyer, P. A., 124, 126 n., 127 n., 156 n., 163 n., 229,233-234,246,248-250 123; right-continuous, 117 minorant, 181 potential theory, v, 1; analogies with martingale theory, 186; analytic tools, of 173-246; modification, separable, 56 "general," 239; "minimum principle" of, 99; Mokobodzki, G., v, 33 n., 221, 227, 234, 241, 1\1'\.(" \ " 250 Co y)'\ bPI') L..O. . Newtonian, 239 ,'M:~·l'"fm'Mtotoneconv~rgen~, 146-148;energyand, probability, conditional, 28; defined, 12; as 142-146; see also Lebesgue's theorem random variable, 27 >t p" 24 probability law, 54; integration of, 15; and mathematical expectations, 12-31; measures p,-measurable sets, 25 and, 21; projective system of, 54 natural increasing process, 136, 141; accessible probability law P, 21 stopping times and, 135-139; difference of, probability space, 13, 22, 27, 30, 52 probability theory, 1, 7-74,202 126 processes, construction of 53-54; equivalent Neveu, J., 32, 211 52; general properties of 52-55; increasing, Newtonian kernel, 103, 140, 178 see increasing processes; martingale as, 104; Newtonian outer capacity, 39, 42 martingale theory and, 156-164; measurable, Newtonian potential, 147, 178, 195, 239 see measurable processes; progressively measnonempty intersection..,37 urable, 68 n.; separable, 55-62; a-fields and, nonmetrizable spaces, 8 65-66; state of, 52 non-negative random variable, notation, explanation of, 2-3, 7, 11, 128-129 product law, 26 product paving, 32-33 product a-field, 9 a,7 product vector space, 226 .,.:lprojective limit, 48 open interval, 58 projective system, 48 open sets, 46-47 pseudo-reduite, of function, 200 4 "sal sampling thjorem, 90-92 Pyke, R., vi orthogonal variabl114 oscillatory discontmuity, 59-62, 68, 100,217 Q,22 outer capacity, 39,42 q-excessive regularization, 217 q-supermedian function, 216 P, 12, 21 path, 52-53, 104, 107; regularity properties of, 93-98; right-continuous, 109; oscillatory R, analytic sets of, 37; real line, 8, 10 Radon law, 51 discontinuities of, 59-62 Radon measure, v, 3, 21, 24-25, 32, 175-177, Pauc, C., 249 179, 185, 203 n., 214, 233, 247-248; compaved set, 32, 34 pletion of, 25; excessive, 184 paving, 32; compact, 32-33; product, 32-33; Radon-Nikodym theorem, 14, 20, 153-154 semicompact, 32, 35, 47, 49, 62, 64

265

<

Index

random variable, 13, 52, 58, 65, 67, 69, 74, 77, 105, 109, 138, 168; convergence of, 14; definition of, 7; elementary, 9; independent, 150; integrable, 29-30; measurable, 30; nonnegative, 77; real-valued, 9-11, 26; symmetric~ 149-.151; uniformly integrable, 16-21, 86, 116-117 Ray, D., 216, 249 Ray resolvents, 216-218 real line, 8, 10 real-valued function, 9, 22 reduite, 182-184, 243 regularity, hypothesis of, 62 regularization, q-excessive, 217 regular measures, 47-51; compact paving and, 50; defined, 47; projective limits of, 48; theory of, 248 resolvent, 188-189; closed, 199; closed Markov, 206; proper, 198; Ray, 216-218; strongly continuous, 211; sub-Markov, 189, 199, 202, 206, 208, 211 resolvent identities, 192 r-excessive function, 195 Riesz decomposition, 89, 97, 181, 185, 195 right-continuous capacities, 45 right-continuous functions, 99-100 right-continuous martingale, 94-100, 110, 115, 124, 135, 163-164 r-potential, 195 r-supermedian function, 190

}(~ Saks, S., 248 Savage, L. J., 149-150 Schwartz, J. T., 14-15, 20, 23 n., 30, 210, 220, 222, 248 Schwartz's inequality, 126 semicompact paving, 32-34, 47, 49, 64 semicontinuous function, 25 semigroup, construction of, 210-218; Feller, 213-214; of kernels, 187; Markov, 210, 215; strongly continuous, 210; sub-Markov, 210, 212; see also l.s.c.; u.s.c. separability, 55, 59, 62-64 separable processes, 55-62; defined, 55-56; example of, 56; universal, 56-59 separating sets, 56-59 separation theorem, 36 n. set-function, 12, 40 sets, see under adjective as in compact set, index set, etc.

Sherman, S., 234 Shilov boundary, 241 n. Shilov set, 240-243 a-algebra, 7 a-compact space, 184, 215 a-fields, 7-9, 20, 24, 28, 30, 32, 35, 39, 48, 50, 53, 66,68,78,139,149,155; enlargement of, 21; examples of, 8; family of, 65, 85 (see also family); generated by collection of functions, 8; independent, 26; product, 9 singleton, 38 Snell, J. L., 151, 250 space, metrizable, 23, 50,52,55,61-62; compact, 59 &.tate space, 52 stochastic boundary function, 89 stochastic Dirichlet solution, 89-90, 97 stochastic intervals, 156 stochastic processes, 21, 34 n., 52-74, 158, 186; defined, 52 stopping times, 165; accessible, 130-140; classification of, 128-140; examples of, 6768; inaccessible, 139 ; Gi} I' 1 sampling theorem and, 99; properties of, 66-67 ; superposition of chains of, 74; systems and chains of, 72-74 Stone-Weierstrass theorem, 2, 23, 216, 221,246 Strassen, V., 243-244, 248 strict positivity hypothesis, 203 subdivision, defined, 230 sublinear function, 219-221, 230 sub-Markov kernel, 173, 179, 183, 186-187, 197, 215 subadditive function, 44 subadditivity, 43 submartingale, 77, 79, 81; nonnegative, 88 sup (j,g), 3 supermartingale, 77-78, 81, 89; continuous, 96, 103; discrete case, 104-105; existence theorem and, 114-128; generalized decomposition of, 123 ;' generation of, 104-108 ; nonnegative, 90; positive, 83; right-continuous, 93-95, 98, 105-106; strong order properties of, 106-107; uniform integrability properties of, 114-122; uniqueness theorem and, 113 supermedian function, 187-190; regularization of, 194-195 v Sur, M. G., 124-125, 249 Suslin sets, 247 symbols, see notation, explanation of symmetric difference, 2

X.

Index

266

thinness, 1 universally measurable sets, 25 time, change of, 108 upcrossings and downcrossings, 60, 82 time set, 52 lL u.s.c. (upper semicontinuous) function, 2, 204topological space, 23; Hausdorff, 33,45-46,51 205,222-223,227,232-233 59,70,248 it- "Y, 11 topology, 2 vector spaces, topological, 2 transitivity, of image laws, 15 t-supermedian function, 195 Watanabe, S., vi, 123,249 well-measurable sets, 156-158 ultrafilter, 119 Williamson, R. E., 151 uniform integrability properties, existence theorem, and, 114-128 Yosida, K., 210-211 uniqueness theorem, 113, 229-232 Zorn's lemma, 10-11, 51, 220, 228 universal completion of (0, :F), 23

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