2 is similar. Interpretation of the • -transforms of the sentences (Vz e (a x b))(3x e a) (3y e b) [ (x, y) = z] and (Vx e a)(Vy e b)(3z e (a x b))[ (x , y) = z] shows that •(a x b) s;;; • a x •b and • a x •b s;;; •( a x b). (b) That •p is a relation on • a t x · · · x • a, follows by interpretation of the • -transform of (Vx e P)(3xt e at ) · · · (3x,. e a,.)[ ( xt , . . . , x,) = x]. To show that, for n = 2, •(dom P) s;;; dom • P, interpret the • -transform of the sentence (Vx e dom P)(3y e a 2 ) [ ( x, y) e P]. The proof of the fact that •(dom P) 2 dom • P is left to the reader (Exercise 3). [x
=
=
,�
=
11.3
Monomorphisms Between Superstructures
81
(c) *f i s a relation o n •a x * b by (b). To show that *f is a mapping, interpret the • -transform of the sentence (Vx e a)('Vy e b)(Vz e b)[ [ (x, y) e f " (x, z) e f] --+ y = z], which is true in V(X). The rest of the proof of (c) is left as Exercise 3. 0 The results in Theorem 3.3 are quite general in nature. To be more concrete we consider, as examples, the interpretation of the sentences (3. 1 ) and (3.2). Remember that the sentence (2. 1) of which (3. 1 ) is the • -transform holds in V(R) because of the fact that there exists a multiplicative inverse of each nonzero element in the field 91, and (2. 1 ) is a formal expression of that mathe matical statement. Clearly (3. 1 ) should be a formal expression of a similar fact about V(* R). To see this, note that the ternary relation P defines a func tion P of two variables since the product of two real numbers is uniquely defined. By parts (b) and (c) of Theorem 3.3 we see that *P is a function from *R2 to *R. Thus for each a, b e *R the number c e *R such that ((a, b ) , c ) e *P is uniquely defined and is called the • -product of a and b. We denote c by a · b or ab. Now (3.1) is true by transfer in V(* R) since (2.1) is true in V(R), and its interpretation establishes the existence for each a #; 0 in • R of a number y e • R so that a y = 1. One can similarly show by trans fer that y is unique. Consider now the interpretation of (3.2). Proceeding as above, we see that (3.2) is equivalent to the ordinary mathematical statement "Given e > 0 in *R there is a � > 0 in *R so that, for all x e *R, lx - ai < � implies l*f(x) - *f(a)i < e. " (The absolute value lxl for x e *R is the extension of the usual absolute value in R.) Notice that here e and � are allowed to be any positive numbers in • R (even infinitesimal). The function *f will be said to be • -continuous at a if it satisfies (3.2), which will be the case, by transfer, if f is continuous at a. In §1.2 we noted that if B was a subset of R then • B was an extension of B (regarded as embedded in • R). This fact is again true in the present context. For if b e .9(X) and a e b then •a e *b by Theorem 3.3(a)(iii). But since a e X we have • a = a and so a e *b, and hence b s; *b. One might expect that this fact is true in general, i.e., that a e b implies a e *b for any entities a, b e V(X), but in general Theorem 3.3(a)(iii) is the best we can do, as shown by the following example. ·
Let J denote the set of closed bounded intervals in R; each = [a, b]. Then J e V2(R). Thus the following statements are true in V(R): ('Vx E J)(3a, b E R )('Vy e R)[a :::; y :::; b +-+ y E x], 3.4 Example
I e J is of the form I = {x e R : a :::; x :::; b, a, b e R}
('V a, b E R)(3 x E J) ('Vy E R) [a � y � b +-+ y E X].
II.
82
Nonstandard Analysis on Superstructures
By transfer, assuming a monomorphism • : V(R) -+ V( * R), we see that if I e • J then there exist numbers a, b e * R so that I = {x e * R : a � x � b}. Even if a and b are standard (i.e., in R), if a -# b such an interval is not identical to an interval in J, since it contains non-standard reals between a and b. Thus •J contains the transform * I { x e • R : a � x � b} of each standard interval I [a, b], a, b e R, and also all other intervals of the form {x e • R : a � x � b} where either a or b or both are non-standard. Notice, in particular, that J is not embedded in • J; i.e., only singleton sets in J lie in • J. This situation is indicative of what happens in general when one forms * b for an entity b of rank higher than one. =
=
The fact that the languages .!l'x and .!l'r contain the existential quantifier 3 allows alternative proofs of many of the results established in Chapter I. In particular, we may use 3 to do the work done by Skolem functions in Chapter I. To illustrate, consider the following proof of the sufficiency of the condition in Proposition 8. 1 of Chapter I, which states that if (s") is a standard sequence and •s" � L e R for all infinite n, then s" converges to L. We present the proof in a hybrid of the languages L� and .!l'R . Translation into the language .!l'R is left to the interested reader. Suppose then that *s" � L for all infinite positive integers n e *N. Let e > 0 be a fixed standard real. Since l*s" - L l is infinitesimal for all infinite posi tive integers, the statement (3.3) is a sentence in 2.R which is true for any infinite positive integer ro. How ever, (3.3) is not the • -transform of a sentence in .!l'R , since it involves the constant ro, which does not name the image *(a) of an element a e V(R). But since (3.3) is true, the sentence (3.4) (3m e *N)(Vn e *N)[n � m -+ IL - * s" l < e] is also true in V( * R) and is the • -transform of (3.5) (3m e N)(Vn e N)[n � m -+ I L - s"l < e] , which is then true in V(R) by virtue of the transfer principle. Since (3.5) is true for any e > 0, we see that s" converges to L. Comparison of this proof with that in Chapter I shows that we have avoided a proof by contradiction, and the use of Skolem functions. An other and more important aspect of this new technique of proof is that we construct a true sentence * «
3.5 Remark
11.4
83
The Ultrapower Construction
from L� to L.�. The construction of * is often accomplished, as above, by writing down a sentence 'I' in .P.x which is true but involves entities, like the w above, which do not occur in the • -transforms of sentences in fi'x, and then appropriately adding the existential quantifier to convert 'I' to a sentence of the form * for some in fi'x. The proof above may seem sur prising since we infer the existence of a standard integer m satisfying (Yn e N)[n � m -+ I s - sn l
(3.6)
<
e]
from the existence of the infinite integer w satisfying (3.3). Since similar proofs will occur in the rest of this book it is important to be able to recognize when a sentence 'I' in .P.x is or is not of the form * for some sentence in fi'x. This question will be dealt with in §11.6. Exercises
11.3
1 . Prove Theorem 3.3(a)(ii). 2. Show that *( n i 1 a 1 ) = ni' = 1 *a ,. 3. Finish the proof of parts (b) and (c) of Theorem 3.3. 4. Use the downward transfer principle to prove the sufficiency of the con dition in Proposition lO. l(a) of Chapter I. 5. Use the downward transfer principle to prove the sufficiency of the con dition in Proposition 10.8 of Chapter I. 6. Use the transfer principle to show that the set N of standard natural num bers is not an element of *�(N). 7. Let • : V(X) -+ V( Y) be a monomorphism. Show that if f e V(X) maps a onto b then *f maps *a onto *b. =
*11.4 The Ultrapower Construction for Superstructu res
·
In this section we show how to generalize the construction of • R in Chap ter I by constructing, for any superstructure V(X), a superstructure V(* X) on an appropriate set • X and a monomorphism • : V(X) -+ V(* X). We begin with an ultrafilter lf/J on an index set I (see the Appendix); both and lf/J will be fixed in the construction of V(* X), but in later sections we I will choose them to have additional properties. Now let V(X) = U:'= o ViX) be a given superstructure. Let S be an entity in V(X). The set of all maps a: I -+ S is denoted by ns; we write a(i) = a, for i E I. The maps a and b in ns are
4. 1 Definition
84
II.
Nonstandard Analysis nn Superstructures
equivalent (with respect to Cf/), and we write a = ,. b iff {i e l : a1 = b1 } e iliJ (the equality is set-theoretic except when S £ X, in which case it is identity). If a = ,. b we say that a1 = b1 almost everywhere (a.e.). The relation = ,. is an equivalence relation on ns. The set of associated equivalence classes is de noted by 0 ,_S, and is called the ultrapower of S (with respect to Cf/). The equiv alence class in n,_s containing a e ns is denoted by [a]. Let V_ 1(X) = 0. the empty set. The bounded ultrapower of V(X) is the set 00
0� V(X) = U n ,. [ V,(X) - V.. - 1 (X)] . n=O
We define the map e: V(X) -+ n� V(X) by e(a) = [a], where a1 = a for all i e I. The proof that = ,. is an equivalence relation on 0 S is similar to the proof of Lemma 1.4 of Chapter I and is left as an exercise. We see immediately from Definitions 1.3 and 1.5 of Chapter I that *R = n,. R where CW is the ultrafilter of §1. 1. The map e is a generalization of the map •: R -+ • R of Definition 1 .9 of Chapter I. 0� V(X) is called a bounded ultrapower since, for each [a] E n� V(X), a, E Vt(X) - I't - l (X), i E I, for some fixed k E N; thus, there is a uniform upper bound to the rank of a1, i e J. We now want to construct from 0� V(X) a superstructure V( * X) over a set * X, and an associated mapping M: 0�V(X) -+ V(*X). We will finally define the mapping •: V(X) -+ V( * X) as the composition of e and M , and show that • is a monomorphism. In the literature, M is called a M ostowski collapsing function. First we must define • X. In analogy with the definition of • R we put ,
(4. 1 ) Now V(* X) is completely determined and we proceed to the definition of M: 0� V(X) -+ V(*X). We define M successively on n,_[V"(X) - V.. - 1(X)] by induction. By (4. 1) 0,.V0(X) = *X, and by definition V0(*X) = * X, and so we define M to be the identity on • X, i.e., M(a) = a, a e n,. V0(X) = * X. (4.2) For higher levels we need the following definition. 4.2 Definition
If [a], [b] e 0�V(X), then [a] e,. [b] iff {i e l l a1 e b1} e CW.
The reader should check that e,. is well defined (exercise). To motivate the definition of M on n,.[ V1(X) - V0(X)] we let X = R and recall from §1.2 the definition of • A, where A is a subset of R. By Defini tion 2.2 of Chapter I (with I = N), • A consists of those elements [a] of
11.4
The Ultrapower Construction
85
•R for which { i e / : a 1 e A} e Cl/1. For our more general situation, the subset A is mapped by e to the element e( A ) = [A] in TI -. V1( R ) . Note that [ A] is not a subset of •R . We want • A to be a subset of •R and to consist of pre cisely those elements [ a] e •R for which [a] e._ [A] . Since • will be the com position of e and M, it follows that we should put M( [A] ) = { [a] e •R : [a] e ,. [A ] } { M( [a] ) e V0( •R ) : [a] e TI-. V0( R ) and [a] e,. [A ] } . The general definition is now clear. =
We define M : TI� V(X) --. V( • X) inductively by for [b] e TI-.V0(X), M( [b]) = [b] n- 1 = e [a M [a b ]) ]) : { ([ ] ( M U TI-. [ Vt(X) - V.. - 1(X)] and [a] e,. [b] }
4.3 Definition
k=O
for [ b] e TI-. [ V,(X) - V.. - 1(X)] , n � 1 . The important properties of M and e are collected together in the fol lowing result. 4.4 Lemma
(i) e and M are one-to-one maps; i.e., a = b iff e(a ) = e(b ), and [a] [b] iff M ( [a] ) = M( [b] ). (ii) e maps X into •x; M maps •x onto •x. (iii ) e maps V, + 1(X) - V,(X) into TI-. [ V, + 1(X) - V,(X)] ; M maps TI-. [ V.. + t (X) - Vn(X)] into V, + 1( • X) - Vn( • X). (iv) a e b iff e(a) e ,. e(b); [a] e,. [b] i ff M ( [a] ) e M( [b] ). (v) e(X) = [ X] and M( [X] ) = •x. (vi) Let [a] , [b] E TI � V( X ) and put c1 = { a ; , b1}, i e /. Then [c] e TI � V(X) and M ( [c ]) = { M ( [a] ), M ([b] ) } . Similar statements hold with { } replaced by < ) and = replaced by e, and also for three or more terms. (vii) I f [b] e,. e(a), a e V,(X) - V, _ 1(X), then [b ] e,. e( V, _ 1(X) ). =
Proof:
We leave the proof of (ii)-(v) and (vii) as exercises.
(i) To show e is one-to-one let a '# b e V(X) . Then e(a ) '# e(b) since ii1 '# b1 for all i e /, and 0 � Cl/1. To show M is one-to-one, we consider only the case that [a] and [b] are in TI � V(X) - •x, and [a] '# [b] in TI � V( X ) . Let U a = { i e I : there exists v1 e a1 with v1 � b 1 } and U b = { i e I : there exists v1 e b1 with v1 � a ; } . If neither Ua nor Ub is in CW, then I - ( U a u Ub) is in C¥1 and a1 = b1
86
11.
Nonstandard Analysis on Superstructures
for almost all i e /. But this is impossible. Assume, therefore, that U, e ft. Choose v1 e a 1 - b 1 for each i e U11 and let v1 be a fixed v10 otherwise. Then M( [v]) e M( [a]) and M( [v]) ¢ M( [b]). The rest is left to the reader. (vi) We prove the first statement and leave the rest to the reader. N ow M( [c]) = { M( [ y]) : y1 e {a,. b1 } a. e . } . If y1 e { a1 , b1} a.e ., let A = {i e I : y1 = a1} and B = { i e I : y1 = b1 }. Then A u B e 'f/, and so either A e 'fl or B e 'if since ff is an ultrafilter. Thus
M( [c] ) = { M( [ y]) : y1 = a 1 a.e .} u {M( [y]) : y1 = { M([a] ), M( [b])}. 0
=
b 1 a .e . }
With • defined as the composition of e and M, we now show that : • V(X) -+ V(•X) is a monomorphism. To do so we need the following funda
mental result; in the proof we use the axiom of choice.
(l.os) If �x " . . . , x") is a formula in fi'x with x 1 , . . . , x, its only free variables, and [a l], . . . ' [a..] E n� v(X), then ·�M( [a l]), M( [a"])) is true in V(• X) iff
4.5 Theorem
.
.
.
•
{ i e I : CI»(a1 (i), . . . , a,.(i) ) is true } e 'fl.
Proof: 1. We first establish the result when «<» is an atomic formula. If cJ) is of the form x e y or x = y, where x and y are either constants or variables, the result is immediate from 4.4(i) and 4 .4(iv). The result for «<» of the form ( x � o . . . , x,.) e x,. + l t (x � o . . . , x,.) = xR + h ( ( x 1 , . . . , x ,.), x ) e x, + h and ( (x 1 , . . . , x, ) , x) = x, + 1 can be proved by induction using 4.4(vi) (Exercise 4). 2. Suppose now that the theorem has been established for the formulas Cl»(x 1 , . . . , x,) and 'l'(x1 , . . . , x,.). We would like to prove it for the formulas -, e�», «<» " '1', «<» v 'I', and «<» -+ '1'. We do so for the first two and leave the proofs for the last two as exercises (Exercise 4); recall, however, that «<» v 'I' is equivalent to -, [(• «<») " (1 '1')]. (i) For -, e�» note that the following are equivalent:
• (-, CI») ( M( [a 1 ] ), , M( [a,] ) ) is true; , M( [a,])) is tr ue; -, • � M ( [a 1 ]), •
•
•
•
•
•
{I e / : CI»(a 1 (i), . . . , a,(i) ) is true } ¢ 'i'; {i e I : -, CI»(a 1(i ), . . . , an(i)) is true } e '¥1 (since ff is an ultrafilter).
11.4
87
The Ultrapower Construction
(ii) For Cl) " 'I' note that the following are equivalent: *(CI) " 'I' )( M([a 1]� M([a8] ) ) is true; *CI)( M( [a 1] ), , M( [a8]) ) A * 'I' (M( [a1 ]), , M([aJ) ) is true; and {i e I : CI)(a1 (i), . . . , aJi)) is true} e <¥1, { i e l : 'l'(a1(i ), . . . , aJi) ) is true} e
•
•
•
•
,
•
•
•
•
3. Suppose the result is true for a formula of the form Cl)(x 1, , x., , y). want to show it is true for formulas of the form (3y e c)CI), (3y e z)fl), (Vy e c)CI), and (Vy e z)CI), where c is a constant and z is a variable. We con sider the case (3y e c)CZ, and leave the case (3y e z)CI) to the reader (Exercise 4). For the quantifier V, replace (Vy e c)CI) with 1 (3y e c)1 CZ, and (Vy e z)CZ, with • (3y e z)--, CI) . Suppose *(3y e c)CI)( M( [a 1] � , M( [a.,] ), y) holds in V(*X), i.e., •
•
•
We
•
•
•
(3y e *c)*CI)( M([a1 ]),
•
•
•
, M([a.,] � y)
holds in V(*X). Thus we can find M( [a] ) e V(*X) so that , M( [a J ), M( [a] ) ) (M( [a] ) e *c) " CI)(M( [a 1 ] � •
•
•
holds in V(* X). Using step 2, this is equivalent to {i e l : a(i) e c A CI)(a 1 (i), . . , a.,(i ), a(i) ) is true} e II'. .
Hence also the larger set {i e 1 : (3y e c)CZ,(a1(i), . . . , a.,(i), y) is true} is in
4.6 neorem
morphism.
The map
•:
V(X) -+ V(* X) defined by •
=
Mo
e
is a mono
Proof: We prove (v) of Definition 3.2 and leave the remaining proofs as exercises. Let Cl) be a sentence in !i'x . Then Cl) has no free variables, so *CI) is
88
II.
Nonstandard Analysis on Superstructu res
true in V(*X) iff {i e 1 :� is true} e 'PI by Theorem 4.5. But the set {i e 1 : � is true} is either I [if � is true in V(X)] or 0 [if � is not true in V(X)], so *� is true if and only if � is true. D Whenever nonstandard analysis is applied in any concrete situation in the rest of this book, we will start with a superstructure V(S) based on a suitable set S, and then use a superstructure V(*S) and a monomorphism • : V(S) -+ V(*S) constructed with an ultrafilter 'PI as in this section. Usually the mono morphism will not be mentioned explicitly, but we will always choose 'PI in such a way that V(*S) has a special property, that of being an enlargement. This will guarantee that *S is large enough to contain "infinite" entities. We turn to this question in the next section. Exercises 11.4
1 . Prove that • is an equivalence relation on ns. Show that the relation e,. of Definition 4.2 is well defined. 3. Finish the proof of Lemma 4.4. 4. Finish the proof of Theorem 4.5. 5. Finish the proof of Theorem 4.6. 6. Show that if a(i) e V,(X) for a fixed n and all i e I, then, for some k ::s;; n and all i e U for some U e 'PI, a(i) e J.t(X) - l't- 1(X). =
2.
11. 5 Hyperfinite Sets, Enlargements, and Concurrent Relations
In §1. 1 we showed that • R was strictly larger than R (regarded as embedded in * R) by exhibiting elements like [ ( 1, 2, 3, . . . ) ] in * R which were not equal to any element of R. The demonstration involved the fact that the ultrafilter 'PI on N was free, i.e., it contained the cofinite filter �N · In the general case it is interesting to determine the conditions under which *X is strictly larger than X. It should be recalled that, by assumption, X contains N and hence is infinite. The following result shows that *X X and hence V(* X) V(X) when 'PI is a principal (nonfree) ultrafilter on I; thus we get nothing new in this case. =
=
5. 1 Lemma If 'PI is a principal ultrafilter on I then • X (as constructed in §11.4) equals X (regarded as embedded in • X).
'PI
Proof: A principal ultrafilter 'PI is generated by a single element i0 e I; i.e.,
consists of all sets U
s;;
I which contain i0 (see the Appendix). If [a] e * X
U.S
Hyperfinite Sets
89
and a10 = a0 then [a] =• [a], where a1 = a0 for all i e I. Thus [a] e X, where X is regarded as embedded in *X. 0 We will next show how to choose an index set and an ultrafilter of subsets of the index set so that the *X constructed as in §11.4 is strictly larger than X, and so that V(* X) has other desirable properties; the most important is that of being an enlargement. We begin by introducing the notion of a hyperfinite or • -finite set. 5.1 Definition If A e V.(X) - V0(X) for some n, we denote by 9.,(A) the set of all finite subsets of A. 9p(A) is in V(X� and we call the image *9.,(A) e V(*X) (with respect to a monomorphism • ) the set of hyperfinite or • -finite subsets of * A. The set of all hyperfinite subsets is the set U:'= 1 *9.,( V8(X)).
Any elementary mathematical result that holds for finite sets extends to a similar result for hyperfinite sets by the transfer principle. An example of a hyperfinite set is the set J c: *N of positive integers less than some j e *N. To see that J is hyperfinite consider the collection J c: 9.,(N) of all finite subsets of N of the form { 1, 2, . . , j} for some j e N (a set of this form is called an initial segment). Then * J c: *9p(N) contains sets of the form {n e * N : n s j} for some j e *N. The following result shows that these hyper finite sets are in some sense the prototype. .
5.3 Theorem
If B e * V.(X), k e N, is a hyperfinite set, then there is an initial segment J = { n e * N : n S j} for some j e * N and a one-to-one, onto mapping f: J -+ B in * V. + 4(X). Proof: Suppose B e *9p(A), where A e V,.(X), n � 1. Now the following statement (in semiformal language) is true in V(X): (VB e 9p( V8(X) ))( 3j e N) ( 3/ e V,. + 4(X) ) [! maps J one-to-one onto B, where J = {n e N : n S j}]
[the reader should check that the sentence in square brackets can be trans lated into a sentence in !t'x (exercise)]. The result follows by transfer. 0 Because of Theorem 5.3 we will often write a hyperfinite set B as B {b 1 , b2 , bJ }, where bt = f(k), k e J, and f is the function of the theorem. It should be noted that the dots in this representation cover somewhat more ground than they do in the standard case, and that this representation is really an abbreviation of the setup in Theorem 5.3. Hyperfinite sets are an important tool in nonstandard analysis by virtue of the fact that many stan dard mathematical structures can be "approximated" by hyperfinite struc tures in a natural way. We will illustrate this fact later in this section. =
•
•
•
,
II.
90
Nonstandard Analysis on Superstructures
5.4 Definition
Entities in V(X), and entities which are of the form * b for some b e V(X), are called standard; all others are called non-standard.
5.5 Examples
1 . Each individual in X £:: • X is standard. 2. In Example 3.4 the intervals I e *J of the form I = {x e *R : a � x � b}, where a < b, are themselves standard entities even though they contain non standard numbers. An interval {x e *R : a � x ::5: P }. where < a < P and IX and p are infinitesimal, is a non-standard entity.
0
5.6 Definidon
The superstructure V(*X) [with respect to a monomorphism : V(X) .... . .. V(* X)] is called an enlargement of V(X) if for each set A e V(X) • there is a set B e *.9�A) such that •a e B for each a e A, i.e., B contains the standard entities in • A. We have already seen that a hyperfinite set of the form {n e •N : 1 � n � j}, where j e • N CXl ' contains every standard natural number. Definition 5.6 is a generalization for arbitrary sets in V(X). We will now show that for a given superstructure V(X) it is possible to choose an index set J and a free ultrafilter .Y on J so that the associated superstructure V( * X), constructed as in §11.4 using J and 1"; is an enlarge ment of V(X). It will follow as a corollary, since X is infinite, that • X is strictly larger than X. The proofs of Lemma 5. 7 and Theorem 5.8 may be skipped on first reading of the chapter. Let J be the set of all nonempty finite subsets of V(X). It follows that a e J iff there is a b e V(X) - V0(X) and a e .9�b) - 0 (why?). If a e J we define J, = {b e J : a £: b}. 5.7 Lemma
The collection :F
=
{A £:: J : there exists a e J such that J, £:: A}
is a free filter on J. Proof: It is easy to sho w that :F is a filter. For example, if A 1 , A 2 e :F there exist a 1 , a2 e J so that A1 2 J,, (i = 1 , 2). Since A 1 n A 2 2 J, , n J,2 = J", u , 2 , A 1 n A 2 e :F. The rest is left as an exercise. To show :F is free, let a e J. Then there is an element b e J so that a n b = 0. Since a r1 Jb, J - {a} � Jb, so :F is free. 0
Now let ..Y be an ultrafilter on J with by Theorem A.5 of the Appendix).
.Y
�
:F
(such an ultrafilter exists
11.5
91
Hyperfinite Sets
5.8 Theorem If V(* X) is constructed from V(X) using "Y and J then it is an enlargement of V(X).
Proof: Let A be a set in V(X). We define a map r: J -+ 9'p(A) by r. = a n A, and let B = M((r]). Then B e *9'p(A). H x e A then J1,.> = {a e J : x e a}, so {a e J : x e a n A } e "Y. Thus [X] e.,. [r] and so •x e B. 0
Robinson's original definition of enlargement (see Theorem 5. 10 below) made use of the notion of concurrent relation and was the cornerstone of his development of nonstandard analysis. 5.9 Definition A binary relation P is concurrent (finitely satisfiable) on A £:: dom P if for each finite set {x h . . . , x.} in A there is a y e range P so that (x,. y) e P, 1 S i S n. P is concurrent if it is concurrent on dom P.
Examples of concurrent relations are the relation S in N and 5. 10 neorem
£::
in 9'p(N).
The following are equivalent:
(i) V(* X) is an enlargement of V(X). (ii) For each concurrent relation P e V(X) there is an element b e range *P so that (*x, b) e •p for all x e dom P.
Proof: (i) � (ii): Let B e *9'p(dom P) be such that, for each x e dom P, •x e B. Since the sentence (Vw e 9'p(dom P))(3y e range P)(Vx e w)[ (x, y) e P] is true in V(X) by concurrence of P, its • -transform is true in V(* X). Thus there exists an element b e range • P so that (z, b) e • P for each z e B, and in particular for each •x with x e dom P. (ii) � (i): Exercise. 0 If Y e V(X) contains an infinite number of entities and V(* X) is an enlargement, then • Y contains entities which are not standard. In par ticular, if A £:: X is infinite then • A properly contains A.
5. 1 1 Coronary
Proof: The relation P on Y x Y defined by (a, b) e P iff a =I b" is con current since Y is infinite. By 5. 10(ii) there is a b e * Y such that b =I •x for all X E Y. 0 "
Corollary 5. 1 1 gives another proof of the existence, in an enlargement V(* R) of V(R � of non-standard numbers, but it holds in much more general situations. 5.U Definition A set !I' of subsets of an entity A e V(X) is called exhausting if, for each finite subset F £:: A , there is an S e !I' with F £:: S.
92
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If !/ is an exhausting set of subsets of A e V(X) and V(* X) is an enlargement, then there is a set C e • !/ containing all the standard entities in • A.
5. 13 Proposition
Proof: Let B be a hyperfinite subset of • A such that •a e B for each a e A. Then there is a C e •g with B � C. D In spite of its simplicity, Proposition 5. 1 3 turns out to be a very powerful tool in nonstandard analysis. The typical application runs as follows. Suppose A is an infinite set with some additional mathematical structure; for example, A could be an infinite graph, or a Hilbert space. Suppose further that A can be exhausted by a family !/ of substructures-finite subgraphs, finite dimensional inner-product spaces, etc.-so that for each S e !/ a certain result can be proved. One wants to establish a corresponding result for A . Using Proposition 5.1 3, we can find a set C e • !/ containing all of the stan dard elements in • A, and by transfer the • -transform of the given result is true for C. The problem then is to show how the validity of the • -transform of the result on C induces the validity of the result on A. This last step can be quite difficult but is often easier than proving the result by standard methods. This method of proof was the basis of the first successful attack, by Bernstein and Robinson [7], on an invariant subspace problem in Hilbert space proposed by Smith and Halmos. We illustrate the technique by proving a result in infinite graph theory due to de Bruijn and Erdos. (See also the related paper by Luxemburg [35].) The application indicates how nonstandard analysis is applicable in areas other than analysis. A graph (A, E) consists of a set A of vertices and a binary relation E on A x A which is symmetric (i.e., ( x, y) e E implies (y, x) e E). If (x, y) e E we say that x and y are connected by an edge. (A, E) is infinite if A is infinite. (A, E) is k-colorable if there exists a map f: A -+ { 1, 2, . . . , k} (the set of"colors") such that if (a, b) e E then f(a) "# f(b), i.e., no two vertices which are connected by an edge are given the same color. If B � A then the subgraph (B, E I B > is defined by "(x, y) e E I B iff x, y e B and (x, y) e E"; i.e., B inherits its edges from E. (De Bruijn-Erdos [ 1 3]) If each finite subgraph of an infi nite graph (A, E) is k-colorable, then (A, E) is k-colorable.
5.14 Theorem
Proof: We work in the superstructure V(A u N). Let !/ denote the set of all finite subsets of A (obviously exhausting). For each F e !/ the graph (F, E I F) is k-colorable, so the following is true in V(A u N): (5. 1 )
(VF e !/)(3fF: F -+ { 1, 2, . . . , k})(Vx, y e F) [ (x, y) e E -+ f�x) "# f�y)].
11.5
93
Hyperfinite Sets
By the definition of enlargement, there exists a B e • !/ so that B ;;;;;;! A. By transfer of (5. 1 ) we see that there is a map (coloring) f8: B -+ *{ 1, 2, . . , k} ( { 1 , 2, . . . , k}) so that if (x, y) e *E then fJ...x) :F fJ... y). We now restrict f8 to A to get a map f: A -+ { 1 , 2, . , k}. f is a coloring since it inherits the property "(x, y) e E implies f(x) :F f(y)" from f8 (check). 0 .
=
.
.
Intuitively, the proof of 5. 14 given above is obvious; we have simply covered A by a • -finite and hence k-colorable graph B and then restricted the col oring. A similar technique can be used to give easy proofs of more intricate theorems in infinite graph theory. In closing this section we note that the results of Chapter I for *R remain valid for an enlargement of V(R). To get more we need to consider the notions of internal and external entities in V(*R); these are introduced in the next section. Exercises
11.5
1 . Show that if j is infinite then J { n e N : n :::;; j} e *&'�N) - &'�* N). 2. Show that in general *&'�A) ;;;;;;! &'�*A) whereas &'(*A) ;;;;;;! *&'(A). =
3. Check the translation into a sentence in .!l'x of the informal sentence in the proof of Theorem 5.3. 4. Show that the family !F in Lemma 5.7 is a filter given that A t o A2 e !F => A 1 n A2 e !F. 5. Prove that (ii) => (i) in Theorem 5.10. 6. Show that if {O« : IX e A} is an open covering of a set S c R but no finite subcollection covers S, then there is a y e *S such that y if. x for all x e S. 7. Give another proof of the existence of infinite natural numbers in an en largement V(* R) of V(R) by using the concurrent relation < . 8. Let A be an entity in a superstructure V(X) - X which is closed under fi nite unions; i.e., if a1 e A (1 :::;; i :::;; n) then U a�1 :::;; i :::;; n) e A. Show that if V(*X) is an enlargement of V(X), there is an element b e *A so that U *a(a e A) !;;;;; b. 9 . (Luxemburg) Let A be an entity of V(X) - X. The intersection monad of A is the set ,u(A) n • a(a e A) in V(* X). A has the finite intersection property (f.i.p.) if a, b e A implies a n b :F 0. Show that V(* X) is an en largement of V(X) iff the intersection monad ,u(A) of each A with the f.i.p. is nonempty. 10. (Luxemburg) Let V(* X) be an enlargement of V(X) and !F be a filter in V(X) - X. Let ,u(!F) be the intersection monad (see Exercise 9). Show that if B e V(X) and F n B :F 0 for all F e !F then ,u(!F) n *B :F 0. 1 1. Show that if !F is a filter in V(X) - X and B is a set in V(* X) such that B n * F :F 0 for all F e !F then it is not necessarily true that ,u(!F) n B :F 0. where ,u(!F) is the intersection monad of Exercise 9 . [Hint: Let X = N, =
II.
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Nonstandard Analysis on Su pe rstructures
!F be the Frechet filter on
N (the collection of complements of finite sub sets of N), and B N.] A standard result states (informally) that a set A e V(X) is finite iff every injective map f: A --+ A is surjective. Formalize this statement and so obtain a similar characterization for the • -finite sets. Let P e V(X) be a binary relation and suppose that in some enlargement V( * X) of V(X) the following is true: for every y e range • P, there exists an x e dom P so that ( * x, y) e • P. Show that there exists a finite set { x 1, , x,} s; dom P such that for all y e range P, there is an i, 1 S i S n, with (x1, y) e P. (Konig's lemma) Let (S, : n e N) be a sequence of mutually disjoint nonempty finite sets and let P be a binary relation on U S,(n e N) such that, whenever x e S, + 1 for some n, there exists a y e S, such that (y, x) e P. Show that there exists an infinite sequence (x, : n e N) such that x,. e s. and (x., x. + 1 ) e P for n e N. (Total ordering) Let X be a nonempty set. A binary relation P on X is a partial ordering if the following holds: (a) P is retlextive, that is, (x, x) e P for all x e X; (b) P is antisymmetric, that is if (x, y) e P and ( y, x) e P then x = y; (c) P is transitive, that is, if (x, y) e P and (y, z) e P then (x, z) e P. A partial ordering P on X is a total ordering if whenever x, y e X then either (x, y) e P or (y, x) e P. Every finite set can be totally ordered. Assuming this result and the fact that enlargements exist, show that any set can be totally ordered. (Rado's selection lemma) Let {A.�. : A. e A} be a nonempty family of finite sets. A choice function over A is a function rjJ :A --+ UA1(..1. e A) so that t/J(A.) e A1 for each A. e A. Let {A7:y e r} be a nonempty family of finite sets. Assume that for each finite subset F s; r there is a choice function rP F over F. Show that there exists a choice function rjJ over r so that for any finite set F s; r, there exists a finite set F' � F with t/J(x) = rbr(x) for all x e F. =
12. 1 3.
•
14.
15.
16.
.
•
11.6 Internal and External Entities; Comprehensiveness
We noted in Remark 3.5 that a basic technique of proof in nonstandard analysis is to establish the validity of a sentence cl> in fi'x by noticing that it is the downward • -transform of a sentence * cl> which is true in � x · Thus it is particularly important to be able to recognize when a sentence 'I' in !.e.x is of the form * cl> for some sentence cl> in !ex . Notice that, for a sentence cl> in !ex. *cl> uses only the names of standard objects and so •ci> involves
11.6
Internal and External Entities
95
only expressions like (Vx e *a)'P , (Vx e y)'l', (3x e *a)'l', and (3x e y)'l'. Thus to check the truth of *cJ) we need only look at elements b in V(* X) which satisfy b e • a for some a e V(X). By 3.2(iv), if c e b and b e • a. then c e * }'t(X) for some k. If b e • a for some a e V(X), we call it internal; otherwise we call b external (Definition 6. 1). A sentence 'I' in !l'.x is not of the form *cJ) if it contains names of external entities, i.e., is an external sentence. A common mistake in nonstandard arguments is to apply the transfer principle to external sentences 'I' in !l'•x · Thus it is important to be able to recognize external entities in V(* X). We will learn in this section that R, N, Z, *R 00 , *N 00 , *Z00 , and m(O) are external subsets of • R. Using these, we can construct many external functions and relations. For example, the characteristic function of an external set is an external function; the relation of nearness � is externaL The properties of external entities cannot be obtained by transfer from those of V(X). For example, it is true that any subset of N which is bounded below has a least element. However, this property is not true of * N 00 , for if n were a least element in • N oo then n - 1 would have to be finite, which is impossible. In this section we first concentrate on internal entities and their properties and then present examples of external entities. The section ends with a dis cussion of comprehensiveness which involves internality.
Definition An entity b e V(*X) is called internal [with respect to • : V(X) -+ V(* X)] if there exists an a e V(X) so that b e • a; i.e., internal entities are elements of standard entities. An entity which is not internal is called external. Similarly, a sentence or formula cJ) in !l'.x is called either standard or internal if the constants in (J) are names of standard or internal entities, respectively. A sentence which is not internal is called external. 6. 1
6.1 Examples
1.
All standard entities are internal (Exercise 1 ).
2. With J the set of closed and bounded intervals in R, every set { x : a � x � b, a, b e *R} e *J is internal; the standard • -intervals are those for which a and b are in R. 3. If 'If denotes the set of continuous functions on R, then each f e *'If is internal and is called a • -continuous function. 4. If P is concurrent, the element b e range •p given by Theorem 5. 10(ii) is internal. 5. The • -transform of any formula cJ) e !l'x is standard. 6. The sentence (V£ > 0 in *R)(Vy e *R)(315 > 0 in *R)(Vx e *R) [ j x - yj < 15 -+ l f(x) - /(y)j < £],
96
Nonstandard Analysis on Superstructures
II.
where f e •rc 1s internal, is an internal sentence and expresses the fact that / is • -continuous on • R. The set of all internal elements of V(* X) is the set • V(X)
6.3 Theorem
U :'= o * Vft(X).
=
Proof: If b e * V(X) then b e * V,(X) for some natural number n � 0 and so b is internal since V,(X) is standard. Conversely, if b is internal then b e • a where a is in V, + 1(X) V,(X) for some n � 1, so a !;;;;; V,(X). Thus *a !;;;;; • V,(X) and b e * V,(X). D
,
-
It is necessary to be able to recognize internal sets. In that regard the fol lowing result is very useful. (Keisler's Internal Definition Principle [24]) Let �x) be an internal formula in !l'.x for which x is the only free variable, and let A be an internal set. Then {x e A : �x) is true} is internal.
6.4 Theorem
eh . . . , eft be the constants in �x); we write �x) , eft, x). Now A, e l > . . . , eft e * J)(X) for some k e N. Thus the sentence
Proof: Let
�e 1
,
•
•
•
(Vx 1 ,
=
, xft, y e V�X) )( 3z e J'l + 1 (X) )( Vx e J)(X) ) [x e Z +-+ [x e y 1\ �x 1 , , x ft, x)] ] •
•
•
•
•
•
in !l'x holds in V(X). Its interpretation in V(* X) says that { x e A : �x) is true} e * l'l + 1(X). D 6.5 Examples
1 . The set Z1 of zeros of an internal • R-valued function f in V(* R) is internal since Z1 = { x e • R : (x, 0) e ! } . 2 . The characteristic function o f a n external set i s external (Exercise 2).
A consequence of property 3.2(iv) of a monomorphism • : V(X) -+ V(* X) is that any element of an internal entity is an internal entity. We use this fact in the proof of the following result. 6.6 Theorem
A
X
B.
If A and B are internal, then so are A
u
B, A
n
B,
A
-
B, and
Proof: We prove the result for A u B and leave the remaining proofs as an exercise. Suppose A, B e * V.. + 1(X) and consider the following true state ment in V(X): (VW, Y E V, + 1 (X) )( 3Z E V, + 1 (X) )(Vx E Vft(X)) [x E Z +-+ x e W 1\ X e Y] .
11.6
Internal and External Entities
97
By transfer, there exists a set C e • v, + 1(X) having exactly the same elements from • V.,(X) as A u B. But by 3.2(iv) all elements of A, B, and C are in • v,(X ) and so C = A u B. 0
,
Having considered internal entities in some detail, we are now ready to demonstrate the existence of external entities. Recall that in Remark 7.8 of Chapter I we showed that there was no set A c R so that • A = R. This fact is not sufficient to show that R is external in the sense of Definition 6. 1 ; we would need to show that R was not an element in the • -transform of an element of V(R). To show the existence of external subsets we use the fol lowing lemmas. 6.7 Lemma If a e V(X ) of the entities in •9(a).
-
X then the internal entities in 9(•a) consist exactly
Proof: Consider the following true statement in V(X) with n
�
1:
(Vx e V,(X)) ((Vy e x) [ y E a] +-+ x e 9(a)] [i.e., for all x e V,(X), x is a subset of a if and only if x e 9(a)] . Its • -transform says that, for all x e • v,.(X), x is a subset of •a if and only if x e •9(a). We see from Theorem 6.3 that if x is an internal set in V(• X), i.e., x e • V(X), then x e • v.,(X) for some n. Such an x is a subset of •a if and only if it is in •9(a). Thus • v(X) n 9(•a) = • v(X) n •9(a) •9(a). 0 =
As an example, we note that the internal subsets of • N are exactly the members of •9(N). 6.8 Lemma
Each nonempty internal subset of the hyperintegers •z which is bounded below (above) has a least (greatest) element.
Proof: If X is an internal nonempty subset of •z then X e •9(Z) by Lemma 6.7. The result in the "bounded below" case now follows by transfer of the sentence
(VX e 9(Z))[(3b e Z)(Vx e X) [ b � x] A X -:1: 0 -+ (3y e X)(Vx e X)[y S x ]] ,
which expresses the fact that each subset of Z which is bounded below has a least element. The "bounded above" case is similar. 0
Theorem In an enlargement V(• R) of V(R) the set • N 00 of infinite natural numbers is external.
6.9
98
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Proof: Suppose that • Nco e &'(* N) is internal. Then by Lemma 6.8 there exists a least b e *Nco · But then b - 1 e *Nco and b - 1 < b (contradic tion). 0 The sets R, N, Z, *Zco (the set of infinite integers), *Reo (the set of infinite reals), and m(O) (the set of infinitesimals) are external in an en largement V(*R) of V(R).
6.10 Coronary
Proof: Note that *Nco = *N - N. If N were internal then *N co would be internal by Theorem 6.6, contradicting Theorem 6.9. Using the fact that the set of integers Z is external (exercise), we see that R is external, since otherwise Z = R n • Z would be internal. Similarly • Z co and • Reo are external. To show that m(O) is external, we note that *Reo = {x e *R : (3y e *R) [ (x, y) e P " y e m(O)]}, where P is defined by "(x, y) e P if y = 1/x." If m(O) is internal then so is *R eo by Theorem 6.4 (contradiction). 0 Clearly, external entities and notions play a very important role in non standard analysis, as we see by noting the occurrence of the set of infinite natural numbers and the set of infinitesimals in many of the results of Chap ter I. The reader might want to review some of the proofs in Chapter I to see just how external sets arise, and how the transfer principle is effective even though it involves only internal sets. In many cases, external entities and notions are useful in recovering stan dard results from internal results. "Limiting" entities corresponding to "converging" families of entities in V(X) can often be identified with internal entities in * V(X), but to recover actual limiting entities in V(X) usually in volves some external operation (one which produces external entities). For example, consider Theorem 14.1 of Chapter I in this light. We constructed the solution t/J(x) of the differential equation
The monomorphism • : V(X) -+ V(*X) is comprehensive if, for any sets C, D e V(X) and any map h: C -+ *D, there is an internal map g: *C -+ *D such that g(*a) = h(a) for a e C. The monomorphism is called
6.1 1 Definition
11.6
99
Internal and External Entities
denumerably comprehensive if the choice of C is restricted so that the cardinal ity of C is that of the natural numbers N. 6. 12 Example Suppose that • is comprehensive, and {An : n e N} is a sequence (in the ordinary sense) of entities in • V,(X) for some integer m. Then there is an internal sequence {Bn : n e •N } such that An = Bn for all n e N.
A monomorphism • : V(X) -+ V(• X), constructed as in §11.4, is comprehensive.
6. 13 Theorem
Proof: Let C, D, and h be as in Definition 6. 1 1 . Each element of • D is of the form M([b]). Let S(M[b ]) be a representative b from the equivalence class [b] We may assume that b1 e D for all i e /. For each i e /, let k1 be the mapping from C to D given by .
k 1 = {(a, S(h(a))(i)) :a e C } and let [k] denote the equivalence class generated by the mapping {(i, k1) : i e J}. We leave as an exercise the proof that M([k] ) is an internal function from •c to •D. If M([a]) e •c and { (i, a1) : i e J} is a representative from the equivalence class [a], then the image of M([a]) under the mapping M([k]) is M([b]), where b1 = S(h(a1) )(i) for i e /. In particular, if a1 = a e C for almost all i e /, then b1 S(h(cx) )(i) for all i e /. Thus M( [k] ) extends h. D =
Exercises
1. 2. 3. 4. 5.
6.
7. 8.
11.6
Show that all standard entities are internal. Show that the characteristic function of an external set is external. Finish the proof of Theorem 6.6. Show that the sets Z, •z oo , and •R oo are external. Show that M( [k] ) defined in the proof of Theorem 6. 1 3 is an internal function from •c to •v. Show that if {xn : n e •N} is an internal sequence and l xn l :s;; 1 /n for all n e N then, for some k e • N 00, l xn l :s;; 1 jn for all n :s;; k. Let {An : n e N} be a sequence in the ordinary sense of internal subsets of • R such that, for any k e N, nAn(l :s;; n :s;; k) "# 0. Assume the monomor phism • is denumerably comprehensive and show that nAn(n E N ) "" 0(a) Show that every nonempty internal subset A of •R with an upper bound has a least upper bound. (b) Show that every nonempty internal subset A of •N has a minimal element.
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9. Show that if P is an internal binary relation on
1 0.
I I.
1 2.
1 3.
c 1 x c 2 then dom P and range P are internal. In particular, c1 x c 2 is internal if c 1 and c2 are internal. Show that st: G(O) - R is an external map. Show that every internal subset of a • -finite set is • -finite. Show that if a and b are internal then the set of all internal functions from a to b is internal. (a) Let sup be the function in V(R) which assigns, to each upper bounded set E c R, its supremum, sup E. The function sup can be extended to a function *sup defined on all internal subsets of • R which are " • -bounded above." Characterize by a sentence in !i'.R the collection 91 of sets which are • -bounded above, and then show that •sup E ::5; *sup F if E s; F are sets in 91. (b) Show that if E is a finitely upper bounded external subset of *R, then • sup £ may have no meaning in *R, but sup{0r : r e E} has a meaning in R.
11.7 The Permanence Principle
In this section we present a principle with many applications called the permanence principle by Robinson and Lightstone [44] or Cauchy's principle by Stroyan and Luxemburg [46]. Throughout the section we suppose that X contains the set of reals R.
7.1 Theorem (Permanence Principle) Let
(i) If 0 in R so that
(i) Let A = {x e *N :1
=
The Permanence Principle
11.7
101
(ii) Given the internal set A defined as i n (i), A � N and A i s bounded above and hence has a largest element I by Lemma 6.8; we can take k = I + 1 . (iii) Let A = { x e •N - {O} : CI»(y) holds for all y with IYI � 1/x} and use (ii). 0 7.2
Coronary (Spillover Principle) Let A be an internal subset of •R.
(i) If A contains all standard natural numbers then A contains an infinite natural number. (ii) If A contains all infinite natural numbers then A contains a standard natural number. (iii) If A contains the positive infinitesim als then A contains a standard positive real number. Theorem 7. 1 can be used to give yet another proof of the fact that if
(s,. : n e N) is a standard sequence and •s,. � L e R for all infinite n, then lim s,. L (see §11.3). Let s > 0 be a fixed number in R. Then ! •s,. - Ll < s for all infinite n. Applying Theorem 7. 1 (ii) with Cl)(b) the internal statement " l •s, - Ll < e", we see that there is a k e N so that !•s, - Ll < e for all b � k in • N and, in particular, Is, - Ll < s for all b � k in N since • s, = s, if b e N. =
This establishes the desired result. The following result has many applications.
(Robinson's Sequential Lemma) Let (s,. : n e •N) be an internal •R-valued sequence such that s,. � 0 for each n e N. Then there is an infinite natural number ro so that s,. � 0 for all natural numbers n � ro.
7.3 Theorem
Proof: The sequence (ns,. : n e •N) is internal. Apply 7.1(i) with Cl)(n) the internal formula "jns,.j � 1" to obtain an ro e • N oo so that js,.j :s;; 1 /n if n :s;; ro. Thus s,. � 0 if n e • N oo and n =:;;; ro, and so s,. � 0 for all n =:;;; ro. 0 One should beware of assertions similar to Theorem 7.3 which sound plausible but are not true. For example, it is not true that if s,. � 0 for all infinite n then there exists a finite k so that s,. � 0 for all n � k as the ex ample s,. = 1/n shows. As an application of Theorem 7.3 we give another proof of the fact (Corollary 1. 1 3.5) that if the sequence (f,.(x) : n e N) of continuous real-valued functions on the interval [a, b] converges uniformly then the limit f(x) is continuous on [a, b]. Let x0 e [a, b]; we need to show that •j(x) � f(x0) if x � x0 • But •j,.(x) � •f,.(x0) for each n e N, and so •Jw(x) � •fw(x0) for some infinite ro by Theorem 7.3. But •fw(x) � •J(x) for all x e •[a, b] by Proposi tion 1 3.2 of Chapter I, and we are through.
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Robinson [41 ] applied Theorem 7.3 in a more significant context in giving a nonstandard construction for Banach limits of bounded sequences. Suppose (s,. : n e N) is a bounded sequence, i.e., l s.l :s; M for some real M > 0. We would like to attach a "limit" to (s.) even though it might not converge in the usual sense. For example, the sequence t. = (s1 + s2 + + s..)/n (n = 1, 2, . . . ) of Cesaro means sometimes converges when (s.. ) does not converge and defines a limit called the Cesaro sum of the sequence (s,.). Any generalized limit should satisfy the properties in the following definition. ·
7.4 Definition
·
·
Let 100 denote the set of standard bounded sequences. A map
L : l co -+ R is called a Banach limit
if
(i) L(au + ln) = aL(u) + bL(t) (a, b e R, u, T e 100), (ii) if a = (s. l n e N) then lim inf s. :s; L( u) :s; lim sup s11, (iii) if a = (s,. I n e N) and T = (t. l n e N) , where t. = s. + " then L(a) = L(T).
To obtain a Banach limit, we let Lf· 1 for ro e • Nco extend the standard summation operators L�= 1 , n e N. Fix ro e • N co • and let L(a) = 0({1/ro) Lf- 1 •sJ for each a = (s,. :n e N) in 100• Then L is a Banach limit.
7.5 Theorem
M
Proof: The mapping L clearly satisfies 7.4(i). Given a = (s.. :n e N), let = sup { l s .. l : n e N}. For a given m e N,
I
1 ro - m
--
Ill
1
Ql
L • s� - - L • sl ro 1 1 1• 111 +
:s;
1 (
==
0.
1·
f
I
.!_ f
ro :::;; _ .!_ •s� ro - m ro 1 .. 111 + 1 ro
)
ro (ro - m)M - 1 ro - m ro
--
1 • 111 + 1
•sl
l + I! ro
f
1 • 111 + 1
• s1 -
.!_ro f
1· 1
•sl
l
mM
+ro
By Theorem 7.3, there is an m e • Nco so that (7. 1)
1
Ill
L(u) == -- L • s... ro - m • • 111 + 1
Fix e > 0 in R. We see immediately from Definition 8. 1 6 of Chapter I that for each n e •N with m + 1 :s; n :s; ro lim infs..
- e
< •s,. < lim sup s..
+ e.
11.7
1 03
The Permanence Principle
By the transfer of the usual properties of an average applied to ( 7. 1 ), lim infs. -
e s;
L(a)
s;
lim sup s., +
e.
Since e is arbitrary, we obtain 7.4(ii). The rest of the proof is left to the reader. 0 Exercises
11.7
I . Prove the real case of (i) and (ii) of Theorem 7. 1 . 2. Assume that A is an internal set in • N such that, for some infinite integer y, if n is infinite and n s; y in *N then n e A. Show that, for some finite m e N , if n e N and m s; n then n e A. 3. Prove that the mapping L of Theorem 7.5 satisfies property (iii) of Defi nition 7.4, i.e., L is invariant under finite translations. 4. Use the permanence principle to show that if f is a standard function and l *f(x) - L l � 0 for all x � I but x -=1- 1 , then limx .... 1 f(x) = L. 5. Let (s., : n e *N) be an internal *R-valued sequence, and suppose that there is an M > 0 in R so that ls., l s; M for all n e N. Show that there is an ro e • N oo so that I s., I s; M for all n s; ro in • N . 6. Show that the assertion in Exercise 5 is not true if " I s., I :s; M" is replaced by "s., is finite." 7. A filter F e V(X) - X has a countable subbasis if there is a countable family {A1 : i e N} of entities in F so that for each F e F there is a sequence i1, , i., with nA1,.( 1 s; k s; n) £ F. Suppose that B is an internal set in V(* X) and F has a countable sub basis. Show that if B n * F = 0 for all F e F then B n J.t(F) -=!- 0. where J.t(F) is the inter section monad of F introduced in Exercise 1 1 . 5.9. 8. Let •: V(R) -+ V(*R) be comprehensive, and let S = {n�: : k e N} be a countable set contained in • N oo . •
•
•
(a) Show that S has a lower bound i n • N oo . [Hint: Regard S as a se quence, i.e., a map h: N -+ *N with h(k) = nt. Use comprehensiveness to extend h to an internal map g: • N -+ • N and apply the spillover principle to the set A = {m e *N :g(k) > m for all k � m } . ] For decreasing sequences nt this was presented by DuBois-Reymond and proved in our context by Robinson. (b) Use the transfer principle applied to g to show that S has an upper bound in *N oo ·
9. Show that if f is an internal function on an internal set A in some super structure V(*X), and f is finite-valued, then there exists a standard n e N so that IJ(x) l :s; n for all x e X. Give an example to show that the assertion is not necessarily true if f is not internal.
1 04
II.
Nonstandard Analysis on Superstructures
10. ( • -Convergence and S-Convergence) An internal *R-valued sequence (s. : n e *N) is (i) • -convergent to L e *R if for each e > 0 in *R there is an m e • N so that n > m implies Is. - L l < e, (ii) S-convergent to L e • R if s. :!:: L for all n e • N fZJ . (a) Show that if s. = • t. where (t.) is a standard sequence converging to L, then (s.) is • -convergent and S-convergent to L. (b) Show that there are internal sequences which are • -convergent but not S-convergent and vice versa. (c) Show that if (s.,) is S-convergent to a finite L e • R then there is an m e N so that s. is finite for n � m and the standard sequence (0s. : n e N ) converges to 0L. (d) Show that if s. = • t. , where (t.) is a standard sequence, then (s.) is S-convergent to a finite L iff there exists an infinite c.o e • N fZJ so that •s. :!:: L for every n e *N fZJ with n :::;; c.o. 11.8 JC-Saturated Superstructures
Theorem 7.5 of the last section is a good example of a result in which a standard entity (a Banach limit) is obtained by performing a standardizing operation on an internal entity [in this case, taking the standard part of the internal sum (1/c.o) L *s 1( 1 :::;; i :::;; c.o)]. Similar applications of nonstandard analysis often occur in more complicated circumstances, and sometimes the internal structure in a given extension V(* X) of a superstructure V(X) is not rich enough to produce a desired result. A specific example arose from a re sult of Robinson, which was that if X is a metric space and B an internal subset of • X in an enlargement V(* X), then the standard part of B is closed (definitions and results will be presented in Chapter III). It was natural to ask whether the result was still true if X was not metric. An example due to H. J. Keisler showed that the answer was negative if V(* X) was only an enlargement of V(X) (36, Example 3.4.3]. Luxemburg [36, Theorem 3.4.2] showed that the result does go through if V(* X) is large enough to satisfy a generalization of the property of an enlargement, valid for internal con current binary relations on an appropriate set A in V(* X). V(* X) is called K-saturated, where " is a cardinal number, if this generalization holds for all sets A in V(*X) with the cardinality of A < " (Definition 8. 1). It is not necessary for the reader to be very knowledgeable about the theory of cardi nal numbers for arbitrary sets in order to apply the theory. In a typical application we will begin with an internal concurrent binary relation on A-then we can assert that the results of the section will be applicable if V(* X) is sufficiently large. Sufficiently large means that V(* X) is K-saturated,
11.8
JC-Saturated Superstructures
105
where K > card A, but this is irrelevant in the application as long as we are assured that K-saturated structures exist (Theorem 8.2). Let V(X) be a given superstructure and • : V(X) -+ V(• X) a monomorphism. We write card A to denote the cardinality, in the standard sense, of a set A . V(• X) is K-saturated if, for each internal binary relation P e V(• X) which is concurrent (Definition 5.9) on some (not necessarily internal) set A in V(•X) with card A < K, there exists an element y e range P so that (x, y) e P for all x e A.
8 . 1 Definidon
H. J. Keisler [2 1 , 22] characterized those ultrafilters d/1 such that the super structure V(• X) constructed from a given superstructure V(X), using d/1 as in §11.4, is K-saturated; he called them K-good ultrafilters. In [21] Keisler established the existence of K-good ultrafilters on the assumption of the gen eralized continuum hypotheses. This assumption was subsequently removed by Konen. Thus we have the following result. Given any superstructure V(X) and cardinal K there is a K saturated superstructure V(• X) and a monomorphism • : V(X) -+ V(• X).
8.2 Theorem
For the proof of this and related results the interested reader is referred to the papers mentioned above and also to the book by Stroyan and Luxemburg [46] , where the desired structures are constructed as limits of ultrapowers. In any applications it will not be necessary to know the details of the proof. It follows from Theorem 5. 10 that if K > card V(X) then V(• X) is an enlargement. In applying Theorem 8.2 it is important to note that the set A of Defini tion 8.1 need not be internal, although the binary relation P must be internal and so the elements of A are internal. For a successful application, however, we do need an upper bound on the cardinality of A which is independent of the particular construction of V(• X). For example, suppose that P is the bi nary relation on • R x •9.,(R) defined by " ( x, B) e P iff the • -finite set B con tains x." Then P is concurrent on any subset A £ • R. However, it is not possible to apply Theorem 8.2 and Definition 8. 1 with A = • R; i.e., it is not possible to find a • -finite subset of •R which contains all numbers of •R, no matter how large K is. For then •R itself would be a • -finite set and hence, by transfer down, R would be finite. The error occurs in trying to apply the result to the set A = • R whose cardinality depends on the con struction of the extension V(• R) and is not fixed in advance. In [36] Luxemburg developed a general theory of monads in enlargements and K-saturated extensions. In the following we present several of his impor tant results.
1 06
Nonstandard Analysis on Su pe rstructures
II.
Let • : V(X) -+ V(• X) be a monomorphism, and let A be an en tity in V(X). The (intersection) monad Jl(A) of A (with respect to • ) is the set
8.3 Deftnidon
Jl(A)
=
n•a(a e A).
Monads Jl(A) are most important when A is a filter �. i.e., when 0 ' �.
F and G in � implies F n G e �, and F e � and G ;;;2 F implies G e �. The next result generalizes the permanence principle. (Luxemburg) Let • : V(X) -+ V(• X) be a monomorphism, and assume that V(•X) is �e-saturated. Fix a filter � e V(X) with card � < �e; then
8.4 Theorem
(a) given an internal set B e V(• X), if • F n B ::1: 0 for all F e �, then JJ(�) n B ::1: 0. (b) given an internal subset A of •� such that every standard element of • � is an element of A, there exists an element E e A such that E s;;; JJ(�). (c) given an internal subset A of •� such that E e • � and E s;;; JJ(� ) implies E e A, there exists an element F e � such that • F e A.
Proof: (a) Define an internal relation P, with domain • � and range con tained in B, by "(F, x) e P if x e B n F." Then P is concurrent on the collec tion of standard elements of •�. and this collection has the same cardinality as �. Therefore there is a y e B so that y e B n • F for each F e �. i.e., y E Jl(�). (b) Define an internal relation P, with domain •� and range contained in A, by "(F, G) e P if G e A and G s;;; F." Then P is concurrent on the collection of standard elements of •� (why?), so there is an E e A such that E s;;; • F for each F e �. i.e., E s;;; Jl(�). (c) Let A satisfy the condition of (c). If A does not contain a standard element • F e •� then the internal set •� A c •� contains all standard elements of •� and so by (b) there exists an element E e •� A with E s;;; Jl(�). But then E e A by the hypothesis on A (contradiction). D -
-
Several exercises in the preceding sections have dealt with situations in which, without saturation, the statement (a) of Theorem 8.4 may or may not hold. The results can be summarized as follows: The statement does not hold in general if B is not internal (Exercise 11.5. 1 1), but does hold if B is standard (Exercise 11.5. 1 0) or if B is internal and � has a countable basis (Exercise 11.7.7). (See Theorem 8.6.) An example due to H. J. Keisler (see Example 2.7.4 in [36]) shows that the statement need not hold if B is internal but V(•X) is only an ultrapower enlargement. We note finally that an internal version of comprehensiveness holds in �e saturated extensions.
11.8
107
K-Saturated Superstructures
8.5 Theorem Let V( * X) be a K-saturated extension of V(X). Assume C is a (not necessarily internal) set of entities in V,.(* X) for some n e N with card C < "· and D is an internal set in V( * X). For any mapping rjJ: C -+ D, there is an internal extension (/): C -+ D of rjJ [i.e., C is internal, contains C, and t/J(a) = (/)(a) if a e C]. If C = { *a : a e C0} we may take C = *C0.
Proof: � t P be the binary relation " ( t/J, lP > e P iff (fJ is an extension of t/J" [i.e., dom tP :;;;;;! dom tP and t/J(a) rjJ(a) if a e dom rjJ] defined on the set of internal mappings with values in D. Let A be the set of all internal mappings fx: { x } -+ t/J(x), x e C. That is, each element of A is a set consisting of exactly one element from rjJ. Then card A card C < K and P is concurrent on A (check). Thus there exists an internal map (/) with values in D which extends each fx , x e C, and so dom (/) = C :;;;;;! C and (/)(a) = rjJ(a), a e C. The rest is =
=
left as an exercise (Exercise 1 ).
D
There is a converse of Theorem 8.5 when cardinal number bigger than card N.
"
=
� t o where �1 is the first
V( * X) is a denumerably comprehensive extension of V(X) (Defi nition 6. 1 1) if and only if V(* X) is � 1 -saturated.
8.6 Theorem
Proof: Exercise. 8.7 CoroUary
D
An extension V( * X) constructed as in §11.4 is �1 -saturated.
Proof: Follows from Theorems 6. 1 3 and 8.6.
D
Corollary 8.7 shows that assuming � 1 -saturation in an application of non standard analysis is not assuming very much. Later in this book we assume a stronger form of saturation (larger K) only in the proof of Theorem 1 .22 of Chapter III (which is not used afterward) and in the proofs of the last few results in §IV.3, where K-saturation is used in a more significant way. Exercises
11.8
1 . Show that if the set C in Theorem 8.5 has the form {*a:a e C0 } then one may take C • c o in the conclusion of the theorem. 2. Prove Theorem 8.6. 3. Let V( * R) be a K-saturated extension of V(R) with card B'(R) < K. Let B be an internal subset of *R and st(B) = { x e R : there exists a y e B with st(y) = x}. Use Theorem 8.4(a) to show that st(B) is closed in R. =
1 08
II.
Nonstandard Analysis on Superstructures
4. (Luxemburg [36]) Suppose that V(* X) is a K-saturated extension of V(X) with " > card V(X). Let A e V(X) contain an infinite number of ele ments. If A c *(&'�A) ) is internal and moreover, E e A for every • -finite subset E c *A with the property that A = {a e V(X) : *a e E}, then there exists a finite subset { a1 , , aft} c A so that {*a1 , , *aft} e A. (Hint: Apply Theorem 8.4 to the Frechet filter of A.) •
•
•
•
•
•
CHAPTER Ill
Nonstandard Theory of Topological Spaces
In Chapter I we showed how the notion of continuity for real-valued func tions of a real variable could be characterized in terms of the nonstandard concept of nearness [! is continuous at x if •J(y) � f(x) for all y � x]. On the real line, nearness and the associated concept of monad are characterized in terms of the distance function, so that x � y if lx Yi � 0. We also char acterized open and closed sets in terms of monads. In this chapter we will show how these notions can be extended to more general settings. In the standard development of topology one usually begins with a set X possessing a collection S" of (open) subsets satisfying the abstract analogues of conditions (i) and (ii) of Theorem 9. 2 in Chapter I. The pair (X, S") is called a topological space. The notions of continuity can then be defined just in terms of the open sets; i.e., a function f: X -+ Y is continuous if f - 1(V) is open in X for every set V which is open in Y. In the nonstandard theory developed here, we will show how the collection S" on X can be used to char acterize nearness and monad and so allow a simple development of the theory of topological spaces analogous to that of Chapter I. One of the most useful results in the nonstandard development is a charac terization of compact spaces (the analogues of closed bounded sets on the real line) due to Abraham Robinson. This development is presented in §111.2, with an elaboration in §III.7. Sections 111.3, 111.4, and 111.5 are devoted to the nonstandard theory of metric, normed, and inner-product spaces, which are of central importance in much of analysis. In §111.6 we show how one may begin with a standard metric space X and construct a (standard) metric space on the nonstandard set • X, leading to the so-called nonstandard hull of a metric space. This con struction plays a central role in some recent applications of nonstandard anal ysis to the theory of Banach spaces by Henson and Moore (see [ 1 6] for a -
1 09
l lO
Ill.
Nonstandard Theory of Topological Spaces
review). The section ends with a discussion of some results in the theory of function spaces, and includes a generalization of the Arzela-Ascoli theorem of Chapter I.
1 1 1 . 1 Basic Definiti ons a n d Results
A topological space is a pair (X, ff), where X is a set and ff is a family of subsets of X satisfying the conditions in the following definition. 1.1 Definition
X if
A family 5 of subsets of X, called open sets, is a topology for
(a) 0. X e ff ; (b) U, V e .r implies U n V e .r (and thus every finite intersection of open sets is open), (c) U 1 e ff (i e I) implies U V ,(i e I) e ff, i.e., every arbitrary union of open sets is open.
Closed sets are complements of open sets. Often we call X rather than (X, 5") the topological space. The usual family of open subsets of R, defined in the proof of Proposi tion 9. 1 of Chapter I, is a topology for R (Theorem 1.9.2). We will presently see that there are many topologies for R as for most sets. With each topology we will associate corresponding notions of convergence and continuity, using only the open sets. In order to develop a nonstandard theory, we first generalize the notions of nearness and monad which were central to the work in Chapter I. We begin with a few basic definitions. 1.2 Definition Let (X, 5") be a topological space. A set U is a neighborhood of a point x e X if U contains an open set V which contains x. The neighbor hood system ..¥" of x is the set of all neighborhoods of x. We denote the system of open neighborhoods of x e X by 5"" . A collection � s;;; ff is a base for ff if each set in ff is a union of sets in � or, equivalently, if for each x e X and each U e 5"" there is a V e ff" n � with V s;;; U. (For example, open intervals form a base for the usual open sets in R.) A collection � is called a subbase for ff if the collection of finite intersections of members of � is a base for !T. Similarly �" s;;; .A: is a (neighborhood) base at x if for each U e ..¥" there is a V e �" with V s;;; U; �" s;;; ..¥" is a subbase at x if the col-
111. 1
Basic Definitions and Results
Ill
lection of finite intersections of members of tJI" is a base at x. If §' and f/ are topologies for X, then ff is weaker than f/ (and f/ is stronger than §') if ff s;;; f/. From now on we work in an enlargement V(*S) of a superstructure V(S), where V(S) contains the standard space X under consideration, so ff e V(S) as well. In this section we will not use the fact that if x e X then x may contain elements. Therefore, we will write x instead of •x for the nonstandard extension of x. 1 .3 Definition The sets in • §' are called • -open subsets of • X. The monad of X E X is the subset m(x) = n • u( U E §'") of •x. A point y E •x is near x e X, and x is the standard part of y, if y e m(x); then we write y � x and x = st(y) . The set of near-standard points is the set ns(* X) = Um(x)(x e X). A point y e • X is called remote if it is not near-standard.
An easy exercise shows that m(x) 1.4 Proposition
=
n • u(U E %").
If /JI" is a local subbase at
X,
then m(x)
=
n • U(U E /JI").
Proof: n • u(U E £f") 2 n • u(U E %") since 111" s;;; %" . On the other hand, for each U E %" there exist Vj E /Jiz{l � i ::s;; n) with n V� l ::s;; i � n) s;;; U, and so n • Vll ::s;; i ::s;; n) s;;; • u by transfer. Hence n • v( V E /JIJ s;;; n•u(U E %").
D
1 .5 Examples
1 . Discrete topology. (X, ff) is discrete if {x} is open for each x e X. In this case m(x) = {x} for each x e X. 2 . Trivial topology. (X, ff) is trivial if ff = {0, X } . In this case m(x) = • X for each x e X. 3. Usual topology on R. The open sets in R as defined in §1.9 constitute a topology. The monads as defined here and in Definition 6.4 of Chapter I are identical [where we assume that *91 and V(*R) are obtained from the same ultrafilter]. This follows immediately from Proposition 1 .4 since the set tJIx of symmetric open intervals about x forms a local base by the definition of open set in R. A subbase for the topology is formed by intervals of the form ( oo, b), (a, + oo) with a b e R. 4. Half-open interval topology on R. Let ff be the topology for R which has as base the set tJI of half-open intervals [a, b) = { x : a ::s;; x < b}, where a and b are real. Here m(x) = { y e * R : x � y, x � y} (Exercise 1). -
,
112
Ill.
Nonstandard Theory of Topological Spaces
5. Finite complement topology. For simplicity let X = N (any infinite set would do), and let f7 be the collection consisting of the empty set and those subsets of N whose complements are finite. It is an easy standard exercise to show that f7 is a topology. Here m(x) = {x} u *N en (Exercise 1). 6. Product topology. Let (X, f/) and (Y, 9') be topological spaces. Then X x Y can be made into a topological space as follows: A set W £ X x Y is open if to each (x, y) e W there correspond sets U e f/", V e 9', so that U x V £ W; i.e., products of open sets form a base for the topology (check that this defines a topology). The resulting topology is called the product topology and is denoted by f7 x 9'. If mr , m9' , and m denote monads in (X, f/), ( Y, 9'), and (X x Y, f7 x 9'), respectively, then m((x, y)) = mr(x) x m9'( y), x e X, y e Y (Exercise 1). The following facts should be noted in comparing the usual monads for
R and monads in a general topological space (X, f/): (a) The concept of nearness is derived from that of monad and not vice versa as in Definition 6.4 of Chapter I. (b) We have defined monads only for standard points in • X. (c) Nearness is not in general an equivalence relation on • X [this is, of course, because of (b)]. The monad m(x) always contains x. That m(x) will in general contain points other than x follows from the following basic lemma, the proof of which requires that V(• S) be an enlargement. 1 .6 Proposition
For each x e X there is a • -open set V e • f/" with V £ m(x).
Proof: The binary relation P on f/" x f/" defined by P( U, V) if V £ U is concurrent. For if U 1 o , U n e f7, then V = U 1 n · · · n Un satisfies P( U�o V ), 1 ::5; i ::5; n. Since V(*S) is an enlargement, Theorem 5.10 of Chap ter II guarantees the existence of an element V e • f/", so that V £ •u for all U e f7, and hence V £ m(x). D •
1 .7 Proposition
•
•
Let A be a subset of X. Then
(i) A is open iff m(x) £ • A for each x e A, (ii) A is closed iff m(x) n • A 0 for each. x in the complement A' of A. =
Proof: (i) Suppose A is open and let x e A. By definition there exists an open set U e f/" with U £ A. By transfer m(x) £ • U £ • A.
111.1
1 13
Basic Definitions and Results
Conversely, suppose m(x) !;;;; *A for x e A. By Proposition 1 .6 there exists a m(x) !;;;; *A. Thus the internal sentence (3 V e *ff,) [ V !;;;; *A] is true and so, by downward transfer, there exists a set V e ff, with V !;;;; A. Thus A is open since A = U V,(x e A). (ii) This follows immediately from (i) and the definition of a closed set: A is closed if A' is open. 0 V e * fl", with V !;;;;
1 .8 Definition A point x is an accumulation point of the set A !;;;; X if every open neighborhood of x contains points of A other than x. We let A denote the set of accumulation points of A; the set A A u A is the closure of A. A is dense in B if A B. =
=
1 .9 Proposition A point x is an accumulation point of A iff m(x) contains a point y e *A different from x.
Proof: If x is an accumulation point of A then the sentence (VU e ff,) U n A) [ y ::1= x] is true for V(X), and hence, by transfer, each U e * fl", contains a point y '# x in *A. This is true, in particular, of the • -open set V of Proposition 1 .6, and so there is a y e m(x) n *A with y ::1= x. Conversely, suppose that m(x) contains a point y '# x in *A. Then, for a fixed U e ff, , * U contains a point y '# x in *A. Thus the internal sentence (3y e *(U n A))[y '# x] is true, and it follows by downward transfer that there exists a y e U n A with y '# x. 0 (3y e
1 . 1 0 Proposition The closure A of A !;;;; X consists of those x e X for which m(x) n *A '# 0. The closure of A is the smallest closed set containing A. Thus A A if A is closed. =
Proof: Exercise.
0
Let f7 and Y be two topologies for a set X with associated monads m_,..(x) and m.Y(x) (x e X). An easy exercise shows that f7 is weaker than Y iff m,.-(x) 2 m.Y(x) for each x e X. We noted in §1.6 that if x and y are distinct standard real numbers then m(x) n m(y) is empty. Therefore, we say that R is a Hausdorff space. This property is not true in general for topological spaces. Properties of spaces which deal with the relationship between monads of distinct points are called separation properties. Some of the more important separation properties are presented next; the most important of these is the Hausdorff property.
1 14 1 . 1 1 Definition
Ill.
Nonstandard Theory of Topological Spaces
The space (X, ff) is
(a) T0 if, for each pair x, y of distinct points in X, there is an open neigh borhood of one not containing the other, (b) T1 if {x} is closed for each x e X, (c) Hausdorff (or T2 ) if whenever x -::/= y in X there are disjoint open neigh borhoods U and V of x and y. There are more separation properties (e.g., regularity and normality) which we will consider in the exercises. 1 . 1 2 Proposition
The topological space (X, ff) is
(a) T0 iff whenever x, y e X and both x e m(y) and y e m(x) then x (b) T1 iff whenever x, y e X and x e m(y) then x = y, (c) Hausdorff iff monads of distinct points in X are disjoint.
=
y,
Proof: We prove (c) and leave the other proofs as exercises. Suppose (X, ff) is Hausdorff and x, y e X are distinct. Then there exist U e �. V e ff., with U ("\ V = 0. Therefore, •u ("\ • v = 0. and since m(x) £ •u and m(y) £ • v. we have m(x) r\ m(y) = 0. Conversely, if m(x) r\ m(y) = 0 then by Proposition 1.6 there exist U e •ff", V e • ff., with U ("\ V = 0. By downward transfer of the appropriate sentence (check), there exist U e .r" , V e ff., with U r\ V = 0. D If (X, ff) is Hausdorff then there is only one standard point st(y) associated with each y e ns( • X). It is defined by st(y) = x, y e m(x). Thus for Hausdorff spaces we have a well-defined map st: ns(• X) -+ X called the standard part map, which has many applications (e.g., see §IV.3 below). 1 . 13 Examples
l . The discrete topology is Hausdorff, and every subset is both open and closed. 2. The trivial topology of a space with two or more points is not T0 • 3. The finite complement topology on N is T1 but not Hausdorff by Prop osition 1.1 2. Also a set is closed in the finite complement topology iff it is finite. For if A is finite then • A = A, and if x e A' then m(x ) ("\ • A = ( { x } u • N rx; ) r\ A = 0. On the other hand, if A is infinite then • A r\ • N rx; -::/= 0 by 6. 1 1 of Chapter I, and m(x) r\ • A -::/= 0 for any x.
So far we have used a topology .r to define associated monads m(x), x e X. Conversely, it is possible to start with a collection k(x), x e X, of subsets of • X with x e k(x), and define an associated family ff as follows: U e ff if
111.1
1 15
Basic Definitions and Results
k(x) s;;; • u for each x e U. An easy exercise shows that !T is a topology. If k(x) (x e X) are the monads of !T then clearly k(x) s;;; k.(x) for all x e X, but set equality does not necessarily hold (see Exercise 6). The sets k(x) will be called pseudomonads; the concept will be used in §II1.8. Let (X, !T) and ( Y, 9') be topological spaces with monads m(x) (x e X) and m(y) (y e Y), respectively. To discuss continuity of mappings f: X -. Y we work in an enlargement containing *X and * Y and thus *!T, *9' and all mappings *f: •x -. * Y, etc. The symbol � will be used for the relation of nearness in both (*X, • !T) and (* Y, • 9'); the context should clear up any ambiguities. The map f: X -. Y is continuous at x e X if to each V e 9'J<xl there corresponds a U e !T"' with f[U ] s;;; V. f is continuous on X if it is continuous at each x e X. A one-to-one map f from X onto Y is a homeomor phism if f and f- 1 are continuous. 1 . 14 Definition
The map f: X -. Y is continuous at x e X iff *f(y) � f(x) x. That is, *f[m(x)] £ m(f(x) ).
l . lS Proposition
for each y
�
Proof: Suppose f is continuous at x e X, and let V by any open neighbor hood of f(x) . Find a corresponding U e !T"' from the definition of conti nuity so that f[U] s;;; V. If y � x then y e •u by 1 .7(i), so *f( y) e • v since *f [* U] £ • V by transfer. Thus *f(y) e • V for each V e 9'/(xJ • i.e., *f( y) � f(x) The converse is left to the reader. D .
Proposition 1 . 1 5 shows that for real-valued functions of a real variable, Definition 1 . 1 4 is equivalent to the e-d definition of continuity. 1 . 16 Theorem
each V e 9'.
The map f: X -. Y is continuous on X iff f - 1[V] e !T for
Proof: Fix x e X, suppose f is continuous, and let V e 9'/!xJ · Then *f[m(x)] s;;; m( f(x) ) s;;; • v by continuity at x and the fact that V is open. It follows that m(x) s;;; *f- 1 [* V] = *(f- 1 [ V]) (check), and so f 1 [ V] is open by Proposition l . 7(i). The converse is left to the reader.
D
-
The reader will have noticed that the proofs of the results 1 .6- 1 . 1 6 are considerably simpler than the proofs of the corresponding results in §§1.9 and 1. 1 0. This is mainly because the richer language of Chapter II allows us to avoid proofs by contradiction which use Skolem functions. If Y is a subset of the topological space (X, !T), then !T induces a topology called the relative topology !Ty on Y. A subset U £ Y belongs to !Ty iff
1 16
Ill.
Nonstandard Theory of Topological Spaces
U = V n Y for some V e ff. It is easy to see that the monads in ( Y, ffr) are given by m(y) = m(y) n • Y, y e Y, where m(y) is the monad of y in (X, ff) (check). The characterizations of relative openness, relative closedness, con tinuity, etc., are the obvious modifications of those we have just proved with m replacing m. Next we define the important notion of a weak topology. Suppose that X is a set and (X; , ffi) (i e I) is a family of topological spaces. We work in an enlargement containing • X and • Y where Y = UX,{ i e 1). We let m�y) (i e I, y e X;) denote the monads of y in (X�o ffi). Let { t/>i: X -+ X; : i e I} be a family of mappings. 1 . 1 7 Definition The weak topology .1" on X for the family { t/>; : i e I} is the topology generated from the subbase f/ consisting of all inverse images of the form t/>;- 1 [U ] , U e ff1; i.e., .1" consists of all sets obtained by taking arbitrary unions of finite intersections of sets in Y.
The weak topology is the weakest topology which makes all the maps t/>; continuous (Exercise 8). 1 . 1 8 Proposition
If m(x) (x e X) is a monad of the weak topology, then
m(x) = { y e • X : * t/> ,{ y) e m� t/> �x) ) for all i e 1}. Proof: Let the right-hand side of the equation be denoted by k(x). If x e X then for i e I the sets t/>1- 1 [ U ] , U e ff� ,(x) • are open neighborhoods of x, so
m(x) £ n { y E * X : y E n •( tJ>;- 1 ( U ] )(U E ff �,( x ) )}(i E I) = n { y e * X : y e * t/> ;- • [ n • u ( U e ff�,( x ) ) ] }(i e l ) = k(x) .
On the other hand, if V e ffx is a neighborhood in the base of §' generated by the subbase f/, then V is a finite intersection of sets of the form t/>;- 1 [ UJ, U; e ff�,(x) · Clearly k(x) £ *4>i- 1 [* UJ for each U1 e ff�,(xJ and so k(x) £ • v. It follows that k(x) £ m(x), and we are through. 0 Let (X ; , ff i ) (i e I) be a family of topological spaces. Then the product X = n X �i e I) is defined to be the set of all mappings x on I with x(i) e Xi for i e I. The product topology ff for X is the weak topology generated by the mappings t/>i : X -+ X i defined by t/>�x) = x(i). 1 . 1 9 Definition: The Product Topology
Basic Definitions and Results
111.1
1 17
To see what • X is, note that each x E X is of the form x: I -+ U X f...i e I) with x(i) e X1 • The • -transform of the collection { X1 : i e / } includes new sets X; for i e • I - I. Thus, by transfer, each x E • X is of the form x : • I -+ *[ U X /.. i e /)] with x( *i) E • X; if i E /, whereas if i is not standard, then x(i) e X; , but X1 need not be the extension of a standard set. If x e X, and m(x) denotes the monad in ff, then by Proposition 1 . 1 8
m(x)
=
{ y e •x : y(i) e mf...x( i) ) for all standard i in */ } .
That is, the monad is determined by just the standard indices in • I . 1 .20 Theorem The topological product of Hausdorff spaces is Hausdorff. Proof: Let X = 0 X; , where the (X;, 9'j) are Hausdorff with monads mf...x ). ff be the product topology with monad m(x). If x, y e X with m(x) n m(y) # 0, let z e m(x) n m(y). Then z(i) e ml,*x(i) ) n mf... * y(i)) for each i e /, and so x(i) = y(i) for each i e I since (X; . 9'i) is Hausdorff, i.e., x = y. 0
Let
We end this section with a result which is valid under the assumption that X is in V(*S) for some S and V(*S) is K-saturated with " > card ff. This result was mentioned at the beginning of §11.8 as a good example of the use of saturation in nonstandard analysis. It will be referred to again in §IV.3. 1 .21 Definition Let (X, ff) be a topological space with monads m(x), x e X. The standard part st(A) of a set A � • X is the set of all x e X for which there exists a y e A with y e m(x). *1.22 Theorem Assume X e V(S) and V(*S) is K-saturated with " > card ff. If B � • X is internal then st(B) is closed. Proof: Suppose z is an accumulation point of o B = st(B). If U e � then there exists a point x e o B with x e U. Since x e o B there exists a y e B with y e m(x), and hence y e • u since U is open. Thus •u n B .;: 0 for all U e ff• . Since V(* X) is K-saturated with " > card ff" , we see from Theorem 8.4{a) of Chapter II that Jl(ff.) n B .;: 0. where p.(ff.) is the intersection monad of the filter ff. (Definition 8.3 of Chapter II). Clearly p.(ff.) = m(z), and so z e B, and we are through. 0 o
Note that if A � •x then st( A ) = st(A n ns(*X) ). Also note that Proposi tion 1 . 1 0 can be interpreted to say that A = st(• A n ns(* X) ), and so Theorem 1 . 22 is a generalization of Proposition 1 . 10. Theorem 1.22 was established
Ill.
1 18
Nonsta nda rd Theory of Topological Spaces
for metric spaces by Robinson using an enlargement [42, Theorem 4.3.3], and in the general case (assuming saturation) by Luxemburg [36, Theorem 3.4.2]. An example due to Keisler shows that Theorem 1.22 is not true if V(*S) is not K-saturated with K > card §" [36, Example 3.4.3]. Exercises 111.1
1 . Verify the statements in Examples 1 .5.4-6. 2. Prove Proposition 1 . 10. 3. Prove (a) and (b) of Proposition 1 .12. 4. Prove that a topology §" is weaker than a topology fJ' on X iff m,-(x) ;;;;;! m.9'(x) for each x e X, where m,- and m.9' denote the monads for §" and !/', respectively. 5. A T1 space is normal if for any two disjoint closed sets A and B there are disjoint open sets U and V with A s;;; U and B s;;; V. A T 1 space is regular if the same condition holds for all A and B, where A is a point (actually a set consisting of a point) and B is a closed set. Give a non standard condition for regularity and normality. 6. (a) Let k(x) be a subset of • X for each x e X. Define a collection §" of subsets of X as follows: U e §" iff k(x) e •u for each x e U. Show that §" is a topology for X. Also show that if k(x) is the §"-monad of x e X then k(x) s;; k(x). (b) Fix an infinitesimal e > 0 in • R and for each x e R let k(x) be the pseudomonad { y e * R : I y x l < e } . Show that a set U is open in R in the usual sense ifand only if, for each x e U, k(x) c: * U. Clearly k(x) • m(x) for each x e R. (c) Let X be any set. Let �'" x e X, be a collection of subsets of X satis fying the following: -
(i) If V E �.., then X E V, (ii) If ¥1 , V2 E �" ' there exists a V E �" with V £ V1 n V2 , (iii) If y e U e �..,. then there is a V e �., with V £ U. Use the sets k(x) = n• u(U e �..,) to define a topology §" as in 6(a). Show that �" is a neighborhood base in §" for each x e X. 7. Finish the proof of Proposition 1 . 1 5. 8. Show that the weak topology is the weakest topology making the corre sponding functions continuous. (See Definition 1 . 1 7.) 9. Let A be a subset of a topological space X. A point x is an interior point of A iff A is a neighborhood of x. The set of interior points of A is denoted by A0• A point x is a boundary point of A if x is not interior to A and not interior to A'. The set of boundary points of A is denoted by oA.
111.1
1 19
Basic Definitions and Results
Show that (a) (b)
X e Ao X E oA
iff m(x) £:: • A, iff m(x) n ·A ::1: 0 and m(x) n •A' ::1: 0.
10. Let A be a subset of a topological space X. Use Exercise 9 and the text
material to establish the following results: (a) oA = A n A' = A - A0, (b) X - oA = A o u (A')0, (c) A = A u o A , A0 = A - oA, (d) A is closed iff A ;;:;! oA, (e) A is open iff A n oA = 0.
1 1 . Let (X A
£::
X
(a) (b) (c)
x Y, 9" x 9") be the product of (X, 9") and ( Y, 9"). Show that if and B £:: Y then
::rx-B
=
A
x
B,
(A X 8)0 = A0 X 8°, o(A x B) = (o A x B)
u
(A
x
oB).
1 2. Let Y be a subset of (X, 9") with relative topology 9"y . If A
that
1 3.
14.
1 5.
1 6.
£:: Y show
(a) A is ffy-closed iff it is the intersection of Y and a ff -closed set. (b) A point y e Y is a ffy-accumulation point of A iff it is a 9"-accu mulation point. (a) Let (X 1 , 9"1), (X 2 , 9"2 ), and (X 3 , 9"3) be topological spaces. Show that a function f: X 1 -+ X 2 is continuous iff, for each subset A £:: X, f[A] £:: /[A]. (b) Show that if f: X 1 -+ X 2 and g: X 2 -+ X 3 are continuous, then the composite function h = g o f defined by h(x) = g(f(x)) for x e X, is con tinuous. Let ff be the product topology on X = TIX1(i e I) where the (X" �) are topological spaces. If A1 £:: X1 for each i e I, show that TIA�i e I) = TIA�i e I), so that the product of closed sets is closed. (a) A sequence (x,. : n e N) in a space (X, ff) converges to x e X if for every neighborhood U of x there is an m so that x,. e U if n � m. Show that (x,.) converges to x iff •x.., e m(x) for all infinite w. (b) Let (x,. : n e N) be a sequence in X = TIX�i e I), where the (X1, ff1) are topological spaces. Show that (x.) converges to x e X iff (
IU.
120
Nonstandard
Theory of Topological Spaces
111.2 Compactness
A cornerstone of topology is the notion of compactness, which is defined as follows. A collection .91 = {A1 : i e I} of sets is a cover of (or covers) X if A s;;; UA1 (i e /). A subcover of .91 is a subcollection of .91 which also covers A. A is a compact subset of a topological space (X, 9") if each open cover, that is, each cover of A by open sets U1 (i e /), contains a finite sub 2. 1 Definition
A
s;
cover. Probably the most useful result in nonstandard analysis is the following pointwise characterization of compactness due to Robinson. l.l Robl1110 n 's Theorem Let (X, ff) be a topological space. Then A s; X is compact iff every y e • A is near a standard point x e A.
Proof: Suppose A is compact but that there is a point y which is not contained in the monad of any x e A. Then each x e A possesses an open neighborhood U" with y ¢ • u" . The covering { U" : x e A } of A has a finite subcovering { U1 , Uft} ; i.e., U1 u · · u Uft ;;;;;;! A. By transfer • U 1 u · · · u • uft ;;;;;;! •A. This contradicts the fact that y e •A but y ¢ •u,, 1 s i s n. Conversely, suppose that A is not compact. Then there is an open cover ing .91 = { U1 : i e /} of A which has no finite subcover. The binary relation P on .91 x A defined by P( U, x) iff x ' U is concurrent (check). By Theorem 5 . 10(ii) of Chapter II there is a point y e • A with y ' •u for all U e .91. If x e A then x e U for some U e .91, but y ' •u so y ' m(x). D •
2.3
•
•
,
·
Examples
1 . In the discrete topology the only compact subsets are finite. 2. All subsets in the trivial topology are compact. 3. In the finite complement topology for N, every subset A is compact. For if A ¢ 0 and y e • A then either y e A or y e • N ao (Corollary 7.6 of Chap ter 1). In the first case y e m( y) and in the . second case y e m(x) for any x e N and, in particular, for some x e A. Recall that a set must be finite to be ,
closed in this topology, so there are compact subsets which are not closed in this non-Hausdorff topology. We use Robinson's theorem to give proofs of the following standard results.
111.2
121
Compactness
l.4 Theorem
If X is compact in the topology � and A
s;;;
X i s closed, then A
is compact.
Proof: Let y e • A. Since X is compact there is an x e X with y e m(x), whence x e A by 1 . 7(ii), so A is compact. D 2.5 Theorem
If (X, � is Hausdorff and A
s;;;
X is compact, then A is closed.
Proof: Let x e A' and suppose that y e m(x), y e *A. Since A is compact, for some :i e A, but then m(x) n m(.i) ::1: 0. contradicting the fact that (X. �) is Hausdorff. D y e m(.i)
2.6 Theorem If (X. �) and ( Y, 9') are topological spaces and f: X -+ Y is continuous, then f [ K ] is compact for each compact K s;;; X.
Proof: Exercise. 2.7 Theorem
D
If (X, � is compact, ( Y, 9') is Hausdorff, and f: X -+ Y is con
tinuous, then (i) f is closed (i.e., takes closed sets onto closed sets), (ii) if f is one-to-one then it is a homeomorphism. Proof: (i) Follows from 2.4-2.6. (ii) We may assume that f[X] = Y. We need only show that f is open (i.e., takes open sets onto open sets). But if U is open in X, then U' is closed. Since f is one-to-one, f[U] = Y f[U'], which is open by (i). D -
The real power of Robinson's theorem is illustrated by the proofs of the following standard results. The standard proofs of these results as given in Kelley [20] are somewhat involved. If (X1 , �1 (i e I) are compact spaces and X flX�i e I ), then X is compact in the product topology �.
2.8 Tyc:honofl''s Theorem
=
Proof: Let y e • X. Then y(i) e • X1 for (standard) i e I and so y(i) is near a standard point x1 e X 1 for each i e I. That is, y(i) e m1(xJ, where m1(xJ denotes the monad of x1 in (X1 , 5" 1). By 1 . 1 9, y e m(x), where m(x) is the monad in §' of the point x e X defined by x(i) = x1 • 0
122
Ill.
Nonstandard Theory of Topological Spaces
Tychonoff's theorem is used in many proofs in analysis. One can usu ally replace these standard proofs by simpler nonstandard ones which use Robinson's theorem directly (for example, see the proof of Alaoglu's theorem, 4.22, below). *2.9 Alexander's Theorem If 9' is a subbase for the topology of (X, ff) and every cover of X by members of 9' has a finite subcover, then X is compact. Proof: (Hirschfeld ( 1 8]) Suppose X is not compact. By 2.2 there exists a y e • X which is not near-standard and so for each x e X there is an open set U" with x e U" and y rl • U "" Since each U" is a finite intersection of members V; of 9', one of the • V; must omit y, so we may as well assume that U" e 9' for each x. Then the covering { U" :x e X} cannot have a finite subcover U1, u. , for in that case •x = • u 1 u · · · u • u. and y e • u, for some i, 1 � i � n (contradiction). D •
•
•
,
Exercises
111.2
1. Prove Theorem 2.6.
2. Let (X, ff) and ( Y, Sf) be topological spaces and suppose that ( Y, Sf)
3. 4. 5.
is compact Hausdorff. Show that f: X -+ Y is continuous iff the graph G1 = {(x,f(x) ) e X x Y: x e X} of f is closed in X x Y. Let X have the topologies .r and Sf, and suppose that (X, ff) is compact Hausdorff. Show that (a) if 9' is strictly contained in .r then 9' is not Hausdorff, (b) if .r is strictly contained in 9' then 9' is not compact. Show that if (X, ff) is compact then there is a hyperfinite set F s;;; •x with X s;; F s;; ns(• X) such that X = st(F). Suppose that (X, ff) is compact. Show that if (A. : n e N) is a sequence of nonempty closed subsets of X which is monotone, i.e., A 1 2 A2 ;2 · · , then nA.(n E N) ::1: 0. The following problem is derived from a result of A. Abian (see [ 1 ] ): Let Pn be a sequence of polynomials and x ., a sequence of variables so that, for each n, p. = p.(x 1 , x 2 , x.) is a function of the first n variables. Let I. be a sequence of closed and bounded intervals in R. Assume that for each n there are values aj e I1 for 1 � i � n such that, for each i � n, pAa� , a� , . . . , aj) 0. Show that there are values a1 e I 1 for 1 � i < oo such that, for each n e N, p.(a11a2, a.,) = 0. (Luxemburg [36]). Let (X, ff) be a regular Hausdorff space (see Exercise 1 .5). If A is an internal set in a K-saturated enlargement of V(X) where " > card .r, and A s;;; ns(*X), then st( A ) {x e X : there exists y e A with x st(y)} is compact. •
6.
•
•
•
,
=
•
7.
•
•
,
=
=
111.3
Metric Spaces
1 23 111.3 Metric Spaces
The most important topologies which occur in analysis are those asso ciated with a metric or distance function. The corresponding spaces are called metric spaces. A metric space is a pair (X, d) , where X is a set and d is a map X into the nonnegative reals satisfying (for all x, y, z e X) (a) d(x, y) = 0 iff x = y, (b) d(x, y) d(y, x), (c) (triangle inequality) d(x, z) � d(x, y) + d(y, z) .
3. 1 Definition x
from X
=
Each metric space (X, d) can be made into a topological space (X, ff,) by specifying that a set U e ff, if, for each x e U, there is an e > 0 in R so that the open e-ball B.(x) = {y e X :d(x, y) < e} £ U. The resulting collection ffd is a topology (standard exercise). When the metric d and associated topology ff, are understood we simply call X rather than (X, d) or (X, ffd) the metric space. Note that the open e-balls about a point x e X form a local base at x. 3.2 Examples
1 . R is a metric space with the usual metric d(x, y) = l x - Y l for x, y e R. 2. R is a metric space with the metric J(x, y) = lx - Yl /( 1 + l x - yi) (check). 3. Let X be any set and define d(x, y) = 1 if x "# y and d(x, y) = 0 other
wise. It is easy to see that d is a metric anti is called the discrete metric. 4. R" is a metric space under each of the following metrics [where x = Xn), Y = ( Y 1 • (x 1 , Yn)] : (a) d 1 (x, y) = Li'= 1 l x ; - Y; l . (p) d 00(X, y) = max { l x 1 y1 1 : 1 � i � n } Properties (a) and (b) of Definition 3.1 are trivial; to check property (c) for metric (a) we have [with z = (z 1 , , zn)] ·
·
·
·
,
·
· •
.
-
•
n
n
i=
i=
•
•
d(x, z) = L l x 1 - z11 � L1 l x 1 - Y; l + I Y1 - z11 = d(x, y) + d( y, z). 1 The triangle inequality (c) for metric ( p) is left as an exercise. 5. Let /00 (also often denoted by / 00 ) be the set of bounded sequences x = (x 1 , x2 , ) Then loo is a metric space under the metric defined by d 00(X, y) = sup {lx1 yJ i e N} with y ( y 1 , y2 , ). Note that d 00(X, y) is finite for any x, y e 100 since, for any i, l x ; - Y ; l � lx 11 + I Y ;I and so .
.
•
.
-
=
.
.
.
sup { lx1 - y1 l : i e N} � s up{lx1l : i e N} + su p { l y1l : i e N}.
1 24
Ill.
Nonstandard Theory of Topological Spaces
To check the triangle inequality (c) we have [with z = (z 1 , z 2 , l x ; - z ;l � l x ; - Y1 l + IY1 - z ; l � sup{ l x 1 - y1l : i e N} + sup{ I Y1 - z 1l : i e N} d00(X , y) + d00(y, z).
•
•
•
)]
=
The result follows by taking sup over i e N on the left. The nonstandard analysis of metric spaces will be carried out in an en largement V(*S) of a superstructure V(S) that contains X. We always assume that S contains the set of real numbers R. In proving abstract theorems con cerning a metric space (X, d) we will write x instead of •x for an element of • X. In concrete examples, it might be important to investigate in more detail the structure of the elements of •X. For example, if X = 100 then we could take S = R, in which case elements of 100 would appear as bounded real-valued functions on the integers. Often the set S in a particular example will not be specified; the reader should be able to fill in the details. By transfer, the • -transform *d of d satisfies the conditions of Defini tion 3. 1 with *d replacing d for all x, y, z e • X. Let (X, d) be a metric space. Two points x and y in *X are near if *d(x, y) � 0. We write x � y if x and y are near and x 1:. y otherwise. The monad of x e *X is the set m(x) = {y e • X : y � x } . Two points x, y in *X are in the same galaxy if *d(x, y) is finite. The principal galaxy of • X is the one containing the standard points, and is denoted by fin(* X). Points in fin(* X) are called finite. 3.3 Definition
An easy exercise shows that for standard points x e X the monad m(x) of Definition 3.3 coincides with the monad obtained from the associated topol ogy !id . The metric monads, however, are defined for all points x e • X. It is also easy to see that the relation � is an equivalence relation. 3.4 Examples
1 . In the metric of Example 3.2.1, x � y iff x - y is infinitesimal. 2. In each of the metrics on R" defined in Example 3.2.4, x � y iff X; - y 1 is infinitesimal for 1 � i � n (exercise). 3. Each element of *I oo is an internal function x: • N -+ • R, and we usually write x(i) = x1 and x (x1 : i e *N). The standard elements in •too are of the form • y, where y = (y1: i e N) is an element of 100 • Each x e *100 is •-bounded in the sense that there exists an M e • R (which could be infinite) so that l x11 � M for each i e •N (exercise). In passing, note that there are external =
111.3
1 25
Metric Spaces
functions z: • N -+ • R which are also • -bounded; an example occurs when l for i e N and z 1 = 0 for i e • N 00 • The real-valued function on rP(N) x 1 00 defined by (A, x) -+ sup{lx11 : i e A} extends by transfer to a • R-valued function on •rP(N) x •too . We again denote the value of this extended function by sup{lx1l : i e A } , where A s;; •N is inter nal and x e •too . Properties of the extended sup function can be obtained by transfer. For example, if A and B are internal subsets of •N and A s;; B then sup{ x ;l : i e A} :S sup{lx1l : i e B} . For each x y e • l oo we have •d 00 ( x , y) = sup{ x1 - y1 l : i e • N } . The monads in •too are easily characterized. We claim that if x, y e •too then x � y iff x1 � y1 for all i e •N. For suppose x � y. Then, for any i e •N, lx1 - y11 :S sup{lx1 - y1 l : i E • N } � 0. The converse is left as an exercise. The finite elements in •too are those x = (x1 : i e •N) for which there exists a finite M (and hence even a standard M) in • R so that lx11 :S M for all i e •N. The value of M depends on x. z1
=
l
,
All of the results of 3. 1 and 3.2 are available for the topological space (X, ff11) associated with a metric space (X, d). We concentrate in this section on some results which are special to metric spaces. The first few revolve around the notion of uniformity. 3.5
Definition Let (X, d ) and ( Y, il) be two metric spaces and A a subset of X.
(a) A map f: A -+ Y is uniformly continuous on A if, given e > 0 in R, there exists a � > 0 in R so that il(f(x), f(y)) < e for all x, y e A for which d(x, y) < �. (b) A sequence of maps f.: A -+ Y, n e N, converges uniformly on A to f: A -+ Y if, given e > 0, there exists a k e N so that il(f.(x), f(x)) < e for all n � k in N and all x e A.
In the following results, (X, d) and ( Y, d) are metric spaces and A is a subset of X. We use � to denote nearness in both •x and • r, letting the context settle any ambiguity. Proposition The map f: A -+ Y is uniformly continuous on A iff •f(x) � •J( y) whenever x, y e • A and x � y.
3.6
Proof: Let f be uniformly continuous on A. Find the � > 0 for a prescribed 0 from 3 5(a) By transfer, •il(•f(x), •f(y)) < e for all x, y e • A for which •d(x, y) < �. In particular, •il(•f(x), *f(y)) < e for all x, y e • A for which x � y. This is true for any e > 0 in R, and so •f(x) � •J(y) for all x, y e • A for which X � y.
e >
.
.
1 26
Ill.
Nonstandard Theory of Topological Spaces
Conversely, suppose *f(x) � *f(y) whenever x, y e • A and x � y. Let 8 > 0 in R be given. Then the internal sentence (3<5 e *R)[c5 > 0 " (Vx, y e • A) [* d(x, y) < c5 -+ * il(*f(x), *f(y)) < 8] ] is true i n V(* S) (choose c5 t o be infinitesimal). That f i s uniformly continuous follows by transfer to V(S). D The sequence f. :A -+ Y converges uniformly on A to f: A -+ Y iff *f8(X) � *f(x) for all n e • N «> and all x e • A. 3.7 Proposidon
Proof: Exercise.
0
If f: A -+ Y is continuous and A is compact, then f is uniformly continuous on A. 3.8 Theorem
Proof: Let x, y e • A with x � y. Then x and y are near a standard point z e A since A is compact, and *f(x) � f(z) � *f(y) since f is continuous at z . The result follows from Proposition 3.6. 0 Theorem If f.: A -+ Y is a sequence of continuous functions which con· verge uniformly on A to f: A -+ Y, then f is continuous.
3.9
Proof: Let x e A and y e • A with y � x. We need to show that *f(y) � f(x). Now *fn(Y) � *f.(x) for each n e N and so, by Theorem 7.3 of Chapter II, *fiiJ(y) � *fiiJ(x) for some w e *N «> . By Proposition 3.7, *fiiJ(y) � *f(y) and *f"'(x) � *f(x), so *f(y) � *f(x) f(x). 0 Next we present the notion of a complete metric space. To do so we need the obvious generalizations of the definitions in §1.8. =
Let (X, d) be a metric space, and let (s8 : n e N) be a sequence of points in X. Then
3 . 1 0 Deftnldon
(i) (s.) converges to s if, given 8 > 0 in R, there is a k e N so that d(s., s) < 8 if n � k, (ii) (s1) is a Cauchy sequence if, given 8 > 0 in R, there is a k e N so that d(sn o s,.) < 8 if n, m � k, (iii) s is a limit point of (s8) if, for each 8 > 0 in R and each k e N, there is an n > k so that d(s. , s) < 8.
111.3
Metric Spaces
1 27
The reader will easily be able to prove that (s.) converges to s iff •s. � s for all n e • N CX> , (s,.) is a Cauchy sequence iff *s,. � •s... for all n, m e • N CX> , and s is a limit point of (s.) iff •s. � s for some n e • N CX> . 3. 1 1 Definition
a point in X.
(X, d) is complete if each Cauchy sequence in X converges to
3. 12 Examples
1 . The set R with the usual metric is complete by 8.5 of Chapter I. 2. Any set X with the discrete metric is complete.
3. R" with each metric of Example 3.2.4 is complete. For example, let (xt ) be a Cauchy sequence in (R ", d1). Then for each i, 1 � i � n, l x� - xl l � d1(xt, x1). Thus, <xr> is a Cauchy sequence for each i and so converges to a point x1 in R. The point x = (x 1 , , x.) in R" is the limit of xt in R " . We now use nonstandard analysis to prove some abstract theorems on completeness. The nonstandard characterization of completeness requires the following notion. •
•
•
Let (X, d) be a metric space. A point y e * X is a pre-near standard point if for every standard 8 > 0 there is a standard x e X with *d(x, y) < 8. 3.13 Definition
3.14 Proposition A metric space (X, d) is complete iff every pre-near-stan dard point y e • X is near-standard.
Proof: Suppose ( X , d) is complete. If y is pre-near-standard, find a se quence s,. e X so that * d(y, s,.) < 1/n. Then (s.) is a Cauchy sequence with limit s and y � •s. � s if n e * N CX> . Conversely, suppose every pre-near-standard point is near-standard, and let (s,.) be a Cauchy sequence. Given 8 > 0, find the associated k e N from Definition 3. 10. Then d(*s,., St) < 8 if n e • N CX> . Thus •s. is pre-near-standard for every n e • N CX> and each such • s, must be near-standard to the same s e X (check). The sequence (s,.) must converge to s. D 3.15 Corollary
complete.
A closed subset (A , d) of a complete metric space (X, d) is
Ill.
1 28 Proof:
xeX
D
closed.
y be a (X, d) is
• A. Then y � x for some x e A by Proposition 1 . 10 since A is
pre-near-standard point in
Let
since
Nonstandard Theory of Topological Spaces
complete. But
Using this characterization, we will show that it is possible to adjoin
(X, d) so that the result is a complete metric densely embedded.
"ideal" elements to a metric space space in which
(X, d) is
3.16 Definition Let (X, d) be a metric space. A metric space (X, d) is a com of (X, d) if (X, d) is complete, there is an isometric embedding 1/J: [i.e., d(x, y) = d{t/J(x� tP(y)) for all x, y e X , whence 1/J is one-to-one],
pletion X -+ X and
1/J[X]
is dense in
X.
3. 17 Theorem Any metric space
Proof:
We let
X'
(X, d) has a completion (X, d).
be the pre-near-standard points in
equivalence classes of
X'
under the relation of nearness
• X, �
and
X
be the
(an equivalence
X are monads m(x) of pre-near-standard d(m(x'), m(y')) = st( *d(x', y')) [note that *d(x', y') pre-near-standard points x', y'] . This metric is independent
relation); thus the elements of points
x' e • X.
is finite for any
Also define
of the pre-near-standard points chosen to represent the elements of
X,
for
y' � y� then * d(x', y') � *d(x� , y'1 ) (Exercise 6). The map 1/J: X -+ X defined by 1/J(x) = m(x) is obviously an isometric embedding. Also 1/J[X] is dense in X. For if m(x') e X , where x' is pre-near
if x'
�
x� and
standard, then given 8 >
d(m(x'), m(x))
=
0 there exists an x e X so that *d(x', x) <
st(*d(x', x)) :s;
8 and then
8.
To show completeness, let ( m(x:J : n
e N)
be a Cauchy sequence in
(X, d),
e X '. Since each x� e X' , there are elements x. e X with * d(x . , x�) < 1/n for each n e N. Given 8 > 0 in R, there exists a k e N so that d(m(x ;.), m(x:J) < 8 and hence *d(x�, x:J < 8 if n, m � k. Then d(x . , x,.) = * d(x. , x .) :s; 2/n + 8 if m � n � k in N by the triangle inequality. Again by transfer, *d(*x. , *x,.) :s; 2/n + 8 if m � n � k in *N. In particular, if ro e *N ,. , *d(x. , •xco) :s; 2/n + 8 if n � k, and so • xCII is pre-near-standard. Therefore *d(x�. • xCII) :s; *d(x�. x,) + * d(x. , * xJ :s; 3/n + 8 if n ;;:: k, yielding d(m(x�). m(* x"')) :s; 3/n + 8 if n ;;:: k. Thus (m(x;.)) converges to m( * x CII). D with x�
As an example, note that the rationals Q form a metric space under the usual metric d(x , y) = l x - Y l . x, y e Q. The completion d) is isomorphic to
(Q,
the real metric space (R, d). Recall that a subset of the real line is compact iff it is closed and bounded.
In arbitrary metric spaces there is a similar relationship between compact-
111.3
1 29
Metric Spaces
ness, completeness, and total boundedness, the last being a generalization of boundedness. A metric space (X, d) is totally bounded if, to each e > 0 in R, there corresponds a finite covering { B.(x1) : 1 :s;; i :s;; n} by open e-balls [each B.(x) = {y e X : d(x, y) < e } ] .
3.18 Definition
3.19 Proposition
A metric space (X, d) is totally bounded if every point of • X is pre-near-standard.
Proof: Suppose (X, d) is totally bounded. Let e > 0 be given and find the corresponding points xi> 1 :s;; i :s;; n, so that X = UB.(xJ(1 :s;; i :s;; n). By trans fer, • X = U* B.(x1)(1 :s;; i :s;; n), and so every point of • X is pre-near-standard. The converse is left to the reader. 0 Theorem A metric space (X, d) is compact iff it is complete and totally bounded.
3.20
Proof: Suppose (X, d) is compact. Then every point y e •X is near a point in X, so (X, d) is complete and totally bounded by 3. 14 and 3. 1 9, respectively. Conversely, suppose (X, d) is complete and totally bounded. If y e • X, then y is pre-near-standard by 3. 19 and hence near-standard by 3.14. 0 One might expect that "totally bounded" may be replaced by "bounded" in this theorem, where boundedness is defined as follows. 3.21 Definition A set A in a metric space (X, d) is bounded if there is a point x0 e X and a number M so that d(x, x0) � M for all x e A.
Example 3.2.2 and the following example show that boundedness is not enough for Theorem 3.20. 3.22 Example Let B1 {x e l00 : d00(x, O) � 1 } be the "unit ball" in (l00 , d00) where 0 ( 0, 0, . . . ). It is easy to see that B 1 is closed and hence is complete when regarded as a metric space with the metric induced by d00 (Exercises 8, 1 4). Also, B1 is obviously bounded. Now consider the element x = (x1: i e *N ) e *B1 which i s zero except at some infinite integer w where x .., 1. Then x is not near-standard. For i f x � *y for some standard y = (y1 : i e N) then 0 = x1 � y1 for at least all i e N , and so y1 = 0 for all i e N. By transfer, *y1 = 0 for all i e *N, and so *y.., � xw . =
=
=
1 30
Ill.
Nonstandard Theory of Topological Spaces
To end this section we consider another compactness criterion, which is especially important in applications. In many situations one can obtain a se quence (x8) of points (in a given topological space X,) which has certain desirable properties, e.g., giving better and better approximate solutions to a set of equations. One would like to assert that a subsequence of the given sequence converges to a point in the space (in order, e.g., to produce an exact solution). Though the criterion of compactness in the sense of §III.2 is not always of help in constructing such a subsequence, if the assertion is never theless always true we call the space sequentially compact. topological space X is sequentially compact if from each sequence (x,.) in X it is possible to select a subsequence which converges to a point x e X.
3.23 Definition A
It turns out that compactness is equivalent to sequential compactness in a metric space. Unfortunately this is not true in general topological spaces, as we shall see in §111.7. 3.24 Theorem A
metric space (X, d) is compact iff it is sequentially compact.
Proof: (i) Suppose that (X, d) is compact and let (x,.) be a sequence in X. By Exercise 9 there is a point x0 which is a limit point of (x,.). We will show that some subsequence of (x,.) converges to x0 • Consider the open ball B 1 = {x e X : d(x, x0) < 1 } . Since x0 is a limit point of (x,.) there is an x,.. e B 1 • Similarly there is an x,.2 in B1 1 2 = {x e X : d(x, x0) < !} with n2 > n1 • Continuing this process inductively, we obtain a subsequence (x,... ) with x... e B 1 11r. = {x e X : d(x, x0) < 1/k}; clearly (x,..) converges to x0 • (ii) Suppose (X, d) is sequentialJy compact. Then it is obvious that (X, d) is complete, so that if (X, d) is not compact, it must not be totally bounded. Thus there exists some e > 0 so that no finite collection {B.(yJ : 1 � i � n} covers X. Let x 1 e X be a given point. Then there is an x2 with d(x� o x2) � e. Similarly there is an x3 with d(x� o x3) � e and d(x2, x3) � e. Continuing in this way, we construct a sequence (x,.) with d(x,. , x,J � e for any n, m e N. Clearly (x,.) can have no convergent subsequence. D The procedure used in part (i) of the proof in going from a limit point to a convergent subsequence does not work in a general topological space. It uses in an essential way the fact that the neighborhood system of x has a countable base. A topological space is said to satisfy the first axiom of countability if the neighborhood system of each point has a countable base. Included in such spaces are the metric spaces. Clearly, a subset A in a metric
111.3
131
Metric Spaces
o r first countable space is closed i ff A contains the limit of any convergent sequence in A. Exercises
111.3
1 . Show that dco(x , y) satisfies the triangle inequality. 2. Show that for the metrics on R" defined in Example 3.2.4, x � y iff x1 � y1 for 1 � i � n. 3. Show that for each x e •teo there is an M e • N such that l x11 � M for all
n e •N.
4. Prove that if x, y are internal sequences and x1 � y1 for all i e • N then sup { l x 1 - y 1 1 : i e •N} � 0. 5. Prove Proposition 3.7. 6. (a) Show that if a, b, c are points in a metric space (X, d) then id(a, c)
d(b, c)i � d(a, b).
(b) Show that if x'
•d(x'1 , y'1 ).
�
x! and y'
�
y! in (•X, •d) then •d(x', y') �
7. Show that if (X, d) is a metric space and each point of •x is pre-near standard then (X, d) is totally bounded. 8. Show that B 1 {x e lco :dco(x, O) � 1 } is closed. 9. Show that a sequence in a compact metric space has a limit point. 10. Let (x") be a sequence in a compact metric space (X, d). Fix w e • Nco . Use the downward transfer principle and the fact that xQ) is near-standard to prove there is a subsequence x", that converges to st(xQ)). 1 1 . Use Theorem 3.24 to prove Robinson's result: If (X, d) is a metric space and A is an internal set in X such that each a e A is near-standard, then st( A ) = { x e X : there exists an a e A with x � a} is compact. (The gen eralization for regular topological spaces (Exercise 2.7) is due to Luxem burg [36].) 12. Prove that a Cauchy sequence in a metric space (X, d) is bounded. 1 3. Use Exercise 12 to show that (X, d) is complete if every finite point in • X is near-standard. 14. Show that (lco , dco) is complete. 1 5. (eo-Continuity, • -Continuity, and S-Continuity) Let (X, d) be a met ric space, A be a subset of • X, and f: A -+ • R be a function. We say that f is eo-continuous ( • -continuous) at x e A if, for each e > 0 in R (• R), there is a c5 > 0 in R (•R) such that l f(x) - f(y) l < e if y e A and • d(x, y) < b. We say that f is S-continuous at x e A if f(y) � f(x) for every y e A with y � x. (a) A = • X and f •g, where g: X -+ R. Show that if g is continuous at each x e X, then f is • -continuous and S-continuous at each x e • X. =
=
1 32
Ill.
Nonstandard Theory of Topological Spaces
(b) Show that if f is ec5-continuous at x e A then f is S-continuous at x e A but not necessarily vice versa. (c) Suppose that f is internal. Show that f is S-continuous at x e A iff f is ec5-continuous at x. (Hint: Use the spillover principle.) (d) Show that there are internal functions f on • R which are • -con tinuous but not S-continuous at zero and vice versa. (Hint: Look for examples on X R with the usual metric). =
16. Let A be an internal set in •x where (X, d) is a metric space and let f: A -+ • R be internal. Show that f is S-continuous at each point x e A iff, for every (standard) £ > 0 in R, there is a c5 > 0 in R such that l f(x) - f(y) l < £ for all x, y e A for which * d(x, y) < c5. (Hint: Again use the spillover principle.) 1 7. Let X be a compact metric space. Suppose that the internal function f: • X -+ • R is S-continuous at each point of • X and finite at each x e X. Let g be defined by g(x) = 0/(x) for X e X. Then g is continuous on X and *g(x) � f(x) for all x e • X. 1 8. Two me tries on X are equivalent if they define the same topology. Show that the metrics d and d' are equivalent if there exist positive (nonzero) constants tx and fJ in R so that txd(x, y) � d'(x, y) � {Jd(x, y) for all x, y e X. 19. Let I = [0, 1] c R and let X be the set of all continuous functions f: I -+ I such that l f(x) - f(y) l � l x - Yl · Define d(f, g) sup { l f(x) g(x) l : x e I} for f, g e X. =
(a) Show that (X, d) is a metric space. (b) Show that (X, d) is compact. 20. Use Robinson's theorem to show that the set of elements x of 1 1 with ll x l l 1 � 1 (the unit ball) is not compact. 21. (Lebesgue covering lemma). If U 1 , , U n is an open covering of a com pact metric space (X, d), then there is an e > 0 in R such that the e ball B.(x) about any x e X is entirely contained in one of the sets U 1, 1 � i � n. •
•
•
111.4 N o r m ed
Vecto r S p aces and Banach S p aces
The space R is not only a metric space with the usual metric; it is also equipped with operations of addition and multiplication, and the distance function d(x, y) l x - Yl involves these operations. In this section we gen eralize this simple example. The metric spaces will have the additional struc=
111.4
1 33
Normed Vector Spaces and Banach Spaces
ture of a vector space, and the metric will come from a generalization of the absolute value. Many theorems and exercises are standard. As in §111.3, the nonstandard analysis will be carried out in an enlargement V(•S) of a suitable superstructure V(S). The choice of S will depend on the context and will not be mentioned explicitly. A (real) t vector space is a set X on which are defined opera tions of vector addition ( + ) and scalar multiplication ( ) (so that we form the sum x + y of two vectors x, y e X and the sc alar mult iple a · x of the vector x e X by a e R). These operations satisfy the following conditions (as usual we often omit the dot in scalar multiplication):
4. 1 Definition
·
(i) X + y = y + X for all X, y E X. (ii) (x + y) + z = x + (y + z) for all x, y, z e X. (iii) There is a vector 8 e X called the zero vector so that x + 8 = x for all
x e X. (iv) a(x + y) = ax + ay if a e R and x, y e X. (v) (a + b)x = ax + bx if a, b e R and x e X. (vi) a(bx) = (ab)x if a, b e R and x e X. (vii) 0 X = 8, 1 · X = X for all X E X. We write ( - l)x = - x, so that x + ( - x) = 8 by (v) and (vii). The set Y � X is a (linear) subspace of X if x, y e Y and a, b e R imply ax + by e Y. An easy exercise shows that the element 0 is unique. A subspace Y of a vector space X is itself a vector space with the inherited operations of addi tion and scalar multiplication. ·
A norm on a vector space X is a nonnegative real-valued function I I I I: X ..... R satisfying (a) l l x l l = 0 iff x = 8, (b) ll x + Y ll S l l x ll + II Y II (tri angle inequality), (c) ll ax ll = l a l ll x ll · A normed vector space (X, I I I D is a metric space if we define the metric d by d(x, y) = ll x - Y ll (exercise). If the normed vector space is complete in this metric it is called a Banach space. A subspace Y � X is closed if it is closed in the topology defined by the norm. The reader should easily be able to prove that the norm function II I I: X ..... R is continuous when X has the topology induced by d. Note also that a closed 4.2 Definition
t Much of this and the succeed i ng section obtains (with some obvious modifications) if the real numbers are replaced by complex numbers in the definition of vector space.
Ill.
1 34
Nonstandard Theory of Topological Spaces
subspace of a Banach space is complete (Corollary 3. 1 5) and hence a Banach space. 4.3 Examples
1 . R• can be made into a vector space in the following standard way: If x = (x 1 , . . . , X8), y = (y 1 , . . . , y. ) , and a e R we define x + y = (x 1 + y1 , . . . , x. + y. ), ax = (ax 1 , . . . , ax. ), and (} = (0, 0, . . . , 0). R" is a normed space under each of the following definitions of a norm (exercise):
(a) ll x ll , = L7= 1 l x; l . (b) ll x ll oo = sup { l x1l : l � i � n } 2. The space 11. The space R"' of infinite sequences of real numbers is a vector space with the following definitions of addition and scalar multiplica tion: If x = (x 1 , x2 , • • • ), y = ( y 1 , y2 , • • • ), and a e R, we define x + y = (x 1 + Y � t X2 + y2 , • • • ) and ax = (ax 1 , ax2 , • • • ) (check). Let 11 be the set of elements x = (x �> x2 , • • • ) in R"' for which ll x ll 1 = L j; 1 l x 1 1 is finite. Then 11 is a linear subspace of R "' and I 11 1 is a norm on 1 1 (regarded as a vector space). For example, to check the triangle inequality 4.2(b) and the fact that 11 is closed under + . we have (with x = (x 1 , x2 , • • • ) and y = ( Y � > y 2 , • • • ) ) R
n
n
L lx; + Y 1l � L1 l x; l + L I Y 1 I � ll x l l 1 + II Y II � t 1= i= l i= l and the results follow by taking the limit as n -+ ao on the left. Properties 4.2(a) and 4.2(c) are immediate. Finally we show that 11 is complete and so is a Banach space. Let (x" : k e N) be a Cauchy sequence in 11 with x" = (x� , xt . . . ). Then given e > 0 there is an n e N so that ll x" - x1 1 1 1 � e if k, l � n. Since Cauchy sequences are bounded there exists a number A so that ll x" l l 1 � A for all k e N. Let w be an infinite integer; by transfer we have • 11 xco ll 1 � A . Now IX:: I � llx" l h for all k, and so by transfer l xr l � A . Let x1 = st(xf). We will show that x = (x1 ) e 1 1 and (x" ) converges to x. For any k and L we have
and so by transfer L
L
L l x 1l � L l x; - xf l + • l l x''% � infinitesimal + A � 2A. i= I i= I This shows that x e 11 . Finally, for any k, l, and L, L
L
L
L
� L + L lx + ll x" - X1 l h · L lx x � L l x 1 = 1 ; - ll 1 = 1 ; - X:: l i = l � - xlj i = l lx; - X:: l
111.4
135
Normed Vector Spaces and Banach Spaces
By transfer, with k = w, we have L
L
l x 1 - xi l + * ll x"' - x ' l h L l x1 - xll � i L i=1 =1
� infinitesimal + * l lx"' - x % .
The right-hand side is � 2e if I � n. Since this is true for any L e N, we conclude that ll x - x'll1 � 2e if I � n. 3. The space l oo is a Banach space under the norm defined by ll x lloo = sup {lx11 : i E N}, where x = (x 1 , x2 , ) (Exercise III.3. 1 4). 4. The space c0 . The space c0 consists of those x = (x1 : i e N) E /00 for which limn - oo x. = 0. It is easy to see that c0 is a closed linear subspace of /00 and hence a Banach space. 5. The spaces B(S) and C(S). Let S be an arbitrary set. We denote by B(S) the set of all bounded functions on S. Then B(S) is a vector space with the usual definitions of addition and scalar multiplication of functions, that is, iff, g e B(S) and a e R, we put (f + g)(x) = f(x) + g(x) and (af)(x) = af(x) for x e. S; we take (} to be the function that is identically zero. B(S) is a Banach space under the norm defined by l l ! ll ctl = sup { l f(x) l : x e S} (Exercise 3). If S is a topological space we define C(S) to be the subset of B(S) consisting of continuous functions. Then C(S) is a closed subspace of B(S) (Exercise 4), and hence a Banach space. •
•
•
Let (X, II II > be a normed space. From now on we will follow the usual convention of denoting the • -transform of the norm I I II o n *X by II II rather than * II II; the context will clear up any possible confusion. We see immediately that the (norm) monad of a point x e * X is the set m{x) = {y e • X : I I Y - x ll � 0}. It is also almost immediate that m(x) = { y e *X : y = x + z, z e m{fJ)}, so that all monads are translates of the monad about zero (Exercise 5). The finite points in •X (Definition 3.3) are those x E • X for which l lx l l is finite. Next we come to the basic notion of linear operator. Let X and Y be vector spaces. A map T: X -+ Y is called a linear operator if T(ax + by) = aTx + b Ty for all a, b e R and x, y e X. The set of all such linear operators is denoted by L (X, Y). Let X and Y be normed vector spaces. (Since there is no possibility of con fusion we denote the norms and zeros on both by II II and fJ, respectively.) A linear operator T: X -+ Y is bounded if the number I I Til = sup { II TxiJ : IJ x ll � 1 } is finite. This number is called the norm of T. Then II Tx l l :s; IITII I I xl l for all x e X (check). The set of all bounded linear operators T: X -+ Y is denoted by B(X, Y). 4.4 Definition
If Y = R (with the usual operations of addition and multiplication and usual norm) then a linear operator T is called a linear functional. In what
1 36
Ill.
Nonstandard Theory of Topo logical Spaces
x and T(x) for the nonstandard extensions •x and * T(x) of x and T(x); •x and * T(x) may, however, have nonstandard elements.
follows, we will often write
T: 11 -+ 11 as follows: if x = (x 1 , x2 , x3 , ) then Tx ( 0, x" x2 , ) . Then T is linear, one-to-one, and bounded (in fact I I Txll = llxll for all x e 11). However, T does not map 11 onto 11 •
4.5 Example Define a map =
•
•
•
Let
(Robinson)
4.6 Theorem
•
•
•
T e L(X, Y),
where
X
and Y are normed
spaces. The following are equivalent: (i)
T is bounded.
(ii) • T: • X -+ • Y takes
finite points to finite points.
• T takes the monad of 0 into the monad of 0. (iv) • T takes near-standard points to near-standard points. In fact, if z e • X is near x e X then • Tz is near Tx. (iii)
Proof: (i) � (ii): Suppose I I Txll � Mllxll for all x e X. By transfer I I * Txll � for all x e • X and (ii) follows. (ii) � (iii): Proceed by contradiction. Suppose x e m(O) but II* Txll '$:. 0. Then the element z = x/llxll e • X is finite with norm 1 (here and in the following
Mllxll
4. 1, 4.2, and 4.4) * Tz = (1/llxl i ) * Tx is not finite since llxll � 0 but I I * Txl l '$:. 0. (iii) � (iv): Let x e X and z e m(x), so x - z e m(O). Then * T(x - z) = Tx - * Tz e m(O), so * Tz is near Tx. (iv) � (i): Procee d by contradiction. If T is not bounded then there exists a sequence (x. e X : n e N) so that llx.ll = 1 but I I Tx. ll > n for n e N (check). Then II* Txcoll is infinite for some infinite natural number ro. Now z = xoJJ I I* Tx"'ll i s near-standard since i t belongs t o m(O), but II* Tzll = J ii * Txcoll is not finite, so z cannot be near-standard. 0 we use freely the transfers of the properties in Definitions
but
continuous at
see that a linear operator is continuous if and onl y 0 (Exercise 6). Therefore we have the following result.
4.7 Coronary
T e L(X, Y)
It is easy to
Proof:
Use
is bounded iff it is continuous.
4.6(iii) and 1 . 1 5.
4.8 CoroUary If
0
T e B(X, Y), then the null space N(T) = { x e X : Tx
closed linear subspace of X.
Proof:
Exercise.
0
if it is
=
0}
is a
111.4
1 37
Normed Vector Spaces and Banach Spaces
One of the most important results concerning bounded linear operators on Banach spaces is the uniform boundedness theorem. The proof is entirely standard. Let X be a Banach space, Y a normed vector space, and § c L(X, Y) a family of bounded linear operators. Sup pose that for each x e X there is a constant M" so that I I Tx ll � M" for all T e §. Then there is a constant M so that II Ti l � M for all T e §, i.e., the operators in § are uniformly bounded. 4.9 Uniform Roundedness Theorem
Proof: Suppose that T e L(X, Y). Note that if II Tx ll � M for all x in the closed ball B,(x0) = { x e X : ll x - x0 l l � e} then T is bounded and II Ti l � 2 M(e. The proof of this fact is left to the reader. Now we proceed by contradiction. Let x0 e X and e0 > 0 be given. Then there is an x 1 e B,0(x0) and a T 1 e § so that II T1 x d l > 1. For otherwise II Tx ll � l for all x e B,0(x0) and all T e §, and then I I Ti l � 2/e0 for all T e § by the remark in the first paragraph. By continuity we can find an e 1 with 0 < e 1 < t. and B,0(x0) ;;2 B,,(x1) so that II T1 x ll > 1 for all x e B, ,(x 1 ). Induc tively we can find a sequence { B•.(xft) : n e N} with B,.(xft) ;;2 B•• • ,(xft + 1) and limft -o oo eft = 0, and a sequence Tft e § so that I I Tftx ll � n for all x e B,.(xft). Now (xft) is a Cauchy sequence since limft -o oo eft = 0. Let x e X be the limit of (xft) (here we use the completeness of X). Then x e B,.(x,), so II Tftx ll > n for all n, contradicting the assumption. 0 As a corollary we can prove the following result. Let X be a Banach space and Y a normed vector space, and suppose that ( Tft : n e N ) is a sequence in B(X, Y) such that for each x e X there is an element y" with limft -o oo Tftx = y" (limit in norm). Then the mapping T given by Tx = y" is in B(X, Y).
4. 10 Theorem
Proof: An easy exercise shows that the map T: X -+ Y is linear. Since II II is a continuous function, limft -o oo II Tftx l l = II Tx ll and thus for each x there exists an M" so that II Tftx ll � M" for all n. By the uniform boundedness theorem there is an M e N with II Tft ll � M for all n e N, so II Tx ll = I im i i Tftx ll � M ll x l l and T is bounded. 0 .
Next we study an important class of bounded linear operators, the compact operators. These operators occur in many applications. There is an extensive analysis of equations in Banach spaces involving these operators; it is called the Fredholm theory.
1 38
Ill.
Nonstandard Theory of Topological Spaces
Y be normed vector spaces. An operator T e L(X, Y) is compact if T[B] is com pact for every norm-bounded set B c X.
4. 1 1 Definition Let X and
(Robinson) T e L(X, Y) is compact iff •T takes finite points to near-standard points.
4. 12 Theorem
Proof: Suppose T is compact and let x e • X be finite, i.e., llxll < M for some M > 0. The ball B {x e X : l l xl l s M} is bounded and so T[B] is compact. Thus every point of •(T[B] ) = • r[• B] is near-standard by Robin son's theorem, 2.2. Since x e •B we conclude that • Tx is near-standard. Conversely, suppose that •T maps finite points into near-standard points, and let B be a bounded set. By Theorem 2.4 we need only show that T[B] s;; K for some compact set K. Let K = { y e Y: y � y' for some y' e •( T[B] )} st(• T[•B]). Then T[B] � K and K is compact by Exercise 111.3. 1 1 . 0 =
=
We see immediately from 4.6 and 4. 1 2 that compact operators are bounded. Theorem 4.1 2 can be used to establish the compactness of many operators, as the following example shows. 4. 13 Example: Integral Operators
(Robinson [42, Theorem 7. 1 .7] ) Let
T: C( [0, 1]) -+ C( [0, 1] ) be defined by Tf(x) =
J01 K(x, y)f(y) dy,
where K(x, y) is a continuous function on [0, 1] x [0, 1]. The reader should check that T is a linear operator. To show that Tf is continuous notice that if jf(x)j S M for all x e (0, 1] then
(4. 1)
ITf(x) - Tf(y)j S
J01 I K(x, t) - K(y, t) l lf(t)l dt
S M max { I K(x, t) - K(y, t)l :(x, t), (y, t) e [0, 1 ]
x
[0, 1]},
and max i K(x, t) - K(y, t)j can be made as small as desired if lx Y i is suffi ciently small by the uniform continuity of K(x, t). Also note that jK(x, t)l s K for all (x, t) e [0, 1 ] x [0, 1] for some constant K, and so, for any x e [0, 1 ], -
(4.2)
I Tf(x)l
S
K max { if( t) j : t e [0, 1 ]}.
T o show that T is compact we need t o show that •Tf i s near-standard for each finite f. Let f e •C([0, 1]) be finite. This means that there is a finite standard M so that lf(t)l s M for all t e •[O, 1 ].
111.4
a
Normed Vector S p ces and Banach Spaces
1 39
From the transfer of (4.2) we see that i • Tf(x)l ::5; KM for all x e • [O, 1], i.e., • Tf is finite, and we may define a function 1/1 on [0, 1 ] by 1/J(x) = st(• Tf(x)), x e [0, 1]. To complete the proof we will show that 1/1 is continuous and •Tf is near • ljl. From the transfer of (4. 1) we have
i • Tf(x) - • Tf(y) i ::5; M max W K(x, t) - • K(y, t)l : (x, t), (y, t) in •[O, 1] x • [O, 1 ]}. Thus • Tf(x) � • Tf(y) whenever x; y e •[o, 1 ] and x � y by the uniform con tinuity of K(x, t) (Theorem 10. 1 0 and Proposition 1 0.8 of Chapter 1). Let e > 0 be a fixed standard real, and let D = {<5 e • R , <5 > O : x , y e •[O, 1 ] and lx - Yl < <5 implies i • Tf(x) - • Tf(y)i < e} . Then D contains all positive infinitesimals by the above remark, and so contains a standard <5 > 0 by Corollary 7.2(iii) of Chapter II. Now if x, y e [0, 1] then ll/l(x) - l/l(y) l ::5; ll/l(x) - •Tf(x)l + i • Tf(x) - • Tf(y)i + i • Tf(y) - 1/J(y)l . The first and last terms are infinitesimal, so that l l/l(x) - 1/J(y)l < 2£ if the <5 is chosen as above; thus 1/1 is continuous. To show that •ljl is near •Tf notice that •ljl(x) � • Tf(x) for all standard x by the definition of 1/1 and the fact that •ljl is an extension of 1/1. If x e •[o, 1] then i • Tf(x) - •ljl(x)l ::5; i • Tf(x) - • rJe x )l + i • Tf( o x ) - • ljl ( o x )l + 1•1/J e x) - • ljl (x ) l, and all terms on the right are infinitesimal by the preceding remarks and the continuity of 1/J. A word of caution here. The reader may think that the above proof is needlessly complicated since we could replace 1/1 by �(x) = st(Tf(x)) for all x in •[o, 1] rather than [0, 1 ], in which case it would be obvious that � is near Tf. Unfortunately the � defined this way is usually external and thus not a standard element in •C([O, 1 ] ). Notice also that an internal finite f e C( [O, 1 ] ) can be quite wild; e.g., f(x) = sin wx, where w is infinite. The set of bounded linear operators can be made into a linear space in an obvious way. If T, S e B(X, Y) and a e R we define (T + S)(x) = T(x) + S(x) and (aT)(x) = aT(x). It is then not hard to see that the operator norm on B(X, Y) makes B(X , Y) into a normed vector space (Exercise 8). 4.14 Theorem Let X be a normed vector space and Y a Banach space. Then
the normed vector space B(X , Y) is complete and hence a Banach space. The set of compact operators in B(X, Y) forms a closed linear subspace.
Proof: Let ( T" e B(X , Y) : n e N) be a Cauchy sequence. Then, for each x e X, T"x is a Cauchy sequence and hence converges to an element Yx by
Ill.
140
Nonstandard Theory of Topological Spaces
completeness of Y. We define T by Tx = lim T.x. Then T is linear (check) and bounded since lim ii T.I I = II TII (check). Finally, we show that T. con verges to T in norm. For given 6 > 0 there is an N so that IIT.x - T..x ll � I I T. Tmll ll x ll < 8 1 1 x ll if n, m � N . Thus I I T.x - Tx ll S 8 1 1x ll if n � N, and so li T. T i l S £ for n � N, and we are through. An easy exercise shows that the set of compact operators is a linear sub space of B(X, Y). To show that it is closed, let ( T.) be a sequence of compact operators converging to an operator T e B(X, Y). If y e *X is finite then it belongs to *B, where B = {x e X : l lx l l S M} for some standard real M > 0. Now note that T.x converges to Tx uniformly on the ball B, i.e., for any £ > 0 there is an m(8) e N so that IIT.x - Tx ll < 6 for all n � m(8) and all x e B. Thus II* T.0x - * Tx l l < 6/2 for n0 � m(8/2) in N and all x e *B. Since T.0 is compact, * TnoY is near a standard z e Y and so II* Ty - z ll < 6 by the triangle inequality. Since 6 is arbitrary, * Ty is pre-near-standard. Since Y is complete, it follows from Proposition 3. 14 that * Ty is near-standard. 0 -
-
The standard proof of the closedness of the set of compact operators usually involves the selection of infinite subsequences with certain desirable properties. The space of bounded linear functionals on a normed vector space X is a Banach space.
4. 15 Coronary
The Banach space of this corollary is used sufficiently often for us to in troduce some notation. 4. 16 Definidon The
Banach space of bounded linear functionals on a normed linear space X is called the dual space of X and is denoted by X'. The dual of X' is denoted by X" and is called the second dual of X. Similarly for X"', etc. It is sometimes difficult to characterize the dual of a given Banach space, but the following example is an easy case.
= l a;, Our aim is to define a mapping T: Ia;, -+ 11 which is linear, 1 - l , onto, and satisfies !ITY I I = I IY I I a:J for y e la;, . Le t y = (y1 : i e N) e Ia;, and define Ty: 1 1 -+ R by Ty(x) = L"! 1 x1y1 for x = (x1) e 11 • Then Ty is linear, and
4.17 Example: 1'1
a:J
I Ty(x) l S sup { l y1 j : i e N} t l x ,l
=
I I Y IIa;, l l x l l h
so Ty is a bounded linear functional on 1 1 with I I TYII S I I Y I I a:J . We next show
111.4
Normed Vector Spaces and Banach Spaces
141
that IITYII � II Y I L., . We may assume IIYII oo > 0. G iven a positive e < II Y II oo • there is an n0 so that I Yno l > II Y I I oo - e. Now define x = (x1) e 1 1 by x1 = 0 for i � no and x .o = Ynoii Yno l · Then ll x ll t = 1 and I Ty(x) l = I Yno l > IIYII oo - e, so IITYII � IIYIIoo · We also see that T is 1 - 1 , since if Ty = (} then II Y I I oo = 0 so y = 0. It only remains to show that T is onto. Let f e 11 . If e" e 1 1 is defined by e" = ( 15j), where 15j = 0 if i � n and 15: = I , then ll e" ll 1 = 1 for all n e N. Put /(en) = Yn e R. Then I Yn l ::5; IIJ II . and so y = (y) e 100 • Now the functional Ty attached to y as in the first paragraph agrees with f on the elements e". A simple limiting argument (check) shows that Ty = f, and so 1'1 = 100 • In the case of a general normed vector space X, it is not at all obvious that X' contains any elements other than 0. The following result, which is basic to the study of duality, shows that X' always contains many elements.
Let X be a vector space and suppose that a given function p: X -+ R satisfies p(x + y) ::5; p(x) + p(y) and p(ax) = ap(x) for each a � 0 e R and x, y e X. Suppose that f is a linear functional defined on a subspace S of X with f(x) ::5; p(x) for all x e S. Then there is a linear func tional F on X which extends f [i.e., F(x) = f(x) for all x e S] and satisfies F(x) ::5; p(x) for x e X . 4.18 Hahn-Banach Theorem
Proof: Let g and h be linear functionals, each defined on a linear subspace of X. We say that g extends h and write h -< g if the domain of g contains the domain of h and g = h on dom h. The relation -< partially orders the set of linear functionals. Consider the set of all extensions g of f which satisfy g(x) ::5; p(x), for x in the domain of g. Applying Zorn's lemma (see the Appendix) to this set, partially ordered by -<. we see that there is a maximal extension F. We need only show that the domain X0 of F is all of X. Suppose this is not the case, i.e., there is a vector y in X but not in X0 . Then F may be extended to a functional g on the subspace X => X 0 consisting of elements of the form ay + x0 , x0 e X0 , a e R, by putting g(ay + x0) = ag(y) + F(x0). Now g is specified uniquely by g(y), and we need to show that g(y) can be chosen so that g(x) � p(x) for all x e X in order to get a contradiction. For x1 , x 2 e X0 we have F(x 2 ) - F(x1) = F(x2 - x1) ::5; p(x 2 - x1) ::5; p(x 2 + y) + p( - y - x.), which yields - p( - y - x1) - F(x1) ::5; p(x 2 + y) - F(x 2). Since the left is in dependent of x2 and the right is independent of x1 there is a constant c e R so that
(i) c ::5; p(x 2 + y) - F(x 2), (ii) - p( - y - x1) - F(x1) ::5; c
Ill.
1 42
Nonstandard Theory of Topological Spaces
for all X to x e X0 • We now put g( y) = c. Then for x = ay + x0 e X the in equality g(x)2 = g(ay + x0) = ac + F(x0) S p(ay + x0) follows by replacing x 2 by xola in (i) if a > 0 and x 1 by x0/a in (ii) if a < 0. 0
4.19 Corollary If X is a normed vector space and x e X, x an x' e X' so that x'(x) = llxll and l lx'll = 1 . Proof: Standard exercise.
::1: 6,
then there is
0
We now show that X can be isometrically and isomorphically embedded in X". 4.10 Theorem Let X be a normed vector space and
define a map T: X -. X" by Tx(x1 = x'(x) for all x' e X'. Then T is a linear and norm-preserving em bedding. If X is a Banach space then T(X] is a closed linear subspace of X" .
Proof: The reader should check that T is linear. That Tx is bounded (as we have implied in the statement of the theorem) follows since I Tx(x')l = lx'(x)l S ll xll llx' ll. and we see that II Txll S l lxll · The result will be established when we show that II Tx l l � l lxll · This is trivial if x = 6, so suppose x ::1: 9. From Corollary 4. 19 there exists an x' e X' so that llx'll = 1 and x'(x) = l l x ll · Thus llxll = l x'(x) l = I Tx(x')l S II Txll llx' l l = I I Tx ll . The rest is left to the reader. 0
Because of Theorem 4.20 we identify X with T (X] and regard X as a subspace of X" in the rest of this section without further explicit comment. We end this section with a consideration of compactness properties in Banach spaces. We have seen in Example 3.22 that the closed unit ball in 100 is not norm-compact. This situation turns out to be typical of all infinite dimensional spaces. In fact one can prove that a closed ball in a Banach space is norm-compact iff the space is finite-dimensional [ 1 4, Theorem IV.3.5]. It follows that no set in an infinite-dimensional Banach space X containing a closed ball can be norm-compact. Since this severely limits the sets which can be norm-compact we look for other topologies on a Banach space in which closed balls are compact. 4.11 Definition Let X be X is the topology whose
a normed vector space. The weak topology on neighborhood system at a generic point x e X is generated by the subbase consisting of sets of the form U(x; x', e) = {y e X : lx'(y) - x'(x) l < e} for some x' e • X . Let X' be the dual space of a normed vector space X. The weak• to pology on X' is the topology whose neighborhood system at a generic point
111.4
1 43
Normed Vector Spaces and Banach Spaces
x' e X' is generated by the subbase consisting of sets of the form V(x'; x, e) = {y' e X' : lx(y') - x(x') l < e} for some x e X (regarded as embedded in X"). Notice that in the definition of the subbase for the weak* topology we take only those x e X and not all x" e X". This turns out to make a crucial difference. An easy exercise, which we leave to the reader, shows that the monads of points x e X and x' e X' in the weak and weak* topologies, respectively, are given by mw(x) = {y e * X : *x'(y) � *x'(*x) = x'(x) for all (standard) x' e X'}, mw.(x') = { y' e *X' : *x(y') � *x(*x') = x(x') for all (standard) x e X}. Using the Hahn-Banach theorem, we can show that the weak and weak* topologies are Hausdorff (exercise). 4.22 Alaoglu's Theorem
The closed unit ball in X' is compact in the weak*
topology. Proof: Let B be the unit ball in X'. We must show that corresponding to every y' e * B there is a point x' e B so that *x(y') � x(x') for all x e X. Fix y' e *B and define a functional x' on X by x'(x) = st(y'(*x)), x e X. Then *x(y') � x(x') for all standard x e X. The linearity of x' is obvious, and, finally, x' E B since lx'(x)l � 0 (11 Y'II II*xll l � l lxl l by transfer (y' E * B so II Y' II � 1). 0 The same result can be proved for a ball of any radius and also follows directly from Theorems 4.22 and 2.6. We obtain as a consequence the following corollary. 4.23 CoroUary
A norm-bounded and weak*-closed subset of X' is compact.
Proof: Use Theorems 4.22 and 2.4.
D
One might expect a similar result to be true for subsets of X in the weak topology. However, it turns out that the unit ball in X is weakly compact iff X is reflexive, which means that X = X" [ 1 4, Theorem V.4.7]. Considering the importance of sequential compactness as emphasized in §111.3, we would like to know when the unit ball B in a Banach space X is weakly sequentially compact. A deep theorem due to Eberlein and Smulian asserts that B is weakly sequentially compact iff B is weakly compact (iff X is reflexive by the above remark). A nonstandard proof ofthis result can be found in [47].
1 44
Ill.
Nonstandard Theory of Topological Spaces
Example We will show that the unit sphere in 100 is not weak* sequen tially compact even though it is weak* compact by Alaoglu's theorem. Consider the sequence e" e 11 (regarded as embedded in l:X,) defined by e" = (��: i e N). Then lle" ll 1 = 1 . Suppose that (e") has a convergent sub sequence (e"k). Define the element x = ( x 1 : i e N) e l "' by xi = 1 if i n,. and k is even, and x, = 0 otherwise. Then e""(x) = 1 if k is even, and 0 if k is odd, so the sequence (e"k(x)) does not converge, i.e., (e"k) does not converge in the weak• topology. Note that by compactness (check) the sequence e" has a weak• limit point y, but we cannot select a convergent subsequence since the neighborhood system at y does not have a countable base.
4.14
=
An extensive study of the structure of Banach spaces using nonstandard methods has been developed by Henson and Moore [ 1 6]. This study uses in an essential way the notion of the nonstandard hull of a Banach space. We present the definition of the nonstandard hull of a metric space in §111.6 to help the interested reader to understand these results. Exercises 111.4
1. 2. 3. 4.
5. 6. 7. 8.
9.
10. 1 1. 1 2.
Show that d(x, y) = ll x - Yl l is a metric. Show that II 1 1 1 and I I lloo are norms on R". Show that B(S) with the sup norm I I lloo is a Banach space. Show that C(S) is a closed subspace of B(S) if S is a topological space. Show that for a normed space all monads are translates of the monad of zero. Show that a linear operator is continuous if and only if it is continuous at 9. Prove Corollary 4.8. Show that the operator norm on B(X, Y) makes B(X, Y) into a normed vector space. Show that the set of compact operators is a linear subspace of B(X, Y). Show that the weak and weak• topologies are Hausdorff. Discuss the relationship between Alaoglu's theorem and the Tychonoff product theorem. Two norms on a space X are equivalent if the corresponding metrics they define are equivalent. (a) Show that the norms 11 · 1 1 and 1 1 1 · 1 11 on X are equivalent iff there exist positive (nonzero) constants rt and P in R so that rt ll x ll � lll x lll � P ll x ll for all x e X. (b) Show that any two norms on R" are equivalent. (Hint: Show that any norm II I I is equivalent to ll · l l oo · To do so you need only show that lll x lll!ll x lloo and ll x lloo!lll x lll are finite for all x e *R". Write x = L7= 1 x,e, and get estimates.)
111.5
1 45
Inner-Product Spaces and Hilbert Spaces
X be a vector space with a topology ff. X is a topological vecwr space if both vector addition (as a map X x X -+ X) and scalar multipli cation (as a map R x X -+ X) are continuous. Let m(a) denote the monad of a e R and p(x) denote the monad of x e X . Show that if X is a
1 3. Let
topological vector space (of more than one dimension) then
(a) p(x) + Jl( Y) = p(x) + y = Jl(X + y) = x + y + Jl(O), (b) m(a)x c m(a)p(x) = aJl(x) = p(ax), (c) ff is Hausdorff iff p(O) n X { 0}, (d) if X is a topological vector space with topologies ff1 and ff2 having monads Jl 1 and Jl2 then ff1 ff2 iff Jl 1 (0) = Jl 2(0). =
=
111.5
Inner-Product Spaces and Hilbert Spaces
In this section we consider those normed spaces and Banach spaces in which the norm is derived from an inner product. Most of the results and proofs of this section are standard. The canonical example of an inner product occurs in Euclidean space R " where the scalar product of x ( x1 , • • • , x") and Y ( y 1 , • • • , y" ) is (x, y) = Li x1y1 • The angle 0 between two nonzero vectors x and y is given by the familiar formula cos 0 = (x, y)/l l x ii ii Y II · The scalar product is generalized to vector spaces as follows. =
=
Let H be a vector space. An inner product on H is a map ( , ): R which satisfies (for all x, y, z in X and a, b e R)
5. 1 Definition
X
x
X
-+
(i) (x, y) = (y, x), (ii) (ax + by, z) = a(x, z) + b(y, z), (iii) (x, x) � 0, and (x, x) = 0 iff x
=
0.
A vector space with an inner product is called an inner product space. A norm on H is obtained by setting llxll = J(x, x) (exercise). If H is complete in this norm it is called a Hilbert space. To prove that II I I is a norm on X one uses the following basic result.
Schwarz's Inequality For any x, y in an inner-product space H, l(x, y) l � ll x i i ii Y I I ·
5.2
Proof: Let x and y be given. For any real A. we have (x + A.y, x + A.y) ll x l l 2 + 2 A.(x, y) + A.2II YII2 � 0. Thus the quadratic expression in A. given by =
Ill.
1 46
Nonstandard Theory of Topological Spaces
l l x W + 2A.(x, y) + A. 2 jj y jj 2 cannot have distinct real roots, and so the discrim inant j(x, y) j l - l l x ii 2I I YII2 :S: 0. D 5.3 Corollary
x, y e H.
(x, y) is continuous on H
Proof: Exercise.
x
H as a function of the variables
0
1,
5.4 Examples
1 . In the linear space R n we define the inner product of x = (x . . . , Xn) and y ( y1 , . . • , Yn> by (x, y) = Li= x1y1• The reader should check that this defines an inner product on R n . From Schwarz's inequality,
1
=
tt� l (JI Y Ct� y:Y'2• x ,y,
:s:
x:
'2
1,
2. The space 1 • Let 1 denote the space of all infinite sequences x = (x 2 x , . . . ) for which Lr;_ 12 x : < oo . If X = (x l , x 2 , . . . ) and y = ( Y � > Y 2 • . • • ) 2 are two such sequences, we define (x, y) = Ll';, x1y1 • To check that (x, y) is finite for x, y e 1 we have 2 '2 '2 '2 '2 y: , x: x: x: j x ,yi j :S: :S:
1
,t1
(t1 Y Ct Y (�1 Y (J1 Y rr; I j x , y,j (L� I x :l'' 2 llxll 1
= and so converges. Using the fact that is a norm, we can now easily check that is a linear space. We will see later that 2 all separable Hilbert spaces are isomorphic to 12 •
Using the inner product, we can introduce a notion of orthogonality in an inner-product space. an inner-product space then x and y in H are orthogonal if (x, y) = 0, in which case we write x .l y. I f S £;;; H then s 1. = { x e H : x .l z for all z e S}.
5.5 Definition If H is
5.6 Proposition For any S
£;;;
H, s 1. is a closed linear subspace of H.
Proof: Let x, y e s1. and a, b e R. Then, for any z e S, (ax + by, z) = a(x, z) + b(y, z) = 0, so s 1. is a linear subspace. To show closure, let x e •s1.
III.S
1 47
Inner-Product Spaces and Hilbert Spaces
and x :::: y e H. Then (y, z) :::: (x, z) = 0 for all z e S by the continuity of the inner product, and so y E s J. . Thus s J. is closed. 0 Since the norm on an inner-product space H is derived from the inner product, we might expect that it has some special properties. It turns out that it is completely characterized by the following law.
5.7
ParaUelogram Law
iff for all x, y e H
A normed space (H, II I I> is an inner-product space
Proof: Suppose H is an inner-product space. Then
llx - Yll 2 + l l x + Yl l 2 = (x - y, x - y) + (x + y, x + y) = ll x ll 2 - (y, x) - (x, y) + II Y II2 + ll x ll 2 + (y, x) + (x, y) + I I Y I I 2 = 2llx W + 2 I IY W . The converse, which we omit, sets (x, y) = ! { l lx + Yll - ll x - Yll }. 0 Using this simple result, we now establish a sequence of results which are fundamental to all further analysis of Hilbert spaces. 5.8 Definition A subset then cxx + (1 - tx)y e K
K of a vector space H is convex if whenever x, y e K for all real ex e [0, 1].
In the proof of the next result we use completeness in an essential way. Theorem If K is a closed convex subset of a Hilbert space H, then there is a unique element x0 e K so that l l x oll S l l x l l for all x e K, i.e., K has a unique element of smallest norm.
5.9
Proof: Let d inf{ llxl l : x e K } . Then for each � > 0 there is an x e K so that d � l l xll < d + �- By transfer, with � infinitesimal, there is a y e • K with II Y II :::: d. We now show that y is near-standard. Since K is complete by Corollary 3.1 5, it is enough to show that y is pre-near-standard (see Proposi tion 3 . 1 4) . Fix e > 0 in R. By transfer from the parallelogram law, =
(5. 1)
l
for any x e K. If x e K then since y e *K, (x + y)/2 e * K , so x + Yll 2 = 4 l l(x + y)/21 1 2 � 4d 2 • It follows from (5. 1 ) that llx - Y l l 2 < 2ll x 2 + 2d 2 4d2 + 'I = 2 l l x ll 2 - 2d2 + , , where 'I is infinitesimal and x e K. But we can find an x e K so that l l x ll 2 < d 2 + s/4, and we get ll x - Yll 2 < s/2 + 'I < e.
Ill.
148
Nonstandard Theory of Topological Spaces
Thus y is pre-near-standard, so y is near some x0 e H. The point x0 e K since K is closed, and llxoll = d by the continuity of the norm. The uniqueness is another application of the parallelogram law (exercise).
0
5.10 Theorem Let E be a closed subspace of the Hilbert space H with
E :F H. There are unique linear operators P: H Px + Qx for all x e H. Further,
x
-+
E, Q: H
-+
El. so that
=
Px = x
iff x e E
and
Qx = x
P and Q are called the projections of H onto E and El., respectively. Proof: For x e H let K x + E {x + y :y e E}. Then K is convex and closed (check). Let Qx be the unique element of smallest norm in K (existing by 5.9), and put Px = x - Qx. Then it is clear that x = Px + Qx and Px e E. To show that (Qx, z) 0 for all z e E, we put Qx = y. Assuming without loss of generality that llzll = 1, we have =
=
=
IIYII 2 � l!Y - az W = (y - az, Y - az) = II YII 2 - 2a( y, z) + l al 2 for every a e R, yielding 0 � - 2a(y, z) + lal 2 • If a = (y, z) this gives 0 � - l(y, z)jl, and so (y, z) = 0. The uniqueness of P and Q follows from the fact that E n El. = {11}. For if x = x1 + x 2 with x1 e E, x2 e El., then x 1 - Px Qx - x 2 and x 1 - Px e E, Qx - x 2 e £ 1. , so x 1 = Px and x 2 = Qx. The rest =
of the proof is left to the reader.
0
The culmination of the preceding sequence of results is the following theorem, which probably has more applications than any other result on Hilbert spaces. 5.1 1 Riesz Representation Theorem To each bounded linear functional
on H there corresponds a unique element y e H so that L(x) x e H, and I l L I I = II Y I I ·
=
L
(x, y) for each
Proof: We may assume that L is not identically zero (otherwise take y = 11). Let E = {x e H : Lx = 0} . Then E is a closed linear subspace (check) and El. :F { 11 } , so we may choose z :F 11 in £ 1. . Then, for any x e H, x - (Lx/Lz)z e E, so (x, z) - (Lx/Lz)(z, z) = 0. Thus Lx (x, [ Lz/(z, z)] z), and we take y = [ Lz/(z, z)] z. The rest is left as an exercise. 0 =
5.1 2 Coronary A Hilbert space H is self-dual; i.e., H
=
H'.
Next we investigate the generalization to Hilbert space of a familiar no tion in Rn, that of an orthonormal basis. In R" the vectors e 1 = ( 1, 0, 0, . . . 0),
111.5
1 49
Inner-Product Spaces and Hilbert Spaces
e2 = (0, I , 0, 0, . . . , 0), . - . , e" = (0, 0, . . . , 0, 1 ) have the property that lle1ll = I , (e; , e1) = 15� (the Kronecker 15-function), and any vector x e R• can be written uniquely as x = L�= 1 a1e1• The set { e1} is called an orthonormal basis. In Hilbert spaces we will see that orthonormal bases exist and that any vector can be expressed in a limiting sense in terms of the orthonormal basis. 5.13 Definition A set S {e1 : i e I} of nonzero vectors in an inner-product space H is orthonormal if e1 .l e1 for i :F j and lle1ll = 1 for all i e I. S is maxi mal (or complete) if it is not properly contained in any other orthonormal set. Given any x e H the numbers x(i) (x, e1) are called the Fourier coeffi cients of x relative to the orthonormal set S = { e1} . =
=
If H is a nontrivial inner-product space (i.e., contains more than the zero vector 0) then there is at least one orthonormal set in H obtained by taking a single nonzero vector x e H and forming the normalized vector e x/llxll· The existence of maximal orthonormal sets then follows from the following more general result. =
Every orthonormal set orthonormal set S c H.
5.14 Theorem
S
c
H is contained in a maximal
Proof: Let f/ be the collection of all orthonormal sets in H containing S, and partially order f/ by set inclusion !,;;;; . f/ is nonempty since it contains S. We use Zorn's lemma (see the Appendix) to show the existence of a maximal orthonormal set. Let rc s;; f/ be any chain in f/. Then the set S = US(S e 'C) is an orthonormal set, for if xl, x2 E S, then X E sl and x2 E sl for some Sl , s2 E Cff. Since rc is a chain, either sl s;; s2 or sl !,;;;; sl . In either case X and y are in some S e Cff, so x .l y. Thus S is orthonormal. By Zorn's lemma there is a maximal orthonormal set. D With a little more work it is possible to prove that any two maximal orthonormal sets can be put in one-to-one correspondence (i.e., have the same cardinality), but we will not need this fact. The reader should prove (exercise) that S is a maximal orthonormal set iff x e H and x .l S implies that x = 0. This fact will be used in the proof of Theorem 5.19. 5.15 Example
set, for if x
=
The vectors e1 (15� :j e N) in 12 form a maximal orthonormal (x1 :j e N) e 12 and (x, e1) x1 = 0 for all i e N, then x = 0. =
=
In the following we will deal only with inner-product spaces H which are (norm) separable (i.e., H contains a countable set which is dense in the
1 50
Ill.
Nonstandard Theory of Topological Spaces
topology induced by the norm). In this case a ny orthonormal set is either finite or countable, for if {e 1 : i e / } is orthonormal and i -=1 j, then ll e 1 - e1i l 2 = (e1 - e1 , e1 - e1) = ll e1 i l 2 + l l e1i l 2 = 2 since (e� o e1) = 0. Conversely, if any or thonormal set in H is either finite or countable then H is separable (exercise). Since the following results are easy if H is finite-dimensional (i.e., contains a finite maximal orthonormal set), we will assume in the following that the inner-product space H contains a countable orthonormal set which we ar range in a sequence ( e1 : i e N ). Without loss of generality we have chosen I = N. Now let x e H and ( a 1 : i e N) be a sequence of real numbers. Then (5.2)
l x - .f a,e,l 2 (x - f a1e1, x - f a1e1) =
•= I
1= 1
i= l
n
n
= l l x ll 2 - 2 L a,{x , e ;) + L i a;l 2 1= 1 i= l
=
l l xW
+
•
•
i= I
i=l
L i a 1 - (x , e1)j l - L i(x, e1)jl.
From this we obtain the following results. 5. 16 Best Approximation Theorem Let
(e1 : i e N ) be an orthonormal se
quence in an inner-product space H. For any x e H,
i.e., the best norm approximation to x by a linear combination of the e 1 is given by choosing the coefficients to be the Fourier coefficients.
Proof: The right-hand side of (5.2) is minimized if a 1 5. 17 Bessel's Inequality For any
=
(x e ;) ,
.
D
x e H,
<Xl
L l <x. e,) i 2 � ll x W. i= I Proof: Exercise.
0
Bessel's inequality has the following interpretation. For any x e H we can consider the sequence (x(i) : i e N ) of Fourier coefficients of x relative to
111.5
1 51
Inner-Product Spaces and Hilbert Spaces
a given orthonormal sequence S = (e; : i E N). Then Bessel's inequality shows that this sequence is in 12 • Thus for a fixed countable orthonormal sequence (e1) we obtain a mapping T: H --+ 12 defined by Tx = (i(i) : i E N). It is easy to check that T is a linear mapping. The next result, which requires that H be complete, shows that T maps H onto 12 • Let (e ; : i E N ) be a countable orthonormal sequence in the Hilbert space H . Then each element of 1 2 is of the form :X for some x e H .
5.18 Riesz-Fischer Theorem
Proof: Let (a; : i E N ) be a sequence in 1 2 so that I� 1 at < oo. Then the sequence x, L�= 1 a;e1 is a Cauchy sequence in H since x,. - x, = Ii"= n + 1 a ;e ; if m > n, and so ll x ,. - x, jll = Ii= .. + 1 a1. Since H is complete, there is an element x which is the (norm) limi t of x, . By the continuity of the inner product, (x, e;) 1im(x, , e1) = a; for any i E N. 0 =
=
The element x which is obtained in Theorem 5. 1 8 is often written x = I� 1 a;e; . One consequence of the next theorem is that if (e; : i E N) is a maximal orthonormal sequence, then the associated map T is one-to-one, and so an x E H can be written in only one way as I � 1 a 1 e 1 • For this rea son a maximal orthonormal set (also called a complete orthonormal set) is sometimes called an orthonormal basis. It should be emphasized that this notion of basis must be understood in a limiting sense and not in the alge braic sense of vector space theory. The orthonormal sequence (e1 : i E N) in the Hilbert space H is maximal iff each x E H can be written uniquely as x = I� 1 (x, e1)e1•
5.19 Theorem
Proof: Suppose (e1) is maximal and x E H . Then by the Riesz-Fischer theorem there is an element y E H so that y = I� 1 (x, e1)e1 and so (y, e1) = (x, e1) for all i e N. But then (x - y, e ;) = 0 for i e N , so x = y by the remark following Theorem 5. 1 4. Conversely, suppose each x e H can be written as x I� 1 (x, e1)e1• If (e) is not maximal there is an x -:f. tJ in H so that (x, e1) = 0 for i E N. But then x = I� 1 (x, e1)e; = tJ (contradiction) . 0 =
( Parseval's Identit ies) If (e1 : i E N) is a maximal orthonor mal sequence in the Hilbert space H then 5.20 Theorem
(i) I � 1 l< x . e 1) ! 2 = ll xW for all x e H; (ii) I,'"; 1 (x, e1)(y, e1) = (x, y) for all x, y E H.
1 52
Ill.
Nonstandard Theory of Topological Spaces
Proof: We leave the proof of (ii) as an exercise. To prove (i) we see by Bessel's inequality that L� 1 i(x , e 1) il � l l xll 2• On the other hand, gi ven e > 0 there is a z I?= 1 (x, e1)e; so that llx - zll < e, whence ll x l l < liz II + e. Thus =
2
0
The results above can now be used to show that 1 is essentially the 2 only separable infinite-dimensional Hilbert space. Given a maximal orthonormal sequence S = ( e ; : i e N) in a separable Hilbert space H, the associated map T: H -+ 12 is one-to-one, onto, and satisfies (x, y) (Tx, Ty) for all x, y e H, and so T is a Hilbert space isomorphism. 5.21 Theorem
=
Proof: Use 5. 1 8 -5.20.
0
We end this section with an application of nonstandard analysis to prove a theorem concerning compact operators in Hilbert space. Much more can be done in this direction. In particular, Bernstein and Robinson [4] first proved that so-called polynomially compact operators have nontrivial in variant subspaces using refinements of the technique used here. We are going to prove that every compact operator on a separable Hilbert space H can be approximated arbitrarily closely by an operator of "finite rank." An operator Q: H -+ H is of finite rank if there is a finite dimensional subspace E c H so that Q x e E for each x e H.
5.22 Definition
Since every separable Hilbert space H is isomorphic to 1 we will identify 2 H and 12 in the following discussion. Thus we will asume that an orthonormal sequence ( e1 ) is given and represent any x e H as either x = Ir; 1 a1e1 or ( a 1 , a2 , • • : ) . First we need the following lemma.
i�
2
If x = (a1 : i e •N ) e •12 is near-standard, then I l a1 i (i e •N, w) is infinitesimal for any w e • N 00 •
5.23 Lemma
Proof: If y = (b1 : i e N) e 12 then Iimk .... oo I l b1l 2 (i e N, i � k) 0, so Li•bd 2(i E •N, i � w) � 0 for any infinite OJ. Now since X E •12 is near-stan 2 dard there is a y e 12 with l l x - • yii 2 = I i a 1 - •b 1 l (i e • N) � 0. By the trans=
111.5
Inner-Product Spaces and Hilbert Spaces
1 53
fer of the triangle inequality,
D] a1 1 2 (i e *N , i � w)] 1 1 2 S:
1 [L ia ; - *b ; l 2(i e *N, i � w)] 1 2 + [L I *b ; l 2(i e *N, i � w)J 1 ' 2 ,
and both terms on the right are infinitesimal.
D
Theorem Let T: H -+ H be a compact linear operator. For each £ > there is an operator Q of finite rank so that l i T - Q ll < £.
5.24
0
Proof: For each k e *N (finite or infinite) we define a projection operator Pt : *H -+ *H by Ptx = (a. , a 2 , , at, 0, 0, . . . ) when x = (a 1 : i e *N). Then Pt is linear and II Ptxll S: l l x l l for any x e *H. Also, 11(1 - Pt)xll2 = Ll a ; l 2 (i E *N, i � k + 1 ), and so, by Lemma 5.23, 11(1 - Pt)xll is infini tesimal for k infinite and x near-standard. It follows that II * T - Pt * T il is infinitesimal for all infinite k. Now let £ > 0 in R be given. The internal set A = {n e *N : II* T - P. * T i l < •
•
•
e} contains all infinite natural numbers, and so contains a finite (standard) integer m by Corollary 7.2(ii) of Chapter II. Thus II* T - P'" • Ti l < £. Trans ferring down shows that li T - P'" TII < £. Finally, the operator Q = P'"T is of finite rank since its range is contained in the subspace E generated by {e1 , , e'"}. D .
.
•
This result can be used as a starting point for the Fredholm theory of compact operators. Exercises
111.5
1 . Show that if ( , ) is an inner product on a vector space H then the map 11 · 11: H -+ R + defined by llxll = .j(x, x) is a norm on H . 2. Prove Corollary 5.3 3. Show that the element x0 of Theorem 5.9 is unique. 4. Complete the proof of Theorem 5.10. 5. Finish the proof of Theorem 5. 1 1 . 6. Show that S is a maximal orthonormal set iff x e H and x l. S � x = 9. 7. Show that if any orthonormal set in an inner-product space H is either finite or countable, then H is separable. 8. Prove Theorem 5. 1 7. 9. Prove Theorem 5.20(ii). 10. Establish the following converse to Lemma 5.23. If x = (a1 : i e • N) e */ , 2 l lx ll2 L l a ; l2 (i E * N) is finite, and r l a ; l2 ( i E *N, i > w) � for all infinite w, then x is near-standard. =
0
1 54 1 1.
Ill. x
The Hilbert cube is the set of all
=
Nonstanda rd Theory of Topological Spaces
(x1 ) e
12 such that
lxd
�
1 /i, i e N.
Show that the Hilbert cube is compact.
1 2.
Let
H
be a Hilbert space and let
bounded linear operators
-+
A: H
B(H) denote the normed space of all H. A subbase for the weak operator
B(H) is formed by the collection of all sets of t he form { A : I( (A - A0 )x , y) l < �}. A0 e .B(H), x, y e H and � > 0 in R. Show that the monad of A0 in B(H) in the weak topology is given by p(A 0) {A e *B(H) : (Ax, y) � (A0x, y) for all standard x, y e H}. 1 3. (Stand ard) A bilinear form on H i s a m a p B: H x H -+ R s uch that B(x, ) is linear for each x e H and B( y) i s li near for each y e H. B is bounded if there exists M e R such that I B ( x , y) l � Mllxii iiYII for all x, y e H. Show that if B is a bounded bilinear form, then there exists an operator T e B(H) such that .B( x , y) = (Tx, y) for all x, y e H. 1 4. Use Exercises 5. 1 2 and 5. 1 3 to show that the unit ball in B(H) is compact topology on
=
·
· ,
in the weak operator topology.
111.6 Nonstandard Hulls of Metric Spaces In this short section we introduce the reader to the concept of the non standard hull of a metric space. This notion was in troduced by Luxemburg
[36]
and has proved to
be
a powerful tool in the nonstandard analysis of
Banach spaces, as i ndicated by the survey paper of Henson and M oore
[ 1 6].
The technique o f nonstandard analysis, a s applied to the theory of Banach spaces, is essentially equivalent to the use of Banach space ultrapowers, a technique which o riginated with Dacunha-Castelle and K ri vine
[ 1 0]
and is
now used extensively. Nonstandard methods, however, are more int uitive and usually easier to apply, especially when they involve concepts, such as the internal card i nality of a •-finite set, which are not easy to express i n t he ultraproduct setting. In this section we will assume that the nonstandard analysis is carried out i n a K-saturated enlargement where
K > �0 •
space. Recall that the principal galaxy
G
S uppose that (X, d ) is a metric
= fin(• X) is the set of points in • X
each of which is at a finite distance from a poi n t in X (regarded as em
b e •X
bedded in • X). If a,
we say as usual that a
denote the equivalence classes of ternati vely, *d(a, b) :::
0}
X
G
(notice that if a
b e G,
=
m(a) and
y
=
m(b)
in
eG
and
we can define
d(x, y) x
�
b
if *d(a, b)
�
fl.
=
0.
Let
X
under the equivalence relation � . Al
{b e G : b E G). Since
is the set of monads, where each monad m(a)
for a e
*d(a, b) is finite for any a,
when
G
st( * d(a, b) )
b
� a then
=
111.6
Nonstandard Hulls of Metric Spaces
6. 1 Proposition
(X, d) is a metric space.
Proof: Exercise.
6.2 Definition
1 55
D
(X, d) is called the nonstandard hull of (X, d ).
We now use saturation to prove that (X, d) is complete [even if (X, d) is not]. Our construction is like that of Theorem 3. 1 7, but here X consists of monads of finite points and not just pre-near-standard points. Suppose that *X lies in a K-saturated superstructure with " > �0 . Then (X, d) is a complete metric space.
6.3 Theorem
Proof: Let ( m(a 1) : i e N ) be a Cauchy sequence in (X, d). Then for each k e N there is an n( k) e N so that *d(a1 , ai) < l/k if i and j are both > n(k); we can assume without loss of generality that n(k ) -+ oo as k -+ oo. Let cp(i) = a; . By Theorem 8.5 of Chapter II, the map cp: N -+ *X can be extended to an internal map <]}: N -+ *X, where N � * N is internal and contains N and so contains some infinite integer. We would like to show that there is some infinite integer m' in N so that *d(a 1 , a'".) < l/k for all i e N with i > n(k), where a'". = <]J(m'). For any k e N the set E(k) = {m e N : *d(a1 , a) < l /k for all i, j e N satisfying n(k) < i � m, n( k) < j � m } is internal and contains N. Therefore E(k) also contains {m e * N : m � mt} for some infinite integer mt o and we may assume that mt + 1 � mt for all k e N. Again by Theorem 8.5 of Chapter II we may extend the sequence (mt) to an internal decreasing mapping from an internal set N c *N into *N. Since mt > k for each finite k e N, there is an infinite w with m"' � w and m"' e E(k) for all k e N. Let m' m"' . Then *d(a1, a'".) < l/k for all i e N with i > n(k). It follows that a'". is finite and (m(a1)) converges to m(a'".). D =
If our metric space (X, d) is a normed vector space with norm I 1 1. the nonstandard hull can be made into a normed vector space in an obvious way. For in this case G consists of all x e *X for which llxll is finite, and so G is a vector space over the reals. We define addition and scalar multi plication of elements in X by
m(x) + m(y)
=
m(x + y),
and
am(x) = m( ax ).
x, y e G ,
1 56
Ill.
Nonstandard Theory of Topological Spaces
Also we define a norm in X by l l l m(x) l l l = st ll x ll . x e G. It is easy to check that d(m( x), m(y) ) ll im(x) - m(y) lil· From Theorem 6 . 3 we see that (X, Ill Ill ) is a Banach space. The details are left to the reader. =
Exercises 11/.6
1 . Prove Proposition 6. 1 ; in particular, show J is well defined.
2. Show that if (X, 11 · 11) is a normed space, then (X, 111 · 111) as a Banach space.
3. Show that there is an isometric embedding of a Banach space into its nonstandard hull. 4. Consider the sequence ( en ) which is 0 for n -# w e * N 00 and 1 for n = w to show that the mapping in Exercise 3 is not onto for 12 • S. Consider the sequence (xn>• where xn = 1/w for 1 � n � w, w e * N 00 and xn = 0 for n > w, to show that the mapping in Exercise 3 is not onto for II.
"111.7 Compactifications
In this section we show how some Hausdorff spaces (X, 9') can be embed ded as dense subsets of compact Hausdorff spaces ( Y, ff). That is, there exists a 1 - 1 map t/J: X -+ range t/1 !;;; Y so that t/1 is a homeomorphism and range t/1 is dense in Y. In this case ( Y, ff) is called a compacti.fi cation of (X, 9'). We usually identify X and range t/1, and so regard X as a subset of Y; we will denote Y by X. A given space X typically has many compactifications. For example, if one adjoins 0 and 1 to (0, 1) one obtains the compact interval [0, 1 ] . Adjoining a single point to both ends of (0, 1) gives a circle. Similarly the plane can be made into a sphere by adjoining a single point. We are interested here in compactifying a space X so that certain continuous functions on X have con tinuous extensions to X. What, for example, should one adjoin to (0, 1] to make sin(l/x) continuous on the resulting compact space? 7. 1 Definition
Let Q be a family of (perhaps not uniformly) bounded, contin uous, real-valued functions on (X , 9'). (X, ff) is called a Q-compacti.fi cation of ( X, 9') if it is a compactification for which
(a) each f e Q has a continuous extension 1 to (X, ff), (b) if x and y are different points in X - X there is an f e Q whose exten sion 1 separates x and y, i.e., /(x) "# 1( y) We sometimes write X0 for X. .
In order to construct a Q-compactification we need to suppose that Q contains sufficiently many functions.
111.7
1 57
Cornpactifications
7.1 Definition A family Q of continuous functions
closed sets if, for each set A that f(x) t1 ! [ A ] .
c
X and each x e X
-
distinguishes points and A, there is an f e Q so
It should be noted that not all Hausdorff spaces X admit sufficiently many continuous real-valued functions to distinguish points and closed sets. There are enough functions if X is completely regular [ 20] . The compactifications of this section will be constructed from • X. The original work on this construction was done by Gonshor [ 1 5 ] , Luxemburg [36], Machover and Hirschfeld [37], and Robinson [43]. Let Q be a family of bounded, continuous, real-valued functions on (X, 9'). Assuming that Q distinguishes points and closed sets, we construct X as fol lows. We call two points y, z e • X equivalent, and write y ...., z, if *f(y) � */(z) for all f e Q. It is easy to see (check) that - is an equivalence relation. The equivalence class containing x e • X is denoted by [ x ], and the set of all equivalence classes is X. Next we show that if x e X, then [x] = m(x), the monad of x. First note that if y e m(x), then y - x since each f e Q is continuous. On the other hand, if U is an open set containing x, then there is an / e Q so that f(x) ¢ / [X - U ] . Thus f - 1 [R - f [X - u] ] is an open set containing x and contained in U. We conclude that [x] m(x). We extend each f e Q to a function on X (again denoted by /) by setting f( [ y] ) = st( *f( y) ), y e *X (check that f is well defined). The set of extended functions is again denoted by Q. The topology !T on X is the weak topology for the functions in Q. Thus U is open in X iff for each [y] e U there is a finite set { !1 , , f,} !;;; Q and a positive number £ in R so that { [z] e X: l !� [y]) - /� [ zJ ) I < £, 1 � i ::; n} !;;; U. In order that we may treat X as an element of the original superstructure V(X) (which will be used in the proof of compactness), we may think of each point in X as a function on Q by the definition [y] ( f ) = f([y] ). Distinct points of X give distinct functions on Q. The standard construction of X is based on such a family of functions on Q. It is often helpful, however, to think of X as a quotient of • X as we have done. Let xa be constructed as above from a set Q of bounded, continuous, real valued functions on (X, 9') which separate points and closed sets. =
•
.
•
7.3 Theorem (Xa, !T) is a Q-compactification of (X, 9').
Proof: Let xa be denoted by X. Define the map 1/J: X ..... X by 1/J ( x ) = [x], m(x) for x e X, so the map 1/J is 1 - 1 by 1 . 1 2(c) since X is Hausdorff.
x e X. Now [x]
=
Ill.
1 58
Nonstanda rd Theory of Topological Spaces
To show 1/1 is a homeomorphism, we must show that t/1 and t/1 - 1 are con tinuous. An easy exercise shows that 1/1 is continuous. To see that .p - t is continuous, we must show that if x e X and V is an open neighborhood of x in !/, then there is a U e ffx so that U n X £:: V (we regard X as contained in X). Let f e Q be such that f(x) j f[ A], where A X V. Then there is an & > 0 in R so that {z e X : lf(z) - f(x)l < &} £:: V (why?); we let U {z e X : lf(z) - f( x) l < & } . To show t/J[X] is dense in X, let [y] e X - t/I [ X ] , and let U e ff be given by U = { [z] e X : l.f.{ [z] ) - f�[yJ ) I < &, 1 � i � n}. We must show that [ x] e U for some x e X Let cx1 f�[y] ), 1 :S i � n. Then the set { x e *X : I.f.{x) cx1 1 < &, 1 :S i :S n} is not empty (indeed it contains y). By downward transfer, the set {x e X : lf�x) - cx1 1 < &, 1 :S i :S n} is not empty, and we are through. To show that X is compact we consider a mapping T on X. For each [y] e X, T( [y] ) is the function from Q into R defined by setting T([y] )(f) = f([y] ) for each f e Q. Let A be the range of T; then T is a 1 - 1 mapping from X onto A. We make T a homeomorphism by letting U be open in A iff r- ' [U] is open in X. Thus a typical neighborhood of an a e A is given by , f.. } c: Q and an & > 0 in R: it consists of those b e A with a finite set { !1 , Ia(/;) - b(/;)l < &, 1 :S i � n. Since X is dense in X, each such neighborhood contains a T([x] ) for some x e X ; i.e., I a(/;) .f.{x) l < 8 for 1 � i � n. To show that X is compact, we need only show that A is compact. Fix b e • A. Let & be a positive infinitesimal in •R, and let Q 1 be a hyperfinite subset of Q such that •J e Q1 for each f e Q. By the transfer principle, there is an x e • X such that l b( f) - f(x) l < 8 for each f e Q 1 . Let c T([x] ). For each f e Q, c(f) T( [x] )(f) � *f(x) � b( *f), so b is in the monad of the standard point c e A. Thus A is compact. Finally, by the construction, each member of Q has a continuous extension to X, and the family of extensions separates the points of X. D =
-
=
.
•
_
•
=
•
-
=
=
It is not hard to see that if Q 1 and Q2 are two families as described above with Q 1 £:: Q 2 , then there is a continuous map � from X0.1 onto xo., such that �(x) = x for all x e X. In this case we write .XO.• � xo.1• It follows that a Q-compactification of X is unique up to a homeomorphism that leaves the points of X fixed (see, for example, [20, Theorem 22] ). Any compactification X of X is a Q-compactification; just let Q = { g x : the function g: X -+ R continuous } , where 9x denotes the restriction of g to X. It follows that if Q consists of all bounded, continuous, real-valued functions on X, then X� is the largest compactification of X, i.e., xa � X for any other compactification X of X. x
111.7
1 59
Compactifications
functions on X, then, following Constantinescu and Cornea [9], we may obtain Q by adjoining to Q0 the family Cc of all continuous functions with compact support (i.e., vanishing off compact sets). If Q0 is empty and so Q = Cc , then XQ can be identified with X u { oo }, where oo is a single point not in X. In this case we have a topology .C/ whose members are all open sets in X, together with all sets U of XQ such that XQ - U is compact in X (check); the space XQ is called the one-point compacti.fication of X. We end this section with a result concerning the Stone-Cech compactifi cation N of the natural numbers N (with the discrete topology).
7.4 Theorem The points in N - N are in one-to-one correspondence with the free ultrafilters on N via the map [w] � �"'1 , where j>j"'1 {A s;; N : =
w E *A } .
Proof: For each A c N , the characteristic function X A i s i n Q , the set of bounded continuous functions on N. It follows that for each equivalence class [w] E N - N either [w] c *A or [w] c *N - *A. If [w] E N - N then w E • N oo , and the family �"'1 = {A s;; N :w E • A} is a free ultrafilter (exer cise). This is the same ultrafilter as { A s;; N : xA( [w] ) = 1 }, where XA has been extended to N. On the other hand, if fF is a free ultrafilter on N, then the intersection monad Jl(fF) n * F(F E §) is a unique element [w] in N N. To prove this, we assume it is false. Then there are at least two distinct equivalence classes [w] and [y] in Jl(fF) and a bounded sequence (s" ) E Q such that a 0S"' ::f: 0 S Y b. We may assume that a < b and choose c E R with a < c < b. Since either {n E N : s" � c} E fF or {n E N : sn > c} E fF, we have a contradiction. D =
=
-
=
Exercises Ill. 7
l. Let (X, ff) be locally compact, and let X denote the one-point compacti fication of X. Let A be an internal set of near-standard points in • X . Use the fact that st[A] is closed in X and a closed subset of X is compact to
show that st[A] is compact.
2. Show directly that the one-point compactification of a locally compact
Hausdorff space is compact. Show that, for w E • N oo , {A s;; N : w E • A} is a free ultrafilter. 4. What is the Q-compactification of (0, 1) when Q { f(x ) x } ? 5. What is the Q-compactification of (0, 1) when Q = { f(x) = x, g(x) sin ( 1 /x)}? 6. Show that X is open in a compactification X if and only if X is locally compact. 3.
=
=
=
1 60
Ill.
Nonstandard Theory of Topological Spaces
*111.8 Function Spaces
Let (X, fl') and ( Y, .9"') be Hausdorff topological spaces and F be a family of mappings from X into Y. This section will be concerned with two questions: (a) For which topologies � on F is the map (jJ,.. : F x A -+ Y defined by (jJ(f, x) f(x) continuous for all subsets A o;; X in a certain family .1f? Such a topology � is said to be jointly continuous with respect to .1f. (b) For which topologies on the space M of all mappings from X into Y is the closure of F compact? =
To answer these questions, we consider two important topologies, the topology of pointwise convergence and the compact-open topology. For a standard treatment the reader is referred to Kelley [20, Chapter 7]. Our treatment follows suggestions of Hirschfeld [ 1 8]. The nonstandard analysis will be done in an enlargement of a structure containing X and Y. Monads in (X, fl') and ( Y, ff) will be denoted by mx(x) (x e X) and my(y ) (y e Y), respectively, but we will denote nearness in both X and Y by � as in §III. l . With each subset A o;; X we associate an important pseudomonad k..t(f) (f E M) on the space M of all maps from X into Y by setting
(8. 1) k,.t(f) = {g E • M : g(x') � f(x) for all x E A and x' E •A with x'
�
x} .
The following result provides a nonstandard answer to question (a). 8.1 Proposition Let � be a topology for F with associated monads m(f) (f E F). Then � is jointly continuous with respect to .1f iff m(f) o;; n { k ..t(f) : A E .1f} for all f E F.
Proof: We need only show that, for each A E .1f, (jJ A is continuous iff m(f) o;; kA(f) for all f e F. But for f e F. and x E A , the monad of (f, x) in * F x •A is m(f) x m,.. (x), where m..t(x) = mx(x) n •A. (jJ,.. is continuous at each (f, x) e F x A <=> *{jJ..t(m(f) x mA(x) ) o;; my({jJA(f, x)) for each f e F, x e A <=> if f e F, x e A, then whenever g e m(f) and y � x, y E • A, we have g(y) � f(x) <=> m(f) o;; kA(f) for each f E F. 0 8.2
Definition
(a) The topology of pointwise convergence &' on M is the weak topol ogy for the family { 4Jx: x E X } of evaluation maps 4Jx : M -+ Y defined by (jJ "(f) = f(x). The monads for &' are denoted by p(f) (f E M). (b) The compact-open topology f(J on M is generated by the subbase con sisting of all sets of the form W(K, U) = {g e M :g[ K] c U}, where K is
111.8
Function Spaces
161
compact i n (X, 9') and U is open i n ( Y, ff). We let c(f) ( f E M) denote the monads of CC. From 1 . 1 8 we see that (8.2)
p(f)
=
{g E * M : g(x)
�
f(x) for all standard points x E X } .
Proposition Let f be the family of compact subsets of (X, ff). Then, for each f E M , kx( f ) £ n { k A (f) : A E f } £ c(f) £ p(f).
8.3
Proof: (i) kx(f) £ kA(f) for any A £ X, and the first containment follows. (ii) Let K be compact in (X, 9') and U be an open set in ( Y, ff) con taining f[K]. If g E n {kA(f) : A E f }, then g E k,d_f), so g(y) � f(x) for all x E K and all y E *K with y � x. Since U is open, g(y) E *U for all y e * K with y � x E K . But this includes all y E * K since K is compact, and so g(* K ] £ * U, i.e., g E * W(K, U). Thus n {kA(f): A E f} £ * W(K, U) for any K and U with f [ K ] £ U, and the second containment follows. (iii) A subbase for 9 consists of sets of the form W({x }, U), and so 9 is
weaker than
CC
and the third containment follows.
0
8.4 Theorem Each topology which is jointly continuous with respect to the family of compact subsets of X is stronger than CC.
Proof: Immediate from 8. 1 and 8.3.
0
8.5 Theorem Assume F c M is closed with respect to 9. Then F is compact in (M, 9) if for each x the set {f(x) : f E F} has compact closure in Y. Proof: Our condition guarantees that, for any x E X, every point in *{f(x):f E F} = {g(x): g E *F} is near a standard point in Y. Given g E *F, let f(x) be defined for each x E X by setting f(x) = y_ , where Yx is a point in Y with Yx � g(x) [such a point is unique since ( Y, Y) is Hausdorff]. Then f E M and f(x) � g(x) for all x E X, i.e., g E p(f). Since g E *F and F c M is closed, f E F. Thus each g E *F is near a standard f E F. 0
The fact that { f(x):f E F} has compact closure for each x E X is an essential ingredient in obtaining a function fe F from a function g E *F. The argument of Theorem 8. 5 does not work, however, for the compact-open topology since the condition g(x) � f(x) for all x E X is not sufficient to guarantee that g E c(f). If, however, g(x') � f(x) for all x E X and x' E X with x' � x, then
Ill.
1 62
Nonstandard Theory of Topological Spaces
g E kx(f) £ c(f) (by Proposition 8.3) and compactness fol lows. A standard condition guaranteeing that this holds is the following from Kelley [20].
The family F is evenly continuous if for each each open neighborhood U of y, there are neighborhoods y so that for all f E F with f(x) E W, we have f[ V] £ U.
8.6 Definition
x
V
y E Y and of x and W of
E X,
8.7 Proposition The family F is evenly continuous iff the following condition holds: Given x E X and y E Y, if g E • F and g(x) � y, then g(x') � y for all x' � x in •x. Proof: Assume first that F is evenly continuous. Fix a neighborhood U E Yy and the corresponding sets V E fl'x and W E Yy given by Definition 8.6. Since g(x) � y, g(x) E • W, so by transfer g[* V] c • U. In particular, g(x') E • u if x ' � x. This last statement is true for any U E Y, , and so g(x') � y if x' � x. To prove the converse, fix U E .'TY and let V and W be •-open sets in */:l'x and • Y Y ' respectively, with V £ mx(x) and W £ my( y). Now if g E • F and g(x) E W, then g(x) � y. By assumption, for all x' E V, g(x') E my(y) � • u. The rest follows by downward transfer. 0 As a corollary we get a generalized Ascoli theorem due to Kelley [20].
8.8 Ascoli Theorem If F c M is closed in CC and evenly continuous, and {!(x) : f E f} has compact closure for each x E X, then F is compact in (M, CC). Proof: Immediate from the discussion preceding Definition 8 .6
.
0
For the rest of this section we assume that ( Y, .'T) is a metric space with metric d. In this context, a notion which is closely related to even continuity is the notion of equicontinuity, which has already been presented in the real variable case in Definition 1.1 3.6.
8.9 Definition A family F c M is called equiconlinuous on X if, for each x E X and each c > 0 in R, there is a V E fl'x such that, for any f E f, if x' E V, then
d(f(x'),f(x) )
< c.
The family F c M is equicontinuous on X iff, for any x E X and any g E *f, g(x ' ) � g(x) whenever x' � x.
8. 10 Proposition
Proof: Exercise.
0
111.8
1 63
Function Spaces
If F is the family { n+ nx : n e N} then F is evenly continuous but not equi continuous on [0, I ]. By Propositions 8. 7 and 8 . 1 0, any equicontinuous family F c M is evenly continuous. If F c M is a family of continuous functions, then the compact-open topology in F is the same as the topology of uniform convergence on compact sets, or the topology of compact convergence. For the latter topology, a typical basic open neighborhood of f e F is of the form {g e F : d(f(x), g(x) ) < e for all x e K} for some compact K � X and e > 0 in R (see [20, p. 229] ). It follows from Theorem 8.8 that if F is an equicontinuous family in M (whence each f e F is continuous), and F is closed in M with respect to the topology of uniform convergence on compact sets with { f(x) : f e F} having compact closure in Y for each x E X, then F is compact with respect to the topology of uniform convergence on compact sets. Moreover, for an equicontinuous family F, the topology of pointwise convergence is jointly continuous on compact sets (exercise), and hence coincides with the topology of uniform convergence on compact sets. Exercises 11/.8
1 . Use Theorem 8.5 to prove Alaoglu's theorem, 4.22. 2. Prove Proposition 8. 1 0.
3. (a) Show that the set of real-valued continuous functions on R (with the usual topology) is closed with respect to the topology of uniform con vergence on compact sets. (b) Show that part (a) is no longer true if we replace the usual topology on R with a topology f/ such that {r} e 5/ for each r ¥ 0 in R, and U is an open neighborhood of 0 if 0 E U and R U is countable. [ Hint: what are the compact sets? Is g continuous if g(O) = 1 and g(r) = 1 for r ¥ 0? ] 4. Show that if ( Y, ff) is a metric space and F is an equicontinuous family, then the topology of pointwise convergence is jointly continuous on com pact sets and hence coincides with the topology of uniform convergence on compact sets. 5. Let C denote the set of real-valued continuous functions on I = [0, 1 ] . Then the map d: C x C --+ R + defined by d(f, g ) max { i f(x ) - g(x) l : x e I} is a metric on C. Show that the compact-open topology on C coin cides with the metric topology. 6. Show that the space C(X, Y) of continuous mappings from (X, 5/) to ( Y, ff) with the compact-open topology is Hausdorff if ( Y, ff) is Hausdorff. -
-
=
CHAPTER IV
Nonstandard Integration Theory
In trying to apply the theory of the Riemann integral we are faced with the following technical problem. Suppose we are given a converging infinite series L:'., 1 fn(x) = f(x) of functions on [a, b] and are asked to calculate f: f(x) dx. The answer is often simple if we can write
Thus we need to find conditions under which integration and infinite sum mation ca n be interchanged. Equivalently [letting gn(x) Li= 1 J,{x)] we need conditions under which, if g(x) limn .... oo g,(x), then =
=
!�� J: g,(x) dx
=
J: g(x) dx
for a sequence {g,(x)} of Riemann-integrable functions on [a, b ]. It turns out that we can reduce the discussion to sequences {g,(x)} which are monotone increasing, i.e., gn + 1 (x) � g,(x) for an n E N [this is the case if /,(x) � 0 for all n E N]. Thus, assuming that {g,(x) } is a monotone increasing sequence of integrable functions and gn(x) converges to g(x) on [a, b], we need conditions which insure that g(x) is integrable and the above equation holds. A result of this type is known as a monotone convergence theorem. Unfortunately, the conditions under which a monotone convergence theo rem holds for Riemann integration are quite restrictive (for example, it holds if the sequence {g,} converges uniformly on [a, b]). This fact led Lebesgue [26] and others to generalize the process of integration in such a way that the conditions for a monotone convergence theorem were considerably re laxed. The procedure was to generalize the concept of the length of an inter val so that one could measure the "length" of a very general subset of [a, b] called a measurable set. The theory. of integration then developed systemati cally from this "measure theory." 1 64
IV. 1
1 65
Standardizations of Internal Integration Structures
An alternative approach was developed by P. Daniel [ 1 1 ] . He began with the general notions of a lattice L of functions on a set X and an integral I on L. As indicated in Definition 1 .2, a lattice of functions is a linear space which is also closed under the operation of taking absolute valves, and an integral I on L is a linear functional which is also positive [ i e , f � 0 implies I ( f) � 0]. Daniel showed that if I satisfied the additional continuity con dition "If { f,} decreases to 0 then I(J,) decreases to 0," (L, I) could be enlarged to a structure (L , i) which satisfied the monotone convergence theorem. Our nonstandard approach to integration follows the Daniel approach except that we begin with an "internal" integration structure (L, I) on an internal set X in some enlargement. We show that, without any continuity assumption, we can construct from (L, I) a standard integration structure (L, i) on the same internal set X, and that structure satisfies the monotone convergence theorem. In §IV.2 we show that the usual measure-theoretic approach can be recovered from any structure ( L , i) satisfying the monotone convergence theorem. The usual Lebesgue theory on R" is developed in §IV.3 by using the standard part map to carry results on *R" down to R". Some important convergence theorems which hold in any structure for which the monotone convergence theorem is valid are developed in §IV.4. A non standard approach to the Fubini theorem, which is an analogue of the iterated integration procedure for the Riemann integral, is developed in §IV.S. Finally, in §IV.6 we apply the nonstandard integration theory developed in the previous sections to study several important stochastic processes, in cluding the Poisson process and Brownian motion. These processes are represented as processes on a *-finite probability space and indicate the usefulness of an integration theory on nonstandard sets. References to the original work on nonstandard integration theory will be given in the body of this chapter, with the exception, as noted in the Pre face, of [27, 29, 32, 33] by the second author. .
IV.1 Sta ndard izations
of
.
I nternal I ntegration Stru ctu res
The Riemann integral for continuous functions on an interval [a, b] (see §1. 1 2) has the properties ( 1. 1 ) ( 1 2) .
f [(Xf(x) + pg(x)] dx = (X f f(x) dx + P f g(x) dx, J: f(x) dx 0 if f(x) � O on [a, b] . �
1 66
IV.
Nonstandard Integration Theory
Implicit in ( 1 . 1 ) is the fact that a linear combination of continuous functions is continuous. It is also true that lfl is continuous if f is continuous. A general theory of integration should specify (A) a class L of "integrable" functions on a space X corresponding to the continuous functions on [a, b] in the above example, and (B) a real-valued function I on L whose value at f e L we denote by If (a numerical-valved function on a set of functions is usually called a functional). Here If corresponds to the Riemann integral of f. In general, the analogues of the properties above should be satisfied. We abstract these properties in the notion of an integration structure. It consists of a lattice of functions and a positive linear functional on this lattice as in Definition 1 .2 below. This definition incorporates the standard (real) and nonstandard (hyperreal) notions of integration structures since we want to consider internal analogues of integration structures when the functions are internal and hyperreal-valued. Our main objective in this section is to show how, beginning with an internal integration structure (L, I) on an internal set X, we can construct a real integration structure (i.. , I) on the same internal set X by a process called standardization. The important fact is that the real integration struc tures so obtained satisfy a closure property called the monotone convergence theorem. This theorem states roughly that a monotone increasing sequence < fn) of functions in L , whose integrals lfn are uniformly bounded, converges to a function f e i.. , and iJ is the limit of (if. ). It is the basic tool in all further developments of integration theory. We begin with a definition summarizing standard notation. 1.1
Definition
are defined by
Let X be a set and E £ X. The functions XE =
{1, 0,
XE• 1 ,
and
0
on X
x e E, X f1 E,
1 = xx . and 0 = X 0 , where 0 is the empty set. If f and g are functions on X, we write f � g if f(x) � g(x) for all x e X; we define rx.f, f + g, Jg, Jjg (if g does not vanish at any point in X), and lf l as usual by assigning the values rx.f(x), f(x) + g(x), f(x)g(x), f(x)/g(x), and l f(x) l at x e X. Definition A set L of real- or hyperreal-valued functions on a set X is a real (hyperreal) lattice if
1 .2
(a) f, g E L implies af + pg E L for all real (hyperreal) a, p, (b) f E L implies lf l E L .
IV.1
1 67
Standardizations of Internal Integration Structures
A real- or hyperreal-valued function I on L is called a real (hyperreal) positive linear functional (p.l.f.) if (c) /((J.f + pg) (1./f + Pig for all f, g e L and real (hyperreal) (J. , p, (d) If � 0 if f � 0. =
The pair (L, I) then forms a real (hyperreal) integration structure on X. The integration structure (L, i) on X is an extension of the integration struc ture (L, I) if L s;; L and iJ = If when f e L. If the sets X and L (and hence all f e L) and the functional / are internal in some enlargement V(* S) of a superstructure V(S), then we say that (L, /) is an internal integration structure. A lattice L always contains 0 (check), and is also closed under the opera tions of taking maxima and minima, defined as follows. 1 .3 Definition
If f and g are (real- or hyperreal-valued) functions defined on
X, we define the maximum and minimum of f and g by
max(!, g) = f v g = (f + g + i f - g i ) /2 , min(f, g) = f " g = (f + g - i f - g j }/2 and the positive and negative parts of f by f + f v 0, f=
=
( -f) v 0.
Clearly, if L is a lattice and f, g e L then f v g, f " g e L. Conversely, if L is a set of functions on X which is closed under linear combinations and for which f, g e L implies f v g and f 1\ g e L, then L is a lattice (Exercise 1 ). Notice that if f, g e L and f � g, then the inequality If � lg follows from 1 .2(d). This fact will be used frequently in the development. The following are examples of real integration structures of real-valued functions. 1 .4 Examples
1 . Let C[ a, b] denote the set of all continuous real-valued functions on the finite interval [a, b] c R. Define the linear functional f! on C[ a, b] by f! f = f! f(x) dx (Riemann integral). Then (C[a, b], f!) is a real integration structure on [ a, b] (exercise). Note that 1 e C[a, b] . 2. Let Cc(R) denote the set of all continuous real-valued functions f on R with compact support, where the support of f is the set supp f
=
{ x:f(x) =I= 0} .
(a) Let f denote the functional on Cc(R) defined by f f f! f(x) dx if supp f s; [a, b]. (The definition of f is independent of the choice of a and =
1 68
IV.
Nonstandard I ntegration Theory
b satisfying this condition.) Then (Cc(R), f > is a real integration structure (exercise). Note that 1 ¢ Cc(R). (b) Let { . . . , x _ 2 , x _ ., x0, x1, } be a countable set of points in R with no limit point. For each f E CJ.R) let l.J L;;. _ 00 f(x1). Then (CJ.R), L) is a real integration structure on R (exercise). •
•
•
=
3. A step function on R is a function f of the form f = Li'= 1 c1XE1, where the sets E1 are disjoint finite intervals (open, closed, or semiopen; this includes the case where the end points are equal and E1 is thus a single point). Let S(R) denote the set of step functions on R. Define the functional $ on S(R) by $ f Li'= 1 cJb1 - a,) if f = Li'= 1 c1xE. and E 1 has the end points a1 and b" a1 � b1• Then (S(R), $) is a real integration structure on R (exercise). 4. With Y = {x 1, , x.,} a finite set, let B( Y) denote the set of all real valued functions on Y. If a 1 , , a.. are fixed real numbers with a1 > 0, 1 � i � n, define the functional L on B(X) by L f = Li'= t aJ(x1). Then (B( Y), D is a real integration structure on Y (exercise). 5. With Y any nonempty set, let B0( Y} denote the set of all real-valued functions on Y, each of which is zero except for finitely many x E Y. If a is a positive real-valued function on Y, let Lo denote the functional on B0( Y) defined by Lo f = Li'= 1 a(xH(x1), where supp f {x1, , x.,}. Then (B0( Y), Lo) is a real integration structure on Y (exercise). If Y is a finite set, this example degenerates to Example 1 .4.4. =
.
.
•
•
•
•
=
•
•
•
The next proposition, easily proved using the transfer principle, shows that each standard real integration structure on a set Y (in particular, each of Examples 1 .4) gives rise to an internal integration structure on • Y by trans fer. We now fix an enlargement of a structure containing Y, with the asso ciated monomorphism • . 1.5 Proposition If (L, I) is a real integration structure on a set Y, then (* L, *I) is an internal integration structure on X = • Y.
Proof: Exercise. 0 There are internal integration structures which cannot be obtained from a real integration structure by using Proposition 1 .5, as the following ex ample shows. 1.6 Hyperflnite Integration Structures Let X be an internal • -finite set , x..,} in an enlargement V(*S) of some superstructure V(S). Let {x1, B..,(X) denote the set of all hyperreal-valued internal functions on X. With {a1 , , a.,} a fixed set of hyperreal nonnegative numbers of the same in ternal cardinality as X, let Lm denote the hyperreal functional on B.,(X) defined by Lm f = Lf£ 1 a,j(x1), where the summation is the extension of finite •
•
•
•
•
•
IV.l
Standardizations of Internal Integration Structures
1 69
summation. Then (B",(X), Lw) is a hyperreal integration structure on X (Exercise 5). Such "hyperfinite" integration structures have recently been used as the starting point in an extensive nonstandard treatment of Brownian motion and other stochastic processes. An introduction to this theory is presented in §IV.6. Now let (L, I) be an internal hyperreal integration structure on an internal set X in an enlargement V(• S) of a superstructure V(S) containing the reals. Our main objective in this section is to construct a real integration structure (L, i) on the same internal set X so that the monotone convergence theorem is valid. (i, i) will be called the standardization of (L, I). To prove the con vergence theorem and other results we need to assume that V(•S) is � 1 saturated. Thus we assume from now on without further explicit comment that any internal structure (L, I) being standardized lies in an � 1 -saturated enlargement V(• S) of a superstructure V(S). L is now defined as follows. 1 .7 Definition Let (L, I) be an internal integration structure on an internal set X. We define the set L0 of null functions to be the set of hyperreal-valued (possibly external) functions g on X such that, for each e > 0 in R, there is a tjJ E L with lg l � tjJ and o I tjJ < e. Further we define L to be the set of real valued functions f on X such that f =
(a) If f =
Proof: (a) Since � -
g - g E L0, we have 1°1( � - 0 in R, there is a tjJ E L with lgil < tjJ (i = 1, 2) and oN < e (why?). From the inequalities =
(
=
0
1 70 1 .9 Theorem
IV.
Nonstandard Integration Theory
The sets L0 and L are real lattices.
Proof: We show only that L0 is a real lattice. The proof that L is a real lattice is left as an exercise [use Lemma2 1 .8(b)]. 2 Let g 1 , g2 e L0 and (X, fJ e R with (X + {J > 0. Given �: > 0 in R there exists a function 1/1 e L so that lg1l � 1/J(i 1 , 2) and N < �:/2(max( I (X I , I fJ! )) . Then j(Xg 1 + {Jg2 j � .ji, where .ji = 21/1 max(I(XI, !fJ!) e L and J.ji < e. Property 1 .2(b) for L0 is obvious. 0 =
If f e L has two representations f = ¢ + g = (/) + g as in Lemma 1.8(a) then °/l/J = 01(/) < oo , so we may unambiguously make the following defini tion.
1 . 1 0 Definition For each f = ¢ + g e L , where
1 . 1 1 Theorem
The functional I is a real p.l.f. on L.
Proof: The linearity of f follows from that of /. If f � 0 then f ¢ + g where ¢ e L, g e L0, and we mly take ¢ � 0 by Lemma 1 .8(b). Thus If � 0. 0 =
1 . 1 2 CoroUary
The pair (L, i) is a real integration structure on
X.
1.13 Definition The structure (L, i) constructed from (L, /) is called the stan dardization of (L, 1).
The next result gives another useful characterization of functions in L. In its proof we use saturation. 1 .14 Theorem A real-valued function f on X is i n L iff for each �: > 0 in R there exist functions 1/1 1 and 1/12 i n LA with 1/1 1 !S; f � I/J 2 ,0l( jl/l . j ) < oo , and 1( 1/1 2 - 1/1 1) < �:, in which case oN 1 � If S o N 1 + �:.
Proof: First assume that f ¢ + g e L with ¢ e L, g e L0, and o I ! ¢ 1 < oo . For each n e N choose ¢. e L with jgj � ¢. and I ¢. < �:jn for some fixed �: > 0 in R. Setting 1/1 1 = ¢ - ¢ 2, and 1/1 2 ¢ +
=
IV.1
171
Standardizations of Internal Integ ration Structures
t/1 1 S / :S: t/1 2 and l(t/1 2 - t/1 1 )
< e,
then t/1 1 - t/Jft S t/J
:S:
t/12
+
t/Jft, and so
olt/1 1 - e/n ::s; 0 /t/J = it ::s; olt/12 + e/n ::s; 0lt/l l + e + e/n for each n e N. It follows that lt/1 1 ::s; it ::s; o lt/1 1 + e. To prove the converse we use saturation. Assume that f is an arbitrary real-valued function on X for which the conditions of the theorem hold. Then there exists an increasing sequence { t/1,. : n e N} (i.e., t/1 ft + 1 � t/1 ft for all n) and a decreasing sequence {t/l� : n e N} in L with t/Jft :s; f ::s; t/1�. 0lt/J,. < oo , and 1(1/1� - t/1,.) < 1 /n for each n e N. We now apply Theorem 11.8.5 with C N, D = L, and t/J: N -+ L and t/J': N -+ L defined by t/J(n) = t/1,. and t/J'(n) = t/1�. Then there are internal extensions (/J: • N -+ L and (/J': • N -+ L. By the permanence principle, Theorem 11.7. l(i), we may find a k e *N rn so that t/1,. and t/1� form increasing and decreasing sequences and t/1,. ::s; t/1� for n ::s; k. Thus, for some infinite w S k, t/1,. S t/10) ::s; t/l'o, ::s; t/1�. and so t/1,. - t/1� ::s; f t/10) S t/1� - t/1,. for all n e N. It follows that f - t/10) e L0 and f e L. D o
=
We now come to a result, called the monotone covergence theorem, which is central to the further development of the subject, both practically and theo retically. The result says, roughly speaking, that L is closed under mono tone limits if the integrals are uniformly bounded. We will later generalize the result to a larger class of functions. (l., i) Suppose that (f. e L: n e N) /,. for all n e N) sequence of functions
1 . 15 Monotone Convergence Theorem for
is a monotone increasing (i.e., J,. + 1 in L for which
�
(a) lim. -+ rn /,.(x) = f(x) exists for all x e X, (b) sup{ it,. : n e N} = lim,. -+ oo if,. < oo .
Then f e L and if = lim,.-+ rn it,..
Proof: We may assume without loss of generality that /,. � 0 (otherwise consider /,. - fd. By l .8(b) we may find representations /,. = t/J,. + g,. with t/J,. e L, g. e L0, and 0 S t/J,. S t/J,. + 1 (check). Let B lim,. -+ rn if,.. The!' given e > 0 in R we may find an m e N so that, for n � m in N, B - e < If,. ::s; B, and hence B - e < lt/J,. < B + e for any e > 0 in R. We now use saturation again. As in the proof of Theorem 1 . 1 4 we can extend the sequence (t/J,. e L : n e N) to ( t/J,. e L : n e *N) so that it is still increasing (if necessary repeat some t/J e L for all n � some k in • N cn ). Thus, for some infinite c.o, t/JO) � t/J,. for each n e N and 0lt/JO) = sup{0lt/J,. : n e N} (Exercise 8). We need only show that f - t/JO) e L0 • Fix t: > 0 in R, and for each n e N choose a t/1,. e L with lg .. l ::s; t/1,. and lt/1,. < e/2". Again by �1 -saturation we may extend the sequence ( t/J . : n e N) =
IV.
1 72
Nonstandard Integration Theory
to (t/l n : n E *N) so that, for some infinite k E *N, t/l n � 0 and It/I n < e/2n for each n � k. Let t/1 = L!= t t/l n . Then It/1 < e and rPn - "' � rPn - t/l n � rP n + 9n
( 1 . 3)
f � ( 1 + e)(r/J w + t/1 )
�
for each n E N , so that
(r/J n - rPw)
-
t/1 � f - rP w
�
e r/Jw + ( 1 + e)t/1
·
We may choose n E N so large that - 2e < l(r/J n
-
rPw) - lt/1
.
Also, l(e r/Jw + ( 1 + e)t/1 ) < el r/Jw + e + e2• Since e is arbitrary, it follows that f rPw E L0 (check). -
D
Our next theorem is a result which is useful in many applications. It gives conditions under which the standard part 0 rP of a function rjJ E L is in L and f(or/J) = o lrjJ . In general we define orP by 0r/J(x) =
{
st(rjJ(x) ), 00 , - oo ,
r/J(x) finite, r/J(x) E • R !. r/J(x) E * R�.
1 . 16 Theorem If rjJ E L takes only finite values, and for some t/1 � 0 in L with o l t/1 < oo we have {x E X : r/J(x) #- 0} £ {x E X : t/J(x) � l } , then rjJ 0 rP E Lo. 0rP E L, and f(or/J) 0lrjJ. =
Proof: For each e > 0 in R, l r/J - or/J I � et/1, and so rP - o rP E Lo. Since l r/J(x) l � nt/l (x) for all infinite n and all x E X, an easy argument using the permanence principle shows that l r/J I � nt/1 for some finite n, and so olr/J < oo . Thus 0 rP = rP + C r/J - r/J) E L and i(0rjJ) = 0 lrjJ. 0
It is important to note that if 1 e L and o 1 1 < oo then the conclusion of Theorem 1 . 1 6 holds under the sole assumption that rjJ is finite-valued, for then we may take t/1 1 . In this case we now show that if f is a real-valued function on X and f = rjJ + g, rjJ E L, g E L0 , then I l r/J I is automatically finite. =
1 . 1 7 Theorem Assume the function 1 E L and 01 1 < oo . Then 1 e L. More over, if f rjJ + g is a real-valued function on X with rjJ E L and g e L0 then I l r/J I < oo, i.e., f E L. =
o
Proof: If 1 E L and / 1 < oo then, by Theorem 1 . 1 6, 1 = o 1 E L. More over, given f rjJ + g as above, we fix t/1 E L with 1 9 1 � t/1 and It/1 < 1. Then o
=
1 73
IV. 1 Standardizations of I nternal Integ ration St ructures
-
o
1 . 1 8 Examples
1 . Let (L, /) ( * C.(R ), •J) be the internal integration structure on X = *R constructed from Example 1 .4.2 using Proposition 1 .5. (a) We first show that if g vanishes off the bounded interval [a, b] c X and takes only infinitesimal values, then g E L0 • We may assume b - a � 1 . For each e > 0 i n R , lg(x)l s
o
=
{
-
1, a ::s; x ::s; b, 0, x ::s; a 1 /n, x � b + 1/n, F. ,a.b(x) = n(x - a) + 1 , a - 1/n ::s; x s a, 1 - n(x b), b ::s; x ::s; b + 1/n, is in L and o I F a.b ::s; 0(b - a) + 2 < oo . Now let w E • N oo • and consider X[a, bJ - F w,a, b · Then for each n E N with 2/n S b - a, 19 1 S
-
=
•.
n(x a + 1 /n) , n(a + 1/n x),
-
-
-
a - 1 /n ::s; x ::s; a, a ::s; x ::s; a + 1/n,
b - 1/n ::s; x ::s; b, b ::s; x ::s; b + 1/n, otherwise.
1 74
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Nonstandard Integration Theory
By transfer !c/J .. 2/n, and we conclude that x1,.,61 e L. An easy calculation shows that Jx[a,b) = o l b - a l. =
2. Let ( L , J) = (•B( Y), •D be the internal structure on X = •y Y = { x 1 , , x.,} constructed from Example 1 .4.4 using Proposition 1 .5. We first show that L0 consists exactly of the functions g which take infinitesimal values. Suppose that g(x1) � 0 for all 1 :=:;; i :=:;; n. Then lg l :=:;; c/J, where c/J = £/2 :D ai> and 0lc/J < f. by transfer. Conversely, if g(x1) = r * 0 for some i, then when lg l :=:;; 4!_, c/J :2:: l r lx1..,11, so lc/J :2:: a11r l 'f 0. Thus g rl L0• It is now easy to show that L, consists of all real-valued functions on X, and that =
•
(i, i)
•
.
(B( Y), D· 3. Let X be any internal set and let x0 e X. Let L consist of all hyperreal =
valued functions which vanish except at x0• Put If = f(x0) for f e L. Then
(L, I) is an internal integration structure. It is easy to check that L0 consists of all functions which vanish except at x0, where they are infinitesimal, and that L consists of all real-valued functions f which vanish except at x0, where
f(x0) is finite, and i(f) = /(x0). Exercises
I f/,1
1 . Show that if L is closed under linear combinations and /, g e L � f v g e L and f " g e L, then f e L � I! I E L. 2. (Standard) Show that the structures in Examples 1 .4 are real integration structures. 3. Let X = 12 and y ( y1) E 12; assume that y1 :2:: 0 for all i e N. Show that (X, I) is an integration structure if we define I x = (x, y) for all x e X. 4. Prove Proposition 1 . 5 . 5 . Show that the structure in 1 .6 is a hyperreal inte»ration structur� . 6. (a) Show that, for functions c/J and ,P in L , l o I l c/J l - o I l c/J I I � o I l c/J - c/J I. (b) Show th!l t if c/J E L n Lo , then °l l c/JI 0 7. Prove that L is a real lattice. 8. In the proof of Theorem 1 . 1 5, show that for some infinite w e •N, c/Jw :2:: c/J., for all n e N , and 0lc/Jm = sup{0lc/J., : n e N}. 9. Show that one cannot in general replace (1 + e)( c/Jw+ 1/1) with (c/Jm+ 1/1) in the right-hand side of Eq. ( 1 .3) in the proof of Theorem 1. 1 5. 10. Let (L, I) be an internal integration structure on the internal set X and suppose that the function f is real-valued and nonnegative. Show that f e L 0 iff f E L on X and iJ = 0. 1 1 . (Comparison Theorem) Let (L, I) and (L', I') be two internal integration structures on the internal set X. Suppose that L0 £ L0 and that for each c/J E L there exists a 1/1 e L' so that lc/J � 1'1/1 and c/J - 1/1 e L0. Show that L £ L• and iJ i'! for all f e L . =
=
=
IV.2
Measure Theory for Complete Integration Structures
1 75
1 2. Use Exercise 1 1 to show that if (L, I) and (L', I') are the • -transfers (•C0(R), • f ) and (•S(R),.$) of the structures in Examples 1 .4.2(a) and 1 .4.3 respectively, then (i, i) = (i', i'). 1 3. In the standardization of Example 1 . 1 8. 1 give an example of a function g
E L0 which takes infinitely large values.
14. (Standard) A collection of subsets of a set X is a ring if A, B E f/ implies that A u B and A - B E f/. A function v : f/ -+ R + is a finitely additive measure on S if v(A u B) = v(A)+ v(B) for A, B E f/ with A n B 0. =
(a) Show that if A, B E f/ then A n B and A D.. B
=
(B - A) E f/.
(A
-
B)
u
(b) Show that if 8 is any collection of subsets of X then there is a unique ring f/ containing 8. (Hint: f/ is the intersection of all rings containing 8.) (c) Show that the set L of all linear combinations of characteristic functions of disjoint sets in f/ is a lattice. (d) Show that if ¢ = Li= 1 a;X..t, E L, we may unambiguously define I¢ L7= 1 a ;v( A ;) and that (L, I) is an integration structure. =
15. Develop the internal analogues of the notions in Exercise 14.
IV.2 Measu re Theory for Co mplete I ntegration Structu res
In the last section we showed that the monotone convergence theorem holds for the integration structure (i, i) obtained from an internal structure (L, I) by standardization. In this section we develop a measure theory for any integration structure (L, i) for which the monotone convergence theorem is valid. Such structures will be called complete. 2. 1 Definition A real integration structure (L, i) on a set X is complete if when ever (f. E L : n E N ) is a monotone increasing sequence for which
(a) lim" _ "" f.(x) = f(x) exists for all x E X, (b) sup{lfn : n E N } = l im n oo lfn < 00 , then f E L and if = lim" _ "" ifn · Throughout this section (L, i) will denote a complete integration structure. Our first objective is to introduce a set M of functions which includes the set L. The functions in M are called measurable functions. Roughly speaking, measurable functions will have the same regularity as functions in L but may not have finite integrals. We will find that products of measurable functions -
IV.
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Nonstandard Integration Theory
are measurable, a useful fact that is not in general true for functions in L. We then extend the functional l to a subset L1 of M , and obtain a real in tegration structure which is an extension of (i, i). We will also study the basic properties of those sets, called measurable, whose characteristic functions are in M. This leads to a discussion of measure theory which is often taken as the starting point for a standard development of integration theory and is important in many areas of analysis; in particular, it is basic to probability theory. We will show that the two approaches are equivalent. Most of the proofs are standard except at the end of the section where we establish connections with §IV. l . The functions in M will be extended real-valued functions; that is, they may take the values + oo and oo. Thus we make the following definition. -
The extended real number system is the set R = R u { - oo , + oo } . By convention oo < x , and x < + oo for all x e R . The rules of arithmetic for R are supplemented by the following rules: If x e R then
2.2
Definition
-
( ± oo ) + ( ± oo ) = x + ( ± oo) = ( ± oo ) + x = ± oo, ( ± oo)( ± oo) = + oo, ( ± oo)( =t oo) = - oo ,
{�
+ oo
x( ± oo) = ( ± oo)x =
+ oo
x/( ± oo ) = 0
if X > 0 if X = 0 if X < 0
for all x e R.
If a set A £ R is not bounded above we define sup A = + oo, and if A is not bounded below we define inf A = - oo, with a similar convention for lim sup and lim inf. As usual, we often denote + oo by oo .
Notice that w e have not defined ( ± oo) + ( + oo ) , ( ± oo )/( ± oo ) or ( ± oo)/( + oo). ,
2.3 Definition L +
denotes the set of nonnegative functions in L. We denote by !Vt + the set of nonnegative R -valued functions h on X such that h 11 j e L for each f e L. If h e M + we define
Jh = sup{i(h A f) :f e L } . J is an R -valued function on M + . We denote by M the set of R -valued function h on X whose positive and negative parts h + h v 0 and h - = h v 0 are both in M + . If h e M and =
-
IV.2
1 77
Measure Theory for Complete Integration Structures
either Jh + or Jh - is finite, we define lh = lh + - lh - . 2.4 Remarks 1.
Since L is a lattice we see that if
h e L then Jh
=
fh .
::::l
L, and it is easy to check that if
2. In defining if + and ih for h e if + , we may assume that f e L + , where L + is the set of nonnegative functions in L. That is, fix h � 0 and suppose that h A f e L for all f e L + . Then if f = ! + - f - E L, we have h A (f + - f - ) = (h A / + ) - f - e L. Similarly, ih = sup{ J(h " f) :f e L + } for h e if + .
3. A n easy calculation shows that ih = sup{ iJ : 0 s; f s; h , f e L } for h e if + . This formula will be used later without explicit comment. 4. Suppose that (i, f) is obtained by standardization from (L, f). For h e if + , ih may be less than the supremum of the integrals I cjJ for cjJ e L, 0 :::;; cjJ S h. For example, let X = {x, y}, and let L be the internal set of *R-valued functions on X. For cjJ e L define lc/J c/J(x) + wcjJ(y), where w e * N 00 • Then each / e L vanishes at y, 1 e if, Jt 1, but sup{0fejJ : ejJ e L, 0 S cjJ S 1 } = oo . a
=
=
I f h1 , h2 e if + and oc e R + , then h1 + h2 , ocht > h1 " h 2 , and h1 v h2 are in if + . Also i(h1 + h2) = ih 1 + lh2 , i(och1 ) = oclh 1 for oc e R, and ih 1 s; i(h2) i f h l s h2 . 2.5 Proposition
Proof: Let f e L + . Then (h l + h2) 1\ f = [{ h l A / ) + (h2 A /)) A j E L .
For oc > 0, (och1 A /) = oc(h1 " ( 1/oc)f) e L. Similarly h1 " h2 and h1 v h 2 e if + . For any f e L + , the reader should check that (h1 + h2) A j S (h1 A /) + (h2 A f). Thus J((h l + h2) A j) S J(h l l\ f) + i(h 2 A j )
S
ih l + ih2 .
Taking the supremum on the left-hand side, we obtain i(h l + h2 ) s ih l + ih2 .
On the other hand, suppose /1 , /2 e L and /1 S h 1 , /2 S h2 • Then /1 + /2 h1 + h 2 , so f/1 + iJ2 = i(ft + /2 ) s i(h t + h 2),
S
and hence ih 1 + ih2 s; i(h 1 + h2). Thus i(h 1 + h2) = ih1 + ih2 . The rest is left as an exercise. 0
IV.
1 78
Nonstandard Integration Theory
Our next result extends the monotone convergence theorem to (M, ]). In considering its meaning remember that J is an extended real-valued function and takes on the value + oo for many functions in M + . Monotone Convergence Theorem for (M +, i) If (hn E A1 + : n E N) is an increasing sequence in M + , then h = sup hn E M + and ih = sup{lhn : n E N} = limn _, oo lhn. 2.6
Proof: Let f E U . Then hn Aj E L for each n, the sequence (hn A j: n E N ) increases to h A j, and sup{ l(hn A j):n E N} � lf) < oo . By completeness, h A J E L and l(h A /) = lim l(hn A /). Thus h E A1 + and lh = sup{f(h A J):f E L} = sup{sup{f(hn A j):f E L} : n E N} sup{Jhn : n E N} . D =
It is now natural to restrict our attention to those functions in M whose integrals are finite. 2.7 Definition We define L1 to be the set of R-valued functions h E M for which lh is finite and Lt to be the set of nonnegative functions in L 1 •
The functions in L1 are extended real-valued functions. For this reason they cannot, in general, be added without encountering difficulties with ex pressions of the form oo - oo. We can, however, restrict ourselves to the real-valued functions in L1 and obtain an integration structure. Later we will show that with any function f E L 1 is associated a real-valued function j E L1 (which equals f almost everywhere; see §IV.4) such that ]f = Jj. 2.8 Proposition The set of real-valued functions in L1 together with J forms a complete integration structure on X, L1 2 L, and ]f = lj if f E L.
Proof: Exercise. 2.9
Remarks
D
1 . To show that a given function h is in L1 it suffices to show that h E M and lhl ::::; g for some g E L t (exercise). 2. If 1 E L, then every real-valued function in L1 is in L (exercise). 3. In general, L1 properly contains L. In Example 1. 1 8.3, L consists of all real-valued functions which vanish except perhaps at x0 , while L1 consists
IV.2
1 79
Measure Theory for Complete Integration Structures
of all R -valued functions f which are finite at x0 , and ]f
=
f(x0).
1 E L . - L.
To proceed we need to make a further assumption on dard development of the subject, to Marshall Stone. l. IO
In particular,
L due,
in the stan
Definition A lattice L (real or hyperreal) is Stonian if t/J e L implies An integration structure (L, /) is Stonian if L is Stonian.
t/J A 1 e L.
l. l l Remarks
1 . If l e L, then L is Stonian. 2. If L is Stonian, then 1 e M + . 3 . Each of the real lattices in Examples 1 .4 is a Stonian lattice. 4. If L is a real Stonian lattice on a standard set Y, then • L is an internal Stonian lattice on X • Y. 5. If L is Stonian, then t/J A a. e L for any IX > 0 since t/J A IX �X( ( 1 /�X) t/J A 1 ). =
=
l. ll
Proposition If (L, I) is an internal Stonian integration structure on the internal set X, then the standardization (i.. , f) is a Stonian integration structure. Proof: Let £ > 0 in R and f e i.. be given. By Theorem 1 . 1 4 there are func tions Y, 1 , Y, 2 e L so that t/1 1 S f � t/1 2 , olt/1 1 < 00 , and ol(t/1 2 - t/1 1) < e. Then t/1 1 A 1 � f A 1 S, t/1 2 A 1 and 01(1/1 2 A 1 - t/1 1 A 1) S, 01(1/1 2 - t/1 1 ) < £, SO fA 1 e L by Theorem 1 . 1 4. D
The above results show that all of the examples of integration structures encountered so far have been Stonian. In the rest of this chapter we will as sume w it ho u t further explicit comment that all integration structures are Stonian. To lead into our discussion of measurable sets we give an alternative cha racterization of measurable function in terms "good" sets which are defined as follows. Definition We let !) denote the collection of all sets A e l.A i.. + .
l. I J
1.14 Proposition If A A e !).
=
s;;
{x e X: f(x) > �X } , where f e L + and
X
for which
IX > 0,
then
1 80
IV.
Nonstandard I n tegration Theory
Proof: By consideri � g ( 1/a)f we may assume a = 1. Then f =j J A 1 E L, and if B = {x E X : f(x) > 0} then A = B. Also 1 A nf E L, 1( 1 A n]) :=:;; f(1 A /) :=:;; if for all n E N, and so X s = lim( 1 A n]) E L by completeness. D -
!Vt + consists of all nonnegative extended real-valued func tions h such that h A nxA E L for each n E N and A E fi'. Given h E M + , lh sup{i(h A nxA) : n E N, A E 2 } .
2. 1 5 Proposition =
Proof: Given / � 0 in L , let A" = {x E X : f(x) > 1/n } , n E N. Then X A . E L
by 2. 14, and the result follows from completeness and the fact that h A j limn � oo [h A nxA. A /] and h A nx A. A J =:;; h A nxA. :=:;; h. D
=
We are now ready to consider the notions of measurable set and measure. These notions were the starting point of the integration theory developed by Lebesgue. He proposed attaching a real number Jl(A), called the measure of A, to a subset A of a set X. The measure of a subset can be thought of as a generalization of the length of an interval on the real line, or the area of a rectangle in the plane. Thus it is natural to require that the measure of a d is joint union of sets is the sum of the measures of the sets, at least for finite unions. Unfortunately it is usually impossible to define Jl on all subsets of a given set X. The best we can expect is that the subsets, called measurable, on which Jl is defined are closed under countable unions and complements, and that the measure is "countably additive". The general definitions of mea surable sets and measure as presented by Lebesgue are as follows. 2. 1 6 Definition
A collection
Jl
of subsets of a set X is called a a-algebra if
(a) X E J/, (b) A E Jl implies that the complement A' of A is in J/, (c) { A1 E Jl : i E N } implies UA1 (i E N) E J/.
Each set in Jl is called measurable, and ( X, J/) is called a measurable space. A nonnegative function Jl : . II -+ R + is called a measure on Jl if Jl(0 ) 0 and =
(d) for each collection { A 1 E J/ : i E N } which is disjoint (i.e., A1 if i # j) we have
n
Ai
=
0
Jl( U A ; (i E N)) = L Jl(A ;) (i E N).
This property is called countable additivity. A measure Jl on Jl is complete if (e) whenever A E Jl with Jl(A) = 0 and B c A, then B E Jl (and thus Jl( B) = 0 since Jl( B) ::; Jl( A B) + Jl( B) Jl( A ) ). The triple (X, JI, Jl) is called a measure space. -
=
IV.2
Measure Theory for Com p lete Integration Structu res
181
2. 1 7 Remarks
l . 0 = X '. 2. If {Ai : i E N } (UA; (i E N) )' E vii .
c
vii then, by De Morgan's law, nAi (i E N ) �
3. Finite unions and intersections of sets in vii are again in vii . 4. If A, B E vii then A - B = A n B' and the symmetric difference A 6 B = (A - B) u (B - A) are in vii . 5. If J.l is a measure on (X, vii ), then for any collection {A n E vii : n e N} we have JJ.( U ! A n) $ Lt JJ.(A"). If A 1 s; A2 then JJ.(A 1 ) ::; JJ.(A2) . (Exercise). 6. The term "complete" for measures is not related to completeness for integration structures. Now we will show how to use a complete integration structure (i, f) on X to introduce a measure theory on X. 2.18 Definition A set A s; X is measurable with respect to (L, i) if X..c E M + . The collection of these measurable sets is denoted by .it. For each A e .it define jl(A) JxA · =
Note that !f 2 . 1 9 Theorem
s;
{ A e .it : JJ.(A)
<
oo } .
.it i s a a-algebra o n X and J1 i s a measure on .it.
Proof: (a) By Remark 2. 1 1 .2, l = Xx E M + . (b) If A E .it then XA E M + and so XA· 1 - X..c E M + . (c) Suppose A j E vii (i E N ) and put A = u � 1 A i and Bn � Ui= 1 A i . Then X ( s : n E N ) is an increasing sequence of functions in M + . Since XA = lim:_ oo Xs XA E M + by the monotone convergence theorem, 2.6, and hence A E vii . (d) In the notation of (c) we have =
• •
by monotone convergence
n = lim L JxA, n - oo i = 1
=
00
L
i= 1
jl( A j)•
D
since the { Ai} are disjoint
IV.
1 82
Nonstandard I ntegration Theo ry
2.20 Examples
1 . Let (i, l) be the standardization of (L, f) = (*Cc(R), • f) on X (see Example 1 . 1 8. 1 ).
=
*R
(a) !i' contains all intervals of finite length, including intervals of infinitesimal length and (the degenerate case) single points [see Example 1 . 1 8. l (c)]. (b) ..H contains each interval on * R (exercise). (c) The set G of finite numbers in • R is in ..H (exercise). (d) The set of numbers infinitesimally close to any a E R is in !i' (exercise). 2. In Example 1 . 1 8.3, !i' consists of {x0}, and ..H consists of all sets.
In the standard developments of integration, one begins with a measure on a a-algebra vii . Using .II , one then defines the notions of measurable function and associated integral. We now present this development. Our eventual aim is to show that if we begin with the ..H and jl obtained from (L, i) then the measurable functions and integrals obtained from the standard development coincide with those obtained from (L, i). In the next few results J.l will be a measure on an arbitrary a-algebra vii . 2.21 Definition An extended real-valued function h on X is measurable with respect to .II if Aa { x E X : f(x ) > IX } E vii for each IX E R. The set of func tions f which are measurable with respect to vii is denoted by M. =
We will see presently that M M in our situation, but a few results must first be established. We want to show that each h E M is the limit of a sequence of functions in M, each of which takes only finitely many values. =
2.22 Definition A function v E M is simple if it takes only finitely many dis tinct real values a1 , . . . , an , and the sets A1 {x E X : v(x) a J E .II (i = 1 , . . . , n). The representation v(x) = Li= 1 a1X..t, is called the reduced repre sentation of v. =
=
2.23 Proposition Each nonnegative function h E M is the limit of a mono tonically increasing sequence ( vn E M : n E N) of nonnegative simple func tions.
Proof:
Define
vn(x)
=
{(k - 1 )/2n, n,
if (k - 1 )/2n � h(x) < k/2", 1 � k if h(x) � n,
:s; n
2" ,
IV.2
Measure Theory for Complete Integration Structures
1 83
(drawing a picture helps here). Then 0 .:::; h(x) - v.(x) .:::; 1/2" if h(x) .:::; n, and v. = n if h(x) > n. Also v. increases monotonically to h. D In the standard development of integration that we are following, the integral of a nonnegative function h E M is defined as follows. 2.24 Definition Let the measure J.l on Jl be given. If v Li� 1 a;XA, is a simple function with each a; � 0, we define the integral of v by J v dJ.l D� 1 a;JJ.( A ;). One can show that the integral is well defined (Exercise 7). If h E M is nonnegative we define the integral of h by =
=
J h dJJ.
=
sup
{ J v dJJ. : v simple, 0 .:::; v .:::; h} .
If h E M and h h + - h - we define J h dJJ. integrals is finite. =
=
J h + dJJ. - J h - dJJ. if one of the
We now show that our development of integration coincides with this standard development. 2.25 Theorem Let (L, i) be a complete integration structure with measurable functions if, and let M be the functions measurable with respect to the a algebra ..H obtained from ( L, i). Then an R-valued function h is in if + iff it is in M + , and Jh J h dji, where Ji is the measure obtained from (L, i). =
Proof: Assume that h E if + . For (X > 0 let A {x E X : h(x) > (X}; fix m > a N and C E .!£. For any n E N , XA " n xc X A ,., c . and =
in
=
A n C
by 2. 14 and 2. 1 5, so
AE
=
{x e X : h A mXc > (X} E !£
..H. Moreover,
{x E X : h(x) > 0}
=
U {x E X : h(x) > 1/n} E ..H,
neN
and so h E M + . Now assume that h E M + , and fix C E !£ and n E N. Then h " nxc is the limit of an increasing sequence of simple functions from L by 2.23. Thus h " nxc E L by completeness, so h E if + . To show that ]h = J h d[J., note that J v d[J. = Jv for nonnegative simple functions and that
J h d[J.
sup{Jv : v simple, 0 .:::; v .:::; h} .:::; sup{ JJ: f E if, 0 .:::; f .:::; h} ih. =
=
1 84
IV.
Nonstandard Integration Theo ry
But if f e L and 0 � f � h, then there exists an increasing sequence (v. : n e N ) of simple functions with 0 � v. � f and lim. .... 00 v. = f, so that
it = lim iv. � . .... 00
J hd{l
.
Hence
2.26 Corollary
if = M and lh = J h d{l for all h e if for which J is defined.
Proof: To show that M £ if let h = h + - h - e M. By Theorem 2.25, h + e M + = if + and h - e M + = if + , and so h e if. To show that if £ M we proceed in the same way, using the fact that if
f, g e if+ and f · g = 0, then f - g e M. To prove this we have {x e X :f(x) - g(x) > IX} =
{{x E X :f(x) > IX}
{ X E X : g(X) < - IX }
if IX � 0 if IX < 0.
Now {x e X :f(x) > IX} e �. Also
{x e X : g(x) < - IX} = {x e X : g(x) � - IX }' = ( n {x e X : g(x) > - IX - 1 /n }( n e N) )' is in � by Theorem 2 . 1 9 . D Let (L, i) be a complete integration structure with associated sets and measure if, �. and ,1. With Corollary 2.26 in mind we will de note the value of J at h e if by the standard notation J h djl. 2.27 Notation
We can now show that the set of measurable functions is closed under many limiting and algebraic operations.
2.28 Proposition If (h. e if : n e N ) is a sequence of functions in if, then the functions h, H, h, fi defined by
are in if.
h(x) = inf{h.(x) : n e N}, h(x) = lim inf h.(x),
H(x) sup { h.(x) : n e N}, H(x) = lim sup h.(x) =
Pro_of: Since {x E X : H(�) > IX} = u := l {x E X : h .(x) > IX}_ we see tpat H e M by 2.26. Then h e M since inf{h . } - sup{ - h. } e M. Finally h sup { inf{ h ., : m � n } }e if and similar fl e if. D =
=
IV.2
1 85
Measure Theory for Complete Integration Structu res
,
Proposition I f f g e M and H is a continuous function on the plane R 2 , then the function h defined by h(x ) H(f(x ) , g(x ) ) is in M. In particular, f + g and fg e M.
2.29
=
Proof: Since H is continuous, the sets U��. { ( u, v) : H(u, v) > ex} are open, and so each can be written as a union of open boxes: =
U ��.
00
=
U { (u , v) : (u , v ) e (a., , b.)
11 = 1
Therefore {x : h(x) > ex } =
..9 ( t
x
( c., , d.,)} .
{x : f(x) e (a., , b.)} n { x : g(x ) e ( c., , d.,)}
is measurable (why?), and so h is measurable.
0
)
The preceding two propositions can be used to show that most functions commonly encountered in analysis are measurable. 2.30 Notation If f e M and J f dP is defined, then fx_. e M for any A e Jt by Proposition 2.29. We put J,. f djl = J fXA dp.
It follows from Proposition 2.29 that if f, g e M and function h defined by h(x)
=
{
f(x), g(x) ,
Ee
Jt then the
x e E, x e E',
is in M (exercise). This fact will be used later without explicit reference. We end this section with several results which hold when the complete integration structure (i.. , i) is the standardization of an internal structure (L, I). We begin by showing that 0 fjJ E M for any f/J E L . 2.31
Proposition If f/J E L then °f/J E M.
Proof: We need only show that o f/J " Xc e L if f/J e L + , and C e !i. The rest follows by considering f/J + and f/J - and rescaling. Given 1: > 0, choose 1/1 1 and 1/1 2 in L with 0 � 1/1 1 � Xc � 1/1 2 � 1 , ol!/1 2 < oo , and 1( 1/1 2 - 1/1 1) < &. Then - 1:1/12 + (f/J 1\ 1/1 , ) � 0f/J 1\ Xc � (f/J 1\ 1/12 ) + 1: 1/12
and
l ( (f/J 1\ 1/1 2 ) - (4> 1\ !/I t ) + 21:1/12)
�
1 (1/1 2 - !/I t ) + 21:11/1 2
Since 1: is arbitrary, the result follows from 1 . 1 4.
0
:s;
6
+
2£1 1/12 .
1 86
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Nonstandard Integration Theory
The following result shows that if h E M + , we may often be able to find a function ¢ E L which is "close" to h in an appropriate sense. Assume that I e L For each h E M + there is ¢ E L so that l
2.32 Proposition
Jh
=
sup{ J(h " n) : n E N }
=
sup{ 0 J(¢ " n) : n E N}
=
o
a
function
/(¢ " w)
for some w E * N oo • Proof: By Theorem 1 . 1 4 we may choose sequences (¢n: n E N ) and (t/ln:n E N) in L such that ¢n � h 1\ n :S t/ln, ¢n � tPn I • and /(t/Jn - ¢n) < 1 /n + for each n e N. Given k � m � n in N, we obtain t/1., " n � h " n � l/Jt " n � t/J., A n and /( (t/J., A n) - (¢., A n)) � l(t/1, - ¢.,) < 1 /m. By N 1 -saturation we may find ¢ E L such that t/1 ., " n � ¢ " n � ¢., " n for every m, n E N with m � n. Clearly l
The function ¢ is sometimes called a "lifting" of h. Given a *R-valued function ¢ E L, where (L, I) is an internal integration structure, it is important to know whether o¢ e L or o ¢ E L1 and whether ]( 0 ¢) 0/¢ . Note that if ¢(x) =:!:: 0 for all x, and /¢ t 0, then o¢ 0 and ](0¢) -::1- 0 /¢. If tjJ � 0 then we always have JC¢) :S o/¢ (Exercise 1 7). =
=
Let ¢ be a finite-valued element of L. Then °t/J E L iff sup { o /( l ¢ 1 - ( 1/n " 1 ¢ 1 ) ): n E N } < oo . 2.33 Proposition
Proof: We may
assume
that ¢ � 0. For each n E N,
{ x E X : ¢ - (1/n A ¢) > 0 }
=
{x E X : ¢ > 1/n } {x E X : ¢ - ( 1 /2n " t/J) > l/2n} £ {x E X : 2n[4> - ( lj2n " ¢)] � 1 }. =
Moreover, o ,p limn -+ oo 0 [¢ - ( 1 /n A ¢)]. If sup {01[¢ - ( l /n A ¢)]:n e N} < oo then, for each n E N, 0[ ¢ - ( l jn " ¢)] e L by Theorem 1 . 1 6, and so o¢ e L by Theorem 1 . 1 5. The converse follows from the fact that if ¢ � 0 is in L and 0 :S o¢ � t/1 E L with o lt/1 < oo , then, for each n e N, ¢ - ( 1 /n A ¢) � t/1 . =
D
In the following treatment "S-integrability", we have replaced Anderson's original definition [2] by a condition which is a direct consequence of the definition of the general integral, and is often easier to apply. 2.34 Definition
A function ¢ e L is
S-integrable
if o ¢ e L 1 and JC ¢)
=
o
I¢.
IV.2
Measure Theory for Complete Integration Structures
187
2.35 Proposition
l(J4JI " 1 /w)
�
0
A function 4J e L is S-integrable iff l(J4JI - ( J 4JI " w) ) � 0 and for each w e *N co ·
Proof: We may assume that 4J � 0. For each n e • N, 4J (4J - (4J " n) )+ ((4J " n) - (4J " 1/n) )+ (4J " 1/n). Assume that J(4J - (4J " w) ) � 0 and 1(4J " 1 /w) � 0 for each w e *N "-' ' Then by the permanence principle /(4J (4J " n) ) and /(4J " 1/n) are finite for some m e N and all n � m in N. Fix n � m i n N. Then 4J " n 5 n 2 (4J " 1 /n), so /((4J " n) - (4J " 1 /n) ) is also finite, whence 14J is finite. Moreover, (4J " n) - (4J " I /n) is finite-valued and { x e X: ( 4J A n) - (4J A 1 /n ) > O } { x E X : 4J > 1 /n } £ / x e X : n(4J A l /n) � l } , and so (04J 1\ n) - C4J 1\ l /n) E L and 0/((4J 1\ n) - (4J 1\ 1 /n)) ]( (4J 1\ n) - e4J 1\ 1 /n) ) b y Theorem 1 . 1 6. Now b y our assumption and Theorem 2 . 6, =
=
=
0/4J
=
lim 0/((4J 1\ n) - (4J 1\ 1/n) )
If, on the other hand, I4l is finite, then the second and third in this string of equalities hold as before. If we also have 0l4J ](04J), then I4J � /( (4J A w) (4J " 1 /w) ), and so 1(4J - (4J " w) ) � 0 and /(4J " 1 /w) � 0 for each w e *N00 • D =
Note that t he condition l( J4JI " 1 /w) � 0 is automatically satisfied for any 4J e L and w E * N oo if I E L and o /(I ) < oo . Exercises I V.2
I. (Standard) Finish the proof of Proposition
2.5.
(Standard) Prove Proposition 2.8. 3. (Standard) Show that if h E M and Jhj 5 g for some g e Lt then h e L , . 4. (Standard) Show that if I e L, then every real-valued function in i t is i n L. 5. (Standard) Show that if(X, ..1(, Jl) is a measure space and A. e ..1(, n e N, then Jl( U f A.) 5 L� Jl(A.), and if A 1 s;; A 2 then Jl(A 1) 5 Jl( A 2 ) 6. Verify the statements in (b)-(d) of Example 2 . 20. 1 . 7. (Standard ) Show that, for a simple function v, J v dJl is well defined in Definition 2 . 2 4 . That is, if v L ail..t, L bilBj • ai � 0 , bi � 0, show that L a iJl( A i ) = L biJl(Bi). 8. Show that the measure ,;. obtained from the standardization (i, i) of an internal integration structure (L, /) is complete (Definition 2 . 1 6(e) ). [ Hint: Use Exercise IV 1 . 1 0] 2.
=
=
1 88
IV .
Nonstandard Integration Theory
9. (Standard) Show that if f, g e M and E e .A, then the function h defined by
h(x) =
{
f(x), g(x),
x e E, x e E' ,
is in M. (Standard) Given a sequence (/n ) of measurable functions, show that the set E of points where lim" ... "' fn(x) exists is measurable. [Hint: Con sider lim sup f" and lim inf /"]. 1 1 . Prove that if (L, J) is an internal integration structure with standardiza tion (L, i), then for each e > 0 and A e !i' there is a ¢ e L with 0 � ¢ � X ..t and J2(A) - a/(¢) < e . In particular, if Jl(A) > 0 there is a ¢ e L with 0 � ¢ � X ..t and /(¢) > Jl(A)/2. 1 2. (Standard) Show that vii consists of those sets C such that C n A e Y for each A e !i'. 1 3. Let S be an internal hyperfinite subset of an internal set X. If .91 is the set of internal subsets of X, define the function v: J
(a) Show that v is finitely additive, i.e., v(A u B) v(A) + v(B) for A, B e .91 and A n B = 0. (b) Show how you may use the theory of §IV. l to define a measure J.l on a a-algebra vii of subsets of X (see 1 .6 in particular) so that .J( => .91 and J.l(A) av(A) for A e d. Note that 0 � J.l( A) � 1 for all A E .Jt. =
=
14. (Nonmeasurable sets) Consider Exercise l3 where X = { n e * N : O � n < w, w e * N 00 } . Define an operation EB on X by n EB m = n + m if n + m < w, and n EB m n + m - w if n + m � w. Call n and m in X equiv =
alent if there is a standard k e N with either n EB k = m or m EB k = n (this is an equivalence relation). Using the axiom of choice, choose one point from each equivalence class to form a set B. Show that B ¢ vii . (Hint: Show that X U [(B EB n) u (B EB (w - n)](n e N)). 1 5. Let (L, /) be an internal lattice. Give an example of a function ¢ e L for which I¢ is finite but ¢ is not S-integrable. 16. Let (L, I) be an internal lattice. =
(a) Show that if f, g e L , g is S-integrable, and 1/l � lgl, then f is S integrable. (b) Show that if f e L is S-integrable and g e l.. satisfies l gl � n for some n e N, then fg is S-integrable. • R are finite, then (c) Show that if f, g are S-integrable and a, af + bg is S-integrable. =
IV. J
1 89
lntegr at1on on R"; the Riesz Representation Theorem
1 7. Modify the proof of Proposition 2.33 t o show t hat for t/> � 0 i n L, }("tf>) � "It/>. (Hint: we may assume oft/> < oo ). 1 8. Use Theorem 1 . 1 4 and 1 . 1 7 to show that if (L, /) is an internal Stonian integration structure, t hen the fu nction 1 is i n L iff 1 e L and 0 /( 1 ) < oo .
1 9.
State and prove Proposi tion 2.35 wi t h the additional simplifying as sum ption that the function I e L and /( l ) < oo . 20. Let (L, I) be t he hyperfinite integration st ructure o f Example 1 .6, and let (i., i) be the standardization of (L, /), with associated .!i', .it, [1, etc. Assume that L ai (i e I) is fi nite. o
(a) Show that A e !2' iff for every E > 0 in B and C of X such that B s; A s; C and
R t here exist internal subsets B) < e.
L a i (i e C
-
( b ) Show that A e if iff t here is an internal set B such that ,U( ( A B) u ( B - A ) ) = 0. (Hint: use � 1 -saturation and the permanence principle.) -
" I V . 3 Integra tion on Rn; the Riesz Representation Theorem
Let X be any open or closed su bset of W and suppose t hat I 0 is a positive linea r functional (p.I.f. ) on the latt ice C,( X ) of co ntinuous functions with com pact support on X (of course CJ X ) = C( X ) if X i s compact). For example, 10(.{) could denote the Reimann integral of f e C,(X) or, more generally, t he Rieman n - Stiel tjes integral of f with respect to a n increasi ng integrator. In part icu lar, I 0(f) could be eval uation of f at some point x 0 e X. We want to use the theory developed in the previous sections to define a measu re space ( X, . ifx , flx l and a co rrespond ing complete integration st ruct ure (Lx, lx) on X w hich i s an ext ensiOn of the st ruct ure (C,(X), / 0 ). Most of these results are easy to prove a nd are left as exercises. The measure fl x will be shown to sattsfy an addi tional condition known as regularity. This and other asso ciated results are more technical, and can be skipped if desired . All of the a bove results taken together yield the Riesz representation t heorem . With minor modifications except in one place, the results and proofs of t his sec tion carry over to the case that X is any locally com pact H ausdorff space . One essen tial difficui i Y arises in t he proof of Lemma 3.8, which, for the gen eral case, req uires U sysohn's lem ma [20]. Also, if X is not compact a "count ability" condition is needed for the general case to show "outer regularity." Without furt her ex .> Jicit comment, the nonstandard analysis in this section will be carried out i n a K-saturated enlargemen t V(* R) of V(R). We assume that " � t-: 1 • For a f ·leral space X we would need x > card .'Y, where :!/ is the collection of or sets i n X . ·
·
IV.
1 90
Nonstandard l ntegra t1on Theory
Let ( L, I ) be the internal integration structure ( * Cr( X ), */ 0) on • X, with (M, ]), (L 1 , ]), . it, !2', jJ. denoting the objects constructed from (L, I) by the procedures of §§IV. l and I V .2. Recall that if G denotes the near-standard elements in • X then the standard part map st: G -+ X ma ps G onto X . The basic idea of this section is to use the standard part map to lift functions from X to • X as follows.
3. 1
Definition For each .R-val ued function f on
J on • x by
i(.x) and for each A �
3.2
=
{��
X we define A
X E G, X f. G,
st(x) ),
= st - 1 ( A )
X we define t he function
n
• X.
Remarks
l . J is constant on the monads of standard points in • X , and zero at all points which are remote (i.e., not near-standard). I n particular, ](x) = 0 if x E • X and the norm of x is infinite. ..-----.,... � .... ,-_; 2. af af, .f + g f + g, f v g f v g, f " g f " g (exercise). X A (exe rcise). 3. X A ,...._,
=
�
=
=
=
=
We now obtain measure-t heoretic struct ures on X with the following definition.
U:/ E M } and define J x by putting J x U ) = when ](/) is defined. For each set A � X with A E Jt, i.e., XA E M x · we set .Ux( A ) jJ.( ;i); the set . ltx = { A � X : A E . H} . We let Lx denote the real-r·a/ued funct ions I i n M x for which J xi is defi ned and finite. 3.3
J( j)
3.4
Definition We let M x = =
Proposition ( L x ,
Jx)
i s a complete in tegra tion struct ure which extends
(CA X ), / 0 ). Moreover, ( X , Jtx .ux ) is a measure space such that I E M x iff I is . fix-measurable, and J I dltx = J xf when J xf is defi ned . •
Proof: That ( L x , J x ) is an integration structure is left as an exercise. To show that ( L x . J x ) ex tends ( Cr( X), 1 0), let / E C(X). By the uniform con tinuity of I. *f(y) ::o: *f( x ) if y ::o: x and *f is zero at any remote point since f has com pact support. Thus J = 0 ( */ ) . By the obvious extension of Exam ple l . l 8. l ( b), f E L and 1/ fl J *j" = 1 0 I . =
=
·
IV.3
191
Integration on R". the Rie;z Rep resent ation Theorem
To show that ( L x , J x l i s com pl ete , let ( f�) be a monotone incre a s i ng sequence of functions in Lx for w hich lim. - oc f.(x) f(x) exists for all x E X and sup i J \,1;, : 11 E N } < x . Th e n ( i. > is a mon otone incr ea s i n g sequence of functions in L �_ a n� sup[J/. : n E NL< rx . A l so li m . - oc].( z ) = f(z) for all z E "' X (chec k ). so f E L 1 and Jf l i m Jf� by the monotone convergence theorem for ( L 1 . ]). T h e re fo re f E Lx and J xf lim._ J x.f•. Th e rest is l e ft to the reader {Exercise 2); the e qu a li t y J f dfl x J xi follows from t h e corresponding fact for si m p l e functions. 0 =
=
=
x
=
When we s t a rt with I 0 be i n g t h e p .l. f. g i ve n b y o rdin ary R iem a n n in t e g r a tio n , then . fix i s called the class of Lebesgue-measurable sets a n d flx is called Lebesgue measure. I n that case we write J f dflx a s J f dx.
3.5 Examples In th e following exa m p l es we consider the case i n which X and I 0 is given by Riemann int egr at ion .
=
R
1 . The characteristic fu n c t i o n of any bounded interval in X i s in Lx (i.e., these intervals are i n . llxl· This follows from E x a mple 1 . 1 8. l (c). The corres po n d i n g result for bo u n ded recta n g l es holds if X R". 2. Ne xt we show that Lx contains the function =
0
< X ,:5;
I,
otherwise,
functions. If A ( 0 I ] t h e n X A and h e nce n x A are in Lx by E xample I . Thus f. nx A " l jj"; E L x by the lattice property. Now the seq u e n ce ( f. > is monotone i n crea s in g and converges to f. An easy calculation shows t h at J x fn $; 2, so the result follows from completeness. 3. If E E . llx is bou nded then flx( E) < oo ( E xercise 3). This ag a i n ge n e r a l i ze s to X R". =
and hence contains u n bounded
,
=
=
The following res ults give more detailed information about .llx a n d fl x and cen t e r about the notions of regularity, which is defined as fo llows . *3.6 N otation Le t . � ·
and
.-1
be the co llec t i on s of su bsets of
compact and open in X, res pec t i v e l y . Recall that, for X
X
R " , V s::::
that are
X is o pe n in X if V X n W for some o pe n W s:::: R". A se t K is compac t in X iff it is compact in R". W e write K -< I if K E .x--· , f E C,(X), 0 ,:5; f $; I, and f(x) = I for a ll x E K. We write f -< V if V E .'T, f E C,{ X), 0 $; f ,:5; I , a n d supp f £ V . T he not ation K -< f -< V me a n s that K -< f a n d f -< V. =
s::::
IV.
1 92
* 3.7 Definition A measure J.l on a a-algebra ..II metric space X is inner regular if
Nonstandard I ntegration Theo ry 2
f
u
ff of subsets of a
(a) JJ.(A) = s up {JJ.(K) : K £ A, K e f}, A e ..II , outer regular if (b) JJ.( A) = inf { JJ.( V ) : A £ V, V E ff } , A E ..II , and regular if it is both inner and outer regular. We first show that ..llx 2 f u ff. To do so we need the following fact about continuous functions. *3.8 Lemma Suppose K e f , V e ff, and K tion f E C,(X) so that K -< f -< V.
c
V. Then there exists a func
Let U be an open set with compact closure 0 such that K £ U £ Y. For any set A c X and x E X, let p(x, A) be the distance from x to A, i.e., p(x, A) inf{ l y xi : y e A}, where H is the norm in X. Then p(x, A) is continuous as a function of x and p(x, A) 0 if x E A. Now define f by f(x) = p(x, U')/[p(x, U') + p(x, K)]. 0 0
Proof: £
=
-
=
*3.9 Proposition If V E ff, then V E ..llx and JJ.x( V) = sup { l 0 f : f -< V }. Proof: Let A e 2 and e > 0 i n R be given. We may choose ljJ 1 , ljJ 2 E L with 0 � 1/11 � XA � 1/1 2 � 1 and /( 1/1 2 - ljJ d < e/3 by Theorem 1 . 1 4 (the inequality 1/12 � 1 uses the fact that L is Stonian). Let f0 = {K E f : K c V}. For each K E f0 let
rxK = inW 1(1/1 1 A *f ) : K -< f } , PK = inWI( IjJ 2 A *f ) : K -< ! } ,
For each K E f0 , PK - rx K � ej3, so f3 - rx s e/3 . By d efini t ion of rx , we m ay choose a standard f e C,(X) with f -< V such that 1(1/1 1 A *f) > rx - e/3 . By K saturation we may choose a K' e • f0 and a 0 there exists an A e 2 so that l(xv) s ft(A 11 V ) + e since lx v = sup{J(x A A Xv) : A E 2} . With the 1/1 1 and f obtained for this e and A as in the first paragraph, we have 01(1/1 1 A *f) � ft(A 11 V) � o/(1/1 1 A *f ) + e, and so
IV.3
1 93
Integration on R"; the Riesz RPpresentation Theorem
0 /( 1/1 1 " *f) � lx v Jl x( V) � 0 1( 1/1 1 " *f) + 2e. Also, by Theorem 1 . 1 6, 0( */ ) j E L, and 0 1(1/1 1 " *f) � ol*f = I o f = Jj � lx v = Jlx( V ) since j � Xv· We conclude that Jlx( V ) = sup{ /0 /:f-< V } . D =
=
*3. 1 0
Proposition
inf{ / 0./ : K -< fl .
If K E .� then K
E
.!1, so K E J/x , and Jlx( K ) =
Proof: Let rx = inf{/0f : K -< ! } . There is a ¢ E L with O � ¢ � 1 , 1/J I * K = 1 , _ and ¢ I * X - K 0 such that /¢ = rx (check). Given f E Cc(X) with K -< f and e > 0 in R, we have =
o
( 1 + e) */ � Xi.
whence Xi. - ¢ E L 0 , Xi. E L, and Ji(K)
* 3.1 1
Corollary If
Proof: Exercise.
*3.12
K E % then Jl x{ K)
=
=
�
¢,
]Xi.
=
0/¢
=
D
rx.
inf { Jl x( V ) : V E ff, V ;;;2 K } .
D
Theorem The measure Jlx on
,,{{ x
is regular.
Proof: (a) We first show that Jlx is inner regular. Let A E J/x . For any e > 0 in R and n E N, choose h E L + so that if lx1 < oo we have ](h " X A) > lx.i - e and if )x.4 = oo we have ](h " X..t) > n. Now choose 1/1 E L so that 0 � 1/1 � h " X..t and 011/1 � ](h " X.-4) - e. Let K st {y E * X : l/f(y) > 0}. Then K is the standard part of an internal set which is near-standard (i.e., contained in G) since 0 � 1/1 � X.4· and K £ A. Thus K is compact by Exercise III.3. 1 1 . Finally, i f lx.4 < oo w e have =
lx..t � lxi.
�
o
11/1 � lx..t
- 2e,
and e > 0 is arbitrary, so (a) is established in this case. A similar argument works if lx.4 = oo . (b) N o w w e show that Jlx i s outer regular. Let A E J/x . The result is trivial if Jlx(A) oo , so suppose that Jlx( A) < oo. First assume that W is open in X and that A £ W £ W £ X and W is compact. Given e > 0 in R we may use (a) to find a compact K £ W - A so that Jlx[( W - A ) - K] < e. Then the open set V = W - K � A and Jlx( V ) - Jlx(A ) < e. In general there exits an increasing sequence < W,.) of sets open in X with X U W,.(n E N), and W, compact and contained in X for each n (exercise). Let A t A n W,. E Jlx , and put B 1 A 1 , Bk = At - A t _ 1 , k � 2, so that the Bt are dis joint and u :� I Bk = A . For each k we may find an open set v,. ;;;2 Bk with =
=
=
=
19
4
IV.
Nonstandard Integration Theo ry
J.tx( J-1) < J.tx(Bt) + e/21 . Then V Uk'= 1 "k is open and J.tx( V) � boo= 1 J.tx( J-1) � boo= 1 J.tx(B1) + e J.tx(A) + e. D =
=
The following result summarizes this section. In its proof we use the nota tion J f dp for integration based on a measure Jl on Jlx .
*3. 1 3 Riesz Representation Theorem Let T be a p.l.f. on C.,(X). Then there exists a a-algebra Jlx on X which contains all open and compact subsets of X and a unique complete regular measure Jlx on Jlx so that T(f) J f dJ.tx for all f E C.,(X). =
Proof: From the previous results, all that remains is to show the uniqueness and completeness of Jlx · To show uniqueness, let Jl be any other regular mea sure on Jlx so that Tf = J f dp for all f e C.,(X). It suffices to show that p( K ) = J.tx( K ) for all K e :K by regularity. Let K e :K and e > 0 in R be fixed. By regularity there is a V 2 K with J.t( V) < p(K) + e. Let f satisfy K -< f -< V. Then
J
J.tx( K ) � f dJ.tx = Tf
=
If dp � I
Xv
dp
=
p( V) < J.t(K) + e .
This is true for any e > 0, so that J.tx( K ) � J.t(K). Similarly J.t(K) � J.tx(K), and the uniqueness follows. The completeness of J.tx follows easily from the com pleteness of {t (see Exercise IV.2.8) and is left as an exercise. D Exercises I Y.J l . Prove the validity of Remarks 3.2.2 and 3.2.3. 2. Show that (Lx , Jx) as defined in Definition 3.3 is an integration structure, and finish the proof of Proposition 3.4. 3. Show that if E e Jlx is bounded, then px(E) < oo . 4. Show that i f X i s an open o r closed subset o f R n and K c X is compact, then there is an open set V in X (i.e., V = X n W for some open W c Rn) such that K £ V and the closure of V is both compact and contained in X. 5. Finish the proof of Proposition 3.9 by showing that if J.tx( V) oo, then J.tx( V) = sup { / 0 f : f -< V } . 6. Assume that X is compact in R n , and deduce Proposition 3.9 from Pro posi tion 3. 1 0. 7. Prove Corollary 3. 1 1 . 8. Show that if X is open or closed in Rn, then there is an increasing sequence < W.. > of sets open in X with X = U W..
IV.4
Basic Convergence Theorems
195
1 0. Show that in the case of Lebesgue integration the function f on
by
f( x)
=
{�
/x ,
R defined
E (0, 1 ), otherwise X
is Lebesgue-measurable but not Lebesgue-integrable.
1 1 . Replace • I 0 with any internal positive linear functional I on *Cc{X) such that I(*f) < oo for each f E Cc{X). Prove that if we define (Lx, J x) on X as in Definition 3.3, then (Lx , J x) is a complete integration structure with J,..J 'If for each f E C,(X). 1 2. Let L\x 1 /n! with n E *N oo be a fixed infinitesimal and let T = {u e *R : u = n i\x, n E *Z} . Let L = *C,(R) and for f E L put I(f) = L f( x) L\ x (x E T) (note that for any f E L the sum is equal to the • -finite transfer =
=
of finite summation).
(a) Show that (L, I) is an internal integration structure. (b) Show that if I' is the •-transfer of Riemann integration in C,(R), then there are internal functions f for which If * I1. (c) If (i, i) is the standardization of (L, I), modify t he procedure in Exercise IV.2. l 4 to prod uce a subset E of * [0, 1 ] which is not in A.
1 3. Let (Lx , Jx). X = R, denote the integration structure on X obtained from the (L, I) of Exercise 1 2 by the procedure of Exercise 1 1 .
(a) Show that the associated .ltx contains all compact and open sets. (b) Show that the associated /J x is regular. (c) Prove, hence, that (Lx. I x) coincides with the Lebesgue integration structure. (Hint: Use Theorem 3. 1 3, especially uniqueness.) 1 4. (Standard) Let D be the unit disk {z : lzl < 1 } in the complex plane, and let C be i ts boundary { z : lzl 1 } . It is well known that for each con tinuous function f on C, there is a unique continuous function h 1 on D D u C such that h1 I C f and h1 I D is harmonic, that is, (o2 h1/ox2) + (iJ2h 1/oi) 0. Moreover, h � 0 i f f � 0. Use Theorem 3. 1 3 to show that for each x e D there is a measure J.lx on C such that h1(x) fc f d!Jx for all continuous functions f on C. =
=
=
=
=
IV.4 Basic Convergence Theorems
In this section we will present several convergence theorems which com plement those which have been presented in §IV.2. Our first concern is to establish analogues of the monotone convergence theorem in which we deal
1 96
IV.
Nonstandard I ntegration Theory
with sequences of integrable functions which are not necessarily monotone. The basic results here are Fatou's lemma and the dominated convergence theorem. Next we present several results concerning various types of con vergence for sequences of measurable functions, including almost uniform convergence and convergence in measure. The proofs are standard; we include these results to fill out the standard theory. Throughout the section we will be dealing with classes M and L 1 of R valued measurable and integrable functions on a measure space (X, ..It, J.l.). Integrals of functions f in M and L 1 will be denoted by f f dJ.t. Before embarking on a presentation of the convergence theorems, we con sider the role played by sets of measure zero in the discussion. These occur frequently enough for us to make the following definition.
A proposition P(x), which depends on x e X, holds J.t-almost everywhere (a.e.) if there is a set E of measure zero so that P(x) is true for all x e E' (the complement of E in X). When the measure J.l. is understood we
4. 1 Definition
write a.e. instead of J.t-a.e.
For example, a function f is bounded a.e. if there is a constant B > 0 so that J.t({x : jf(x)j > B}) = 0. Similarly, we say that f g a.e. if there is a set A s;; X with J.t(A) = 0 a�d {x :f(x) =F g(x)} s;; A. If J.l. is a complete measure or f and g are measurable, we need only specify that J.t({x :f(x) =F g(x)} = 0. The relation of equality a.e. is easily seen to be an equivalence relation (Exercise 1 ). The basic fact is that sets of measure zero can be ignored as far as integra tion is concerned, as indicated by the following results. =
4.2
Theorem
(a) If f e M is zero a.e. then f f dJ.t = 0. (b) If f e M + and f f dJ.t = 0 then f = 0 a.e.
Proof: Let
E=
{x :f(x)
"#
0}; then E e ..lt.
(a) Suppose first that f e M + and J.t(E) = 0. Letting v.. = nl£, we have e M + and f v.. dJ.t nJ.t(A) = 0. With h = lim v.. it follows from Theorem 2.6 that h e M + and f h dJ.t sup { f v.. dJ.t : n e N } 0. Finally f � h, and hence 0 � f f dJ.t � f h dJ.t 0, so that f f dJ.t = 0. For general f we write f ! + - f - . If f = 0 a.e. then / + and ! - are both zero a.e., and the result follows by linearity of the integral. (b) The sets E.. {x :f(x) � 1 /n} are in .I( and E = U E.. (n e N). Since f � ( 1/n)XE. • we have 0 = J f dJ.t � ( 1/n)J.t(E..) � 0, so J.t(E..) = 0. H ence J.t(E) = 0 by countable additivity. D v..
=
=
=
=
=
=
IV.4
1 97
Basic Convergence Theorems
4.3 Corollary I f f
,
g
e M and f g a.e. then J f dp = J g dp. =
Proof: If E {x :f(x) g(x)}, then J fxx - E dp J f dp J f'x.E dJl J (Jl,E dp = J g dp. =
=
=
=
J 9Xx - E dJ.L
=
=
4.4 Theorem
If .f e M and J III dp <
oo ,
0
by 4 2( a) .
,
then f is finite a.e.
Proof: Let E { x : lf(x)l oo }. Then E e Jt (check) and nxE � III. and so � J III dp < oo for any n e N. We conclude that p(E) 0. 0 =
np( E )
=
=
M ost of the results in §IV.2 can be improved by replacing assumptions which hold everywhere by corrresponding assumptions holding almost every where. We ill ustrate this by proving a final version of the monotone conver gence theorem. 4.S Lebesgue's Monotone Convergence Theorem
Let In (n e N) and g belong to M. If J, � g a.e. where J g dp > oo, and fn � /, + 1 a.e. for all n e N then fn converges a.e. to a function f e M and lim n _ "" J /, dp J fdp. -
,
=
Proof' By combining the countably many sets (where J, < g, f.. > h + 1) into one set E of measure zero, we may set each J, and g equal to 0 on E without changing the integrals. We may also assume that 0 � g(x) > oo for all x (check), so oo < J g dp � 0. The result now follows from the monotone convergence theorem applied to f.. g. 0 -
-
-
4.6 Fatou's Lemma
If (I") is a sequence of nonnegative measurable func
tions, then J (lim inf /,) dp � lim inf J In dp.
Proof: If g" inf J; (i � n), then g" e M + and (g" : n e N) is an increasing sequence which converges to lim inf f.. . Also, if n � m, then g" � f,. , so J Y n dp � J J,. dp; hence J g,. dp � lim inf J /, dp. Therefore J (lim inf fn) dp lim" _ J g" dp � lim inf J J, dp by the monotone convergence theorem. 0 =
=
:r
4.7 Lebesgue's Dominated Convergence Theorem
Suppose that (fn ) is a se quence of measurable functions which converges a.e. to a measurable func tion f . If there is nonnegative function g e L1 so that IJ,I � g a.e. for each limn - ao J fn dJl. n e N, then I E L and J ldp 1 =
Proof: Fix a set E e Jt with p(E) 0 so that (fn ) converges to f except possibly on the set E, and Ifni � g except possibly on the set E. I f j" =
=
IV.
1 98
Nonstandard Integration Theory
a nd ii 9Xx - F,... then the seq uence <.l> of meas ll:rable _ _ function s converges everywhere to f, If.\ � ii on X, and fi nally J f dp 1 f d p and J .f. dp_ J fn dp b � Coroll � ry 4.3. Since l f l � g and f E M, f E L 1 , as is each of the functions .1:. N o w g + .1: � 0, and s o by Fatou's Lemma
.f.Xx - £• ] fl.x - E• =
=
=
=
f g dp fj dp fw +
=
=
Hence J / dp 0, we obtain
�
+
lim in
j ) d11 � lim inf fw + .i,. ) d 11
{J d
g p+
J .f. dpJ J g dp + lim inf J.f. dp. =
lim inf J .f. dp . Similarly, appl ying Fatou's lemma to ii
-f�
J g dp - Jj dp = J(g - j) dp � li m i nf J(g - j ) dp J g dp - lim sup J.1: dp. =
Thus lim sup J .f. dp
�
J j dp , and the result follows.
D
The rest of this section will center on various convergence properties of sequences of measurable functions without special concern for the conver gence of their integrals. The first of these is the famo us resul t of Egoroff which states that a.e. convergence "almost" implies uniform convergence. To be specific we introduce the following definition.
A sequence <.1�) converges almost uniformly if for each e > 0 there exists a set E E .If with p(E) < e so that ( f.) converges uniformly on
4.8 Definition
E' .
4.9 Egoroff's Theorem If p(X) is finite and (J.) converges a.e. to f on X then (f.) converges almost uniformly to f. Proof: For each k and n define the set Ek n E .A by Ek n = n: = . { X : l fm( x) f(x) l < 1/k } . Notice tha t if E is the set on which (f.) converges then for each k we have U Ek n (n E N) 2 E. For fixed k we have Ek• s;;; Ek m if n � m , and so lim. � oo p( Ek .) p( U Ekn(n E N) ) � p( E) p(X). Thus, for a given e > 0, we see that with each k e N is associated an nk e N so that p( Ei •. ) < e/2 k . If F = n Ek . k (k E N) then p(F') � L:"= 1 Jl(Ei •.) < L:"= 1 ej2k e. Fi nally we show that (f. ) converges uniformly on F. Let e > 0 be given and find a k so that 1 /k < e. Then \fm(x) - f(x)\ < e for all m � nk if x E Ekn" · Since F s;;; Ek"" we have uniform convergence on F. 0 =
=
=
IV.4
Basic Convergence Theorems
1 99
Another type of convergence which is important in probability theory is that of convergence in measure.
< fn> of measurable real-valued functions on X f if for every real e > 0 we have lim• - oo Jl( {x : lf. - fl � e} ) 0. Similarly (f.) is Cauchy in measure if for each e > 0 we have lim..... - oo Jl( {x : lf.(x) - f,(x) l � e } ) 0.
4. 1 0
Definition
A sequence
converges in measure to a real-valued function =
=
It is easy to see that if (f.) is convergent in measure to f then it is Cauchy in measure. Recall that Egoroff's theorem has been established only for sets of finite measure (see Exercise 2). The following result shows that, in general, almost uniform convergence is stronger than both convergence a.e. and convergence in measure. 4. 1 1
Theorem If a sequence (f.) converges to f almost uniformly then it converges a.e. and in measure.
Proof: For each k E N let (/.) converge uniformly to f on Fk where Jl(F�) < 1 /k. Then (f.) converges on F where F UFk( l � k < oo) and Jl(F') � Jl(f�) < 1 /k for each k E N, so that Jl(f') 0. Thus (/.) converges a.e. =
=
To prove convergence in measure let e > 0 be given and choose k with 1/k < e. Since f. converges uniformly on Fk , there is an m such that {x : IJ.(x) f(x)l � e} � F� for all n � some m depending on k. Thus Jl({x : lf.(x) f(x) l � e } ) < 1/k < e for all n � m, and the result follows. D The following example shows that a sequence can converge in measure but fail to converge at any point. 4. 12
Example Represent each n E N as n k + 2"', m � 1, 0 � k < 2"', and define J.(x) on (0, 1 ] to be x1t2 - m ,(k + t w m1 (the reader should draw some pictures). Then for any x E (0, 1 ] and any n0 there is an m1 � n0 and an 1 . Thus f. does not converge at m2 � n0 so that J� ,(x) = 0 and f,2(x) =
=
any point. On the other hand, given e > 0, the Lebesgue measu re of {x : lf.(x)l > e} � 2/n, so that f. --+ 0 in measure. I n this example it is possible to select a subsequence of (f.) which con verges a.e. This is true in general, as we now show. 4. 1 3
Theorem If (f.) converges in measure to f, then there is a subsequence (f.k) which converges almost uniformly and hence a.e. to f.
IV.
200
Nonstandard I ntegration Theo ry
Proof: Given k we can find an n,. so that Jl({x : lf.(x) - f(x)l � r 11 }) < 2 - " for n � n" . We may assume that nu 1 > n 11 • Now let E11 = {x : l f,.,.(x) - f(x) l � r " } . Given £, let m be chosen so that r • + 1 < £. I f X ' Ut= m E.. A then 1/.,.(x) - f(x) l < r�< for k � m, so /,.,.(x) converges uniformly to f(x) on A'. But p(A) :s;; LtCXI= m p(E11) :s;; LtCXI= m 2 - t = 2 - "' + 1 < £, and the result follows. D =
Exercises
I V.4
1. (Standard) Show that the relation = on the set of functions on a mea sure space (X, ..II, p) defined by f = g iff g a.e. is an equivalence relation. 2. (Standard) Show that Egoroft''s theorem does not hold for Lebesgue measure on all of R. 3. (Standard) Show that if for each n e N , /,. e L 1 and L:0= 1 I 11.. 1 dp < oo , =
then the series LCXI= 1 f. converges absolutely and almost everywhere to an integrable function f and I f dp = LCXI= 1 I /, dJ.t. 4. (Standard) Show that if lim, ... <XI I ]! - !.. I dp = 0 then /, converges to f in measure.
In the following problems, (L, f) will be an internal integration structure and (l, i) the complete integration structure of §IV. l with associated mea surable structure of §IV.2.
5. Show that if g e L0 then g � 0 P,-a.e. (Hint: Assuming g � 0, for any e > 0, there is a t/1 e L with 0 :s;; g :s;; t/1 and It/1 < e . Use Proposition 2. 33, Exer cise 2. 1 7, and the fact that { x : g :s;; 1/n } � {x : t/J � 1/n} � {x : t/1 � 1/2n}) 6. (Lifting of Measurable Functions) Assume that 1 e L. A function f is in M iff there exists a c/J e L such that oc/J f fi-a.e. If f is bounded then c/J can be obtained with the same bound and I f dP, = olc/J. (Hint: Use Pro position 2.32 and Exercise 5.) Any function c/J e L satisfying these condi tions is called a lifting of f. 7. (Lifting of Integrable Functions) Assume that 1 e L. Show that f e L 1 iff f has an S-integrable lifting c/J, in which case I f d{l = lc/J. =
a
IV.S The Fubini Theorem
A familiar process in the theory of Riemann integration for functions of several variables is that of iterated integration. If, for example, f(x, y) is a continuous function on the set [a, b] x [ c, d] in R x R then we have the equality
J: f f(x, y) dx dy J: (f f(x, y) dy) dx f (J: f(x, y) dx) dy. =
=
IV.S
201
The Fubini Theorem
The purpose of this section is to establish a nonstandard version of this equality in the contexts of the earlier sections of this chapter. The general result is known as the Fubini theorem, after its originator, G. Fubini. The nonstandard version is then applied to establish a Fubini theorem for inte gration structures on Euclidean spaces. First some notation. We will be dealing with integration structures (internal or standard) on product spaces U x V (internal or standard). These structures will typically be denoted by (Lu x v . lu x y). We will also be given integration structures (Lu , lu) and (Ly , l y) on U and V, respectively. Given a function f e Lu x y we may find that f( u, ) e Ly for u e U, in which case I v f is a func tion of u. If g = J vf is also in Lu then we denote its integral lug by lulvf (a slight abuse of notation since we are suppressing variables). ·
5.1 Definition Let (Lu , l u), (Ly , l y), and (Lw . l w) be integratio n structures on U , V, and W U x V, respectively. If the integration structures are stan =
dard, we say that a function / e Lw has the strong Fubini property with respect to I U • ly , and lw if (i) f(u, · ) e Ly for all u e U and /( · , v) e Lu for all v e V, (ii) lvf is in Lu and luf is in Ly , (iii) l wf = l ul v f = lvl uf·
If ''all" in (i) is replaced by "almost all" (i.e., the conditions hold a.e.), and (ii) and (iii) hold if luf and l vf are set equal to zero when not otherwise defined, then we say that f has the Fubini property. If the integration struc tures are internal and (i), (ii) and (iii) hold without exception, we say that f has the internal strong Fubini property. To begin we need the following basic result. 5.2 Lemma Let (Lu . lu), (Ly , ly), and (Lw , l w) with W U x V be real complete integration structures on U, V, and W, respectively. Suppose that each function f. e Lw in the sequence { f. : n e N} has the Fubini property with respect to I u• I y , and I w. and Un } is a monotone increasing sequence convergi ng to a real-valued f. Also suppose that sup { lwfn : n e N } < oo . Then f has the Fubini property with respect to I u. I y , and I w· =
Proof: Exercise.
D
We next establish results concerning the standardizations (Lu . iu). (Lv , iv ), and (f.w . iw) of internal integration structures (Lu . I u), (Ly , I y}, and (Lw , I w)
IV.
202
Nonstanda rd Integration Theory
on the internal sets U, V, and W = U x V, respectively, in an � 1 -saturated enlargement. These will be used to establish results on Euclidean spaces via the results of §IV.3. We assume that the function 1 (i.e., the function which is identically 1) is in Lw and that o I w 1 < oo. This will allow us to apply Theorem 1 . 1 6 when 4J e Lw by taking t/1 = 1 . We also assume that each func tion in Lw has the internal strong Fubini property (as in the case, for example, with Riemann integration of continuous functions). In particular, 1 is in Lu and Lv and olv1 < oo and 0 1 v 1 < oo . Suppose that 4J is a finite-valued function in Lw . Then o4J has the strong Fubini property with respect to iu . fv . and iw .
5.3 Lemma
Proof: Since, by assumption, 4J (u, ) e Lv for each u e U , we see that 4J(u, - ) e Lv by ! heorem 1 . 1 6. Similar I}', using Theorem 1 . 1 6 whe �e neces sary, we have l v( 0 4J ) = 0 1 v 4J in L u . lu(oc/J) = 0 l u c/J in Lv . and lw("4J) 01 w( 4J ) 0 1 ulv(4J) fu0 1 v(4J) iuiv("4J). The same argument with U and V reversed yields the result. D ·
a
=
=
=
=
For the next lemma we use the fact (Exercise IV. l. 1 0) that if h is real-valued and nonnegative, then h is a null function (Definition 1 .7) with respect to an integration structure (L , J) iff h e L and i(h) = 0. Suppose that h is a bounded real-valued null function on W. Then h has the Fu bini property with respect to iu , iv , and iw .
5.4 Lemma
h + - h - and using the fact that the Fubini property is preserved under sums (exercise). Then we have 0 s; h s; K for some standard in teger K. Since h is null there is a decreasing sequence <4J. : n e N) of functions 4J. e Lw with h s; 4J. s; K for all n, and lim 0/w{4J.) (n e N) = 0. Since h is real-valued there is a real-valued H E Lw to which the sequence (04J.) monotonically decreases, and 0 ::::;; h s; H. Now H also has the strong Fubini property by Lemmas 5.3 and 5.2 (appro priately modified), and fw(H) = 0. I t follows from Theorem 4.2 that for almost all u E U (in the measure induced by Lu . iu ), TvH(u, - ) 0, whence h(u, · ) is null on V. Therefore iufvh = 0. The same argument works with U and V reversed, and we conclude that the Fubini property holds for h. D
Proof: We may assume that h � 0 by considering h
=
=
Our main theorem generalizes a result of H. J. Keisler [25, p. 7]
5.5 Nonstandard Fubini Theorem Let (Lu , I u). (Lv , I v ), and (Lw , I w) be internal integration structures on the internal sets U, V, and W U x V, respectively, with 1 in Lw and 0 l w l < oo. Assume that every finite-valued =
IV. S
The Fubini Theorem
203
function ¢ in Lw has the internal strong Fubini property with respect to I u , I v • and I w · The '! any f E M !f for which Jw lfl < oo has the Fubini property with respect to I u • I v . and I w ·
Proof: Using the fact that the Fubini property is preserved under sums and writing f f+ - f - , we may assume that f is positive. Also, we may assume that f is bounded by first proving the result for f 11 n and using Lemma 5.2 to pass to the limit. Su ppose then that f E L w is a bounded nonnegative function. Then f has a decomposition f ¢ + h with ¢ E Lw bounded and h a bounded null function (check). Now f o,p + (¢ - o¢) + h, and since the null function (¢ - 0¢) + h is real-valued, the theorem follows from 5.3 and 5.4. D =
=
=
We will now apply Theorem 5.5 to prove a Fubini theorem for integration structures in Euclidean spaces. In the following, X and Y will denote closed and bounded (and thus compact) subsets of R " and R '", respectively, and Z = X x Y. Notice that 1 belongs to C(X), C( Y), and C(Z ). Given positive linear functionals I x . I y , and Iz on C(X), C( Y ), and C(Z), we obtain inte gration structures (C( X ), I x ), (C( Y}, Iy}, and (C(Z), Iz). These structures have • -transforms on • X, * Y, and •z, namely, (* C(X), *Ix). ( * C( Y), *Iy}, and (*C(Z), *Iz), respectively. For example, * C(X) is the set of all • -continuous functions on • X. U sing the techniques of §§IV. l and IV.3, we find that these internal structures induce integration structures (Lx . lx). (Lr . lr). and (Lz . iz) on • X, • Y, and • Z, which in turn induce integration structures (Lx , J x). (L y , J y}, and (Lz , J z) on X, Y, and Z, respectively. The latter structures extend (C(X), Ix). (C( Y}, Iy}, and (C(Z), Iz). The reader should recall (Remark 2.9.2) that every real-valued function in (Lz) 1 is in Lz . We remark that for f E C(Z) the equality of the iterated integrals always nolds [34, 1 6B, p. 44]. If that common value is Iz then the strong Fubini property holds for f.
5.6 Standard Fubini Theorem Assume that X and Y are compact. Suppose that each f E C(Z) has the strong Fubini property with respect to I x . I y , and I z . Then each f E M z such that J zlfl < oo has the Fubini property with respect to J x , J r , and J z .
Proof: I t suffices t o prove the result for f bounded and hence i n Lz . The assumptions of Theorem 5.5 are satisfied with • X = U, • Y V, and • Z W, since the strong Fubini property for each f E C(Z) transfers to the internal strong Fubini property for each ¢ E * C(Z) . Let f E Lz . Then J E Lz has the Fubini property with respect to fx . lr , and fz . If nx 1 0X 2 then J ( x 1 , y) J (x 2 , y) for all y E. • Y. Thus there is a standard set A c X such that J ( x , ) E Lr for all x E • X - A. Also A is null in • X so A is =
=
=
=
·
IV.
204
xf E x E E
(x, rf(x,
Nonstandard Integration Theory
(x,
null in X. If X - A then � = j ·2 o n • Y so J rf(x, · ) = J .J ). Set � A. Since J rf(x, · ) J ) for E • X - A, we have J rf( , ) 0 for Jy ( , ) Lx and l xl rf ixirj JJ J7j. The same argument with x
x
=
·
·
=
=
=
=
the roles of X and Y reversed gives the result.
x
·
0
·
We have established the Fubini theorem for the case that X and Y are compact subsets of R" and R m , respectively. The extension of this result for the case that X and Y are both open or both closed in R" and R m is a standard exercise, which we leave to the reader ( Exercise 3).
Exercises I Jl.5 1 . Prove Lemma
5.2.
2. Show that the Fubini property is preserved under sums.
3. Use Theorem 5.6, Exercise IV.3.8, and the obvious extension of Lemma 5.2 (for the case of R -valued functions) to establish Fubini's theorem for integrable f on X x Y, when X and Y are both open or both closed in R" and R m , respectively. 4. (Nonstandard version of Tonelli's theorem) In the notation of this sec tion, assume that 1 with I w 1 < oo , and the other assumptions of Theorem 5.5 hold. Show that if f M�, then
E Lw E E Mv iuEf(·,u) E Mv, f( ·, v) E M u uE iv f(u, · ) e Mu, iwf iu vf viuf f w J f = Jul vf = Jul v i i i J ul v i i i I J wf I < lvlu f · a
(a) f(u, · ) (b) (c)
for a.e. u and
U, and
V,
for a.e.
i =i · 5. In the conl e � t of ExeEci�e 4, show that if � M and eitht:r of the !e pe ated oo and w is finite, then i!_l t'lgrals or =
6. State and prove a standard version ofTonelli's theorem extending Theorem 5.6 for the case that f � 0. 7. (Standard) (a) Let f be the function on [0, 1 ] x [0, 1 ] defined by (x, y) '#
(0, 0),
=
(0, 0) .
(x, y)
Use trigonometric substitutions to show that (with Lebesgue integration)
So' [So' f(x, y)dy]dx i· So' [So' f(x,y)dx]dy � =
=
-
.
Conclude that f is not Lebesgue integrable on [0, 1 ]
x
[0, I ].
IV.6
205
Applications to Stochastic Processes
(b) Let f be the function on S = [ - 1 , 1]
x
[ - 1, 1] defined by
(x, y) ::1: (0, 0) (x, y)
=
(0, 0)
Show that the iterated integrals of f over S are equal, but f is not integrable.
•tV.6 Applications to Stochastic Processes
In this section we present a few examples which show how the theory of integration structures and the associated measure theory as developed in §§IV. l and IV.2 can be applied to problems in probabili ty theory, and in particular to stochastic processes. The essential idea is to extend the concepts of elementary probability theory on finite sample spaces to situa tions in which the sample space is a • -finite set in some enlargement. By transfer, this allows us to use the techniques of calculation and also the conceptual simplicity of the finite cases to deal with probabilistic situations in which the sample spaces are intrinsically infinite. The standard treatment of the problems we present below, and especially Brownian motion, can be a little complicated. Following the nonstandard treatment of coin tossing and Poission processes in [27], a nonstandard approach to the theory of Brownian motion was developed by Robert Anderson in [2]. This work has since led to a sequence of papers on nonstandard probability theory (see, for instance, the survey article [39] and other related papers [ 1 9]) and, in particular, has resulted in the solution of some difficult questions in the theory of stochastic processes by Keisler [2 5] and Perkins [33]. We begin with a very quick survey of probability theory. This theory was developed in order to provide a mathematical foundation for the study of problems in which the outcomes of certain experiments or measurements cannot be determined with certainty. To illustrate, we consider two typical examples from elementary probability theory: 1. A die is tossed at random and the upturned face is recorded. 2. A marksman is shooting at a target, and the resulting hole in the target
is noted. Each shot is subject to unpredictable effects of wind.
In example 1 the words "at random" are meant to convey that no device (for example, weighting the die, or influencing it with magnets) is in opera tion. We are interested in the likelihood of a particular number or set of
206
IV.
Nonstandard Integration Theo ry
numbers occurring on any given toss. It is almost evident that one is more likely to toss an element from the set {2, 4, 6} than that the number 3 will turn up. If the die is tossed n times and even numbers tum up n1 times, then ntfn "should" turn out to be quite close to ! (and "should" approach ! as n -+ oo. Thus, the ratio ! is a measure of the likelihood of an even number turning up and is called the probability of that event. To attack the problem mathematically and, in particular, to attach a meaning to the word "should" used above, we consider the set { 1, 2, 3, 4, 5, 6 } consisting of all the possible outcomes of the experiment. Because of the randomness we argue that each face is equally likely to turn up, and so the probability of any outcome is t. Using the idea that the probability of an even number turning up is the sum of the probabilities of a 2, 4, or 6 turning up, we see that the probability of an even number turning up is ! . Similarly, we can attach a number P(A) between 0 and l to any subset A of { 1 , 2, . . . , 6} which will be a measure of the likelihood of a number in A turning up and will be called the probability of the event A. Given the random nature of the experiment, P(A) will be JA J/6, where JAI is the number of elements in A. For multiple tosses of the die, we must consider a product space and corresponding probabilities. In example 2 the analogue of the set { 1, 2, . . . , 6} in example 1 is the set of points in the target. Each point has zero probability of being hit, but sets with positive Lebesgue measure are assigned a positive probability of being hit. A typical event is the event A of hitting a particular set in the target. The probability of the event A should again be a number between 0 and 1 which could be approximately determined by performing the experiment many times. In general, an abstract model for problems in probability is constructed as follows: (a) We construct a space n, called the sample space, whose points consist of all of the outcomes of the experiment. In example 1, n consists of the possible six faces (or, equivalent, the numbers from 1 to 6). In example 2, n consists of all of the points in the target (b) The events to which we wish to assign a probability are subsets of n. Given events A and B, A' is the event that A does not occur, and A u B is the event that A or B occurs. More generally we require the set of events to be closed under complementation and countable unions and thus be a a-algebra f!.
(c) The probability of an event A is a number P(A) satisfying 0 � P(A) � 1 . Since tS i s a a-algebra w e can and do require P to be a measure. on f! . In particular, P(U i A,.) = L;;, 1 P(A,.) for disjoint events A,. . This generalizes the procedure for constructing P used in example 1 . Now we have the following definition.
IV.6
207
Applications to Stochastic Processes
6. 1 Definition A probability space is a measure space (0, t!, P), where P is a measure on (0, If) satisfying P(O) 1 . The u-algebra tf is called the collection of events and P the probability measure. =
When the space 0 is finite and tf is the set of all subsets of 0, then P is completely determined by its values on the points in 0. A particularly im portant situation, as represented above, occurs when all the points of 0 are assigned equal probabilities (the equiprobability model ), in which case P(A) IAI/IOI. In the examples we will consider below, the nonstandard models will be hyperfinite analogues of the equiprobability model. Following a standard convention for probability theory, we will use w to denote elements of 0. Thus w will no longer be used for elements of * N 00 • In applications, we are usually concerned with functions defined on the sample space 0. For instance, in example 2 the target might be divided into three concentric regions A 1 (the central circle), A 2 , and A 3 , and the marks man could score 1 0, 5, or I depending on whether he hit A 1 , A 2 , or A 3 • The expected average score of the marksman if the shooting is performed many times would be 1 0P(A 1) + 5P(A 2) + P(A3). This leads us to the following definition. =
6.2 Definition A random variable is a real-valued measurable function X on the probability space (0, If, P). The expected value E(X) of X is J X dP(when the integral is defined).
In many situations we are more interested in a particular random variable than in the underlying probability space (0, If, P) on which it is defined. The probabilistic information involving a random variable X is contained i n its distribution Px . which i s a probability measure defined on the collection .1( of Borel-measurable subsets of the real line R (i.e., the smallest u-algebra containing the open sets in R) by the formula Px(A) = P({w e O : X(w) e A } ), A e .1(. It turns out that Px is completely determined by its value on all intervals in R. Thus in many applications the properties of a random vari able are defined in terms of the function Fx(x ) = Px( ( - oo , x]), which is called the distribution function of X. When X takes on only finitely many values {a1 , , an} then Px is completely determined by the values Px(a 1) P({w E O: X(w) ai}). i = 1 , . . . ' n. We are now ready for the definition of a stochastic process. •
•
•
=
=
6.3 Definition A stochastic process is a family { X, : t e I} of random variables all defined on a common probability space (0, If, P). I is called the parameter set.
208
IV.
Nonstandard Integration Theory
In the following examples I will be either the positive integers (for infinite coin tossing) or a subset of the real line (for the Poisson and Brownian motion processes). In the case of the coin-tossing and Poisson processes, the random variables will take values in the integers. A fundamental notion in probability theory, and especially important for stochastic processes, is the notion of independence. 6.4 Definition A
if for any
collection X 1 ,
•
•
•
, X,.
of random variables is independent
x . , . . . , x,.
P({ ro E n : X 1 (ro) � x 1 ,
.
.
.
•
X,.(ro) :!::
II
x ,.
} ) = n P( {ro E n : X,(ro) � X;}). I=
1
If the X1 are integer-valued we can replace the inequalities � by equality. Suppose, for example, a coin is tossed n times and x. , 1 � k � n, is the random variable which records a 1 or - 1 if the outcome of the kth toss is a head or a tail, respectively. Here n is the set of sequences of 1's or - l's of length n and contains 2" elements. It is clear that, for any k, P({ ro E n : X.(ro) 0 }) is 2" - 1/2" 1/2. More generally, if X; is fixed as 1 or - 1 for 1 � i � k, then P({ro E n : x Il l = x 1 ' ' X Ilk = x. } ) = 2" - t/2" 1 /2• , so the X" are independent. A common practice is to define a stochastic process {X,} by properties involving the distribution functions of certain combinations of the X,, for example the increments X, - x• . The Poisson and Brownian motion pro cesses are ones in which the increments over a finite number of disjoint intervals are independent. One last notion, which is central to probability, is that of conditional probability. Suppose, for example, that we want to compute the probability of a 5 turning up in example 1 given the extra information that an odd number will tum up. The answer is clearly l· In general the probability of A given B is denoted by P(A I B) and is computed as follows. =
=
.
.
•
=
The conditional probability of the event A given the event B is given by P(A I B) P(A (") B)/P(B) if P(B) ::/:: 0.
6.5 Definition
=
In the standard approach to the problems to follow it is sometimes difficult to define a suitable space (0, 8, P) on which the process is defined. One advantage of the nonstandard approach is that this step is relatively easy. 6.6 Example
(Infinite Coin Tossing)
In the elementary theory of probability (for finite sample spaces), one encounters the experiment of tossing a fair (i.e., unbiased) coin a finite number
IV.6
209
Applications to Stochastic Processes
of times. If the coin is tossed n times, then, as just remarked, a sample space for the experiment can be taken to be the set n. of all sequences (e1 , e2 , , e.), where e1 is either + 1 or - 1 depending on whether a head or a tail is obtained on the ith toss; thus n consists of 2• points (sequences). Specifying any event, for example the event of obtaining exactly two heads in n tosses, is the same as specifying a subset A of n. Since the coin is fair, it is argued that each sequence is equally likely, and so the probability P.(A) of an event A is measured by P.(A) (1/2•) 1 A I , where I A I is the cardinality of A. Suppose now that a coin is tossed an infinite number of times. We may define an associated stochastic process {X. : n e N} by putting X.(w) + 1 or - 1 depending on whether a head or a tail occurs on the nth toss. Thus {X,.} is a discrete parameter stochastic process in which the X. take on the values 1 and - 1 . We would now like to define a probability space (Cl, tl, P) on which this process is defined. In the standard theory one takes n to be the (infinite) set of all infinite sequences (e1 , e2 , . . . ) of + 1 's and - 1's. Now, however, the specification of the set of events and the probability of each event is not so clear. It is required that the set of events form a u-algebra tl of subsets of n, and that the probability is a countably additive measure P on tl with 0 s; P( A ) s; P(O ) 1 for each A e tl. Also, tl should contain any event A, which depends on only a finite number of tosses, for example the event of getting two heads in the first 10 tosses, and P should assign to this event A the probability obtained by using only the finite theory. In the standard theory, the existence of an tl and P satisfying these conditions is a consequence of a general theorem of Kolmogoroff. We now show that the nonstandard theory provides an appropriate tl and P, and that these have conceptual as well as calculational advantages. Our sample space n is the internal set of all internal sequences (e�o e 2 , e, ) of + 1 's and - 1 's of length ,, where 'I C! and C is an infinite integer. The lattice L is the set of all hyperreal-valued internal functions on n; we define l( c/J) ( 1/2") L c/J(e1) if cP e L. As noted in 1 .6, (L, l) is an internal integration structure. We denote by t1 the collection of internal subsets of n and put P(A) I(x11) for A e tl. The associated collection i of measurable sets will be the collection of events, and the measure P on i coming from (L, i) as in §IV.2 will define the probability. The reader should check that 0 s; P(A) s; 1 for each A e tl, and that any internal set A is in i with P( A ) st( i A I /2"), where 1 · 1 is the internal cardinality, i.e., the •-transfer of the standard car.: dinality function. It is not hard to show that if A is an internal set in t1 which depends on only the first n tosses then P(A) equals the probability obtained using the finite theory on n • . Thus (O, i, P) is an alternative to the standard space mentioned above. We can use (0, i, P) to compute the probabilities of events depending on an infinite number of tosses. As an example, let A. be the event "The first •
•
•
=
=
=
•
=
=
=
=
•
•
,
210
IV.
Nonstandard Integration Theory
n - 1 tosses are tails, then the nth toss is a head" in n. Then
The event
A
U:'= 1 A2ft
P(A 2 ft) =
1 /2
2
".
corresponds to the standard event of getting at least one head in an infinite number of tosses, the first one occurring at an =
P(A) =
even-numbered toss. and
�i ( 1 /2 2 ft)
=
!.
Note that the internal se t
is the even t of getting at least one head in '1 tos�s, the first one occurring at an even-numbered toss, and we also have P(B) = 2 st( '; 1 ( 1 /2 ft) st(! - 1/(3 · 2") ) = !.
B = U:'= 1 A 2ft
D
=
We may now consider the original stochastic process { X,. : n
process on
(e1 1
•
•
•
(O, i, P)
, e,).
by putting
X(co) = { 1 1
according as
P({(l) e O : Xft(co) = eft}) =
For each n,
e,. =
st(2" - 1/2")
e N}
{1
=
- 1 or 1 . Similarly any finite set of the X ,.'s is independen t.
t
1
in
for
as a
co = eft =
(The Poisson Process)
6.7 Example
The Poisson process is a stochastic process which is intended to model
situations in which isolated events occur randomly in time. Im agine, for instance, an experiment in which we record the time
t�0
of arrival of each
telephone call to an office. We can define a set {N(co, t) : t e (0, oo)} of random variables by specif_lin_s that, for any particular co in the as yet unspecified
(0, l, P) (co should represent a particular selection from the set of ways the calls come in), N(co, t) e 'luals the �!_Umber of incoming calls in the time interval (0, t]. Then, for s < t, N( co , t) - N(co, s) equals the number of calls in the interval (s, t ] . In many situations it is found that N(co, t) has the probability space
following properties, which define a Poisson process, and in particular, force the measurability of the process.
(6. 1 ) for each
(6.2)
if
s
co, N(co, t) � 0, N(co, 0) = 0,
and
N(co, t) is integer-valued,
co e 0, then N(co, s) :::;;; N (co, t) and
and
N(co, t)
is right continuous (see Exercise 3(a) for definition),
· · · < tft e R the random variables N( t 2) - N( , t 1 ), , N( , t ft) - N( tft _ 1 ) are independent [i.e., the N( , t) have independent increments)],
(6. 3) for each t1 < t 2 < · ,
·
•
•
•
·
· ,
·
(6 .4)
P({co: N(co, s + t) - N(co , s) = k}) = e - Ar((.l.. tf/k!).
The assumption (6. 3) says that what happens in one time interval is not affected by what happens in a disjoint time interval. The assum ption
(6.4)
says in particular that the probability of n calls occurring in the interval
(s, s + t]
t of the interval rate of the process.
is independent of s and depends only on the length
and the parameter l in the manner indicated. We call A. the
We now present a nonstandard model for discussing the Poisso n pr�ss. _ (O, ti, P) on
In doing so we will specify an appropriate probability space which the process can take place.
IV.6
Applications to Stochastic Processes
21 1
As in Example 6 . 6, let '7 = C! be an infinite factorial in •N. For simplicity we choose a standard positive rational number A as the rate for our process, and we let y be the infinite integer A'7. Divide the interval [0, '7) into '72 inter vals [0, 1 /'7). [ 1/'7. 2/'7)• . . . • [('7 - o;,, , ), and let 0 be the internal set of all internal ways that y distinguishable points can be put into the '72 intervals [k/'1. (k + 1 )/'1). That is, 0 consists of internal sequences w (ro1 : 1 � i � y ) with I ::::; w1 � '7 2 for each i. By transfer, the internal cardinality 101 of 0 is '12Y. Again we use the counting measure to induce an internal integration structure on 0. Let L denote the internal lattice of all internal • R-valued functions on 0, and for f E L define If = ( I /'72Y) �)!Y, f(ro,). Then (L, / ) is an internal integration structure on 0. We let the set of internal subsets of 0 (internal events) be denoted by I. If A e I then x 4 e f. and we define the internal probability of A by P( A ) IX A· The standardization (L, i) of (L, / ) leads to the measure space (0, i, P), where i is the collection of measurable sets and P is the measure on i obtained from (L, f) by the methods of §IV.2. Since P(O) = I , we see that 0 � P(A) � I for all A e i, and (0, i, P) is a probability space. Also P(A) = P( A) for all A e 8. We now define an internal stochastic process {N, : t e / } on 0 which is an internal analogue of the Poisson process. Here I is the internal set { k/'7 : I ::::; k � '72}. For any w e 0 and t e f, we define N,(ro) to be the number of points which the outcome w places in the interval [0, t). For the event w = (ro1), a point "lies" in [k/'1. (k + 1 )/'1) if roi k + I for somej, l � j � y. Note that N,(ro) can be an infinite integer for some w even for finite t. Also note that since 'I is an infinite factorial, any positive, standard rational number is of the form k/'1 e f. We want first to compute the P and P probabilities of the internal set A { ro : N,2(ro) - N,1(ro) k }, i.e., the probabilities that k points fall in [t 1 , t ), 2 where t2 - t 1 is finite and k is an ordinary natural number. Let s = t2 - t 1 • For simplicity we assume that t 1 and t 2 are rational numbers. Then there are exactly S'7 of the '72 intervals inside [t1 , t 2 ), and the P-probability of any one of the y points being put in [ t 1 , t 2) is S'l/'7 2 = s/'1 = As/y Now by (transfers of) elementary counting and independence, =
=
a
=
=
=
.
P(A )
=
=
�
( ) ( y) y! ( y ) (l y ) k ! y• (y - k) ! ) e - J.• . (y )• (
y! As k I (y k)!k! y _
(As)• s
k!
•
t
-
As Y
y
�
-
_
As y - k
As Y
(As)•
k!
This establishes the analogue of (6.4) for N, .
_
=
As - k
P( A)
212
IV.
Nonstandard Integration Theory
The analogue of (6.3) can be established in the same way. Let t1 + 1 - t1 = s1 and s s1 + · + s., where t1 < t 2 < · · · < t. , n is an ordinary natural num ber, and t1 e I. Assume s is finite. If A 1 = {w : N,j t 1(w) - N,1(w) k1}, where the k1 are ordinary natural numbers, with k = k1 + · · · + k. - 1 and i = 1 , . . . , n - 1 , then =
·
·
=
P(A 1
� • • =
=
•
�
A. - 1)
( )( ) ( ) ( ) ( )( )
A.s1 t• A.s2 t2 . . . A.s. _ 1 t" - • A.s y - t y! 1 y y y k1 ! k2 ! . . · k. - 1 ! (y - k) ! y A.s - t A.s Y (y s 1)t • (A.s 1)h . . . (A.s.- 1)t" - • y! 11k (y - k )!yt 1 ! k2! k. - 1 ! y y _
(ysl)t• . . . (A.s. - 1)t"- • e - .u � P�(A 1 ) · · · P�(A n - 1 )· - k 1 I• kn - 1 ·1 We want to use { N,(w) : t e I} to define a standard Poisson process. Unfor tunately, for some w e n, NJ.w) will take infinite values even for a finite t e I. Another possibility is that for some w e n there can be many points falling in an infinitesimal interval. We will show in the next paragraph that these abnormalities happen only on a set of P measure zero in ll; we can define a standard Poisson process on the remainder. Given w e n, we order the distinguishable points b1 by the order in which they fall in the line •R. Thus b1 � b1 + 1 and b1 = b1 + 1 if and only if b1 and b1 + 1 are in the same interval [k/'l. (k + 1 )/'7). Again fix j > 0 and k � 0 in N. Given t0 e I and t > 0 with t finite and t0 + t e I, let C,0 be the event "b1 e [t0 , t0 + 1/'7)", and let D,0 be the event "If j + 1 :s; i � j + k, b1 e [ t0, t0 + t), and b1 +H 1 ¢ [t0 , t0 + t)." Let y ' = y - j. Given C10 , the conditional probability of getting a given point of the remaining y' points in [t0 , t0 + t) is _
t'l 2 '1 - t0'1
----=---'--
=
t '1 - to
--
=
A.t t0A.
A.t
--- = ---Y -
y' + j toA. · Therefore, for all finite t0 , and hence for all t0 < T for some infinite conditional probability A.t t A.t y-t y' P(D ,o I c,o) = 1 ' (y ' k)!k ! y ' + j - toA. - y + j - toA.
(
(A.t)t
� k! e
- .l.r
)(
r,
the
)
.
On the other hand, ...Lr o < t P(C,�) � 1 , and so L ro < r P(D,01 C10 ) P(C,0) � (A.t)te - .l.l/k!. That is, the P-probability of having exactly k more distinguishable
IV.6
213
Applications to Stochastic Processes
points in the interval of length t after the jth point is (A.t)"e - A'/k!. Since co
(A.t)"
L
l = O , leN
--
k!
e - Ar = e - u .
eA r = 1 '
the P-probability of having only a finite number of distinguishable points in any finite interval [0, t] is 1 . Moreover, since Iim, .... 0 e - At 1, the P-prob ability of having point bi + 1 infinitely close to b1 is 0. Since this is true for eachj � 1 in N, it follows that the P-probability of having two distinguishable points in the same monad is 0. We now let E c 0 denote that set of measure zero consisting of those w for which N,(w) is infinite for some finite t or for which two or more distinguishable points fall in the same monad. Since E e 8, we define a new prob�bility space (n, i, PJ by putting n 0 - E, j = {A : A c;;; n, A e llf}, and P(A) = P(A) for A e tf. We now use N,(w) to define a process {N, : t E R} on (fi, i, P). For w E fi and t E R + we put N ,(w) sup NJ.w) (s � t, s E J ). By the above remarks, N ,( w) is finite and integer-valued for any w e fi and t e R, and N ,(w) NJ..w) for some s E J, s � t. We leave it to the reader to show that N,(w) is right continuous (Exercise 3) and that (6.3) and (6.4) are satisfied. Thus {N,} is a Poisson process on (fi, i, P). =
=
=
=
6.8 Example
(Anderson's Construction of Brownian Motion)
Brownian motion is a stochastic process which is intended to model the behavior of a particle (for example, a small particle suspended in water). The particle is subject to random disturbances (for example, collisions with the water molecules) which cause its position to change with time. For simplicity, we consider the one-dimensional case, and denote the random position of the particle on the real line at time t � 0 by X(t). Again for simplicity we follow the particle only for a unit time interval. Then {X, : t E [0, 1 ] } is to be a stochastic process on an as yet unspecified probability space (0, tf, P). A (standard) Brownian motion { X, : t E (0, 1 ] } must satisfy the following conditions: (6.5 )
X0
=
0,
� s. < r. are points in [0, 1] then (6.6) if s 1 < t 1 � s 2 < t 2 ::;;; the random variables X(t d - X(s 1 ), X(t 2 ) - X(s 2 ), , X(t.) - X(s.) are independent random variables, which we denote by X, , - Xs,, etc., •
•
•
•
•
•
(6.7) if t > s are points in (0, 1] then P({w E O : X,(w) - X J..w) � ex}) = 2 1/J(cx/�). where Y, (x) = ( 1 /$ ) r� 00 e - "11 du .
214
IV.
Nonstandard Integration Theory
Condition (6.5) locates the particle at the origin at t = 0. Condition (6.6) says that the probability of a change in position of the particle in any time interval (s� o tt] is unaffected by the changes in position in other disjoint intervals. Condition (6.7) indicates how closely the position of the particle at time t can be determined if its position at time s is known. The probability distribution function t/J(x) is known as the normal distribution with mean 0 and variance 1. One should note that 1/!(x/a) = ( lja J21i) J� oo e - u212"2 du, which is the normal distribution with mean 0 and variance u 2 • In [2], Robert M. Anderson used the measure space construction of §IV.2 to obtain, among other things, a nonstandard representation of Brownian motion. We give here a brief account of some of his results, which is neces sarily incomplete since we refer to his nonstandard version of the central limit theorem (Theorem 6. 1 1), which is crucial to the development. The central limit theorem is one of the deeper results in probability theory and to prove it here woule lead us too far from the main theme of these examples. A Brownian motion can now be defined as follows. Fix '1 ,!, an infinite factorial in * N; and let (0, t!, P) be the internal space for infinite coin tossing of Example 6.6 (with n being all sequences w (wl> . . . , w.>. and w1 = _+_1 or - l) constructed from the internal integration structure (L, /). Let (0, t!, P) be the corresponding standardization of (0, t!, P) constructed from (L, i) as in Example 6.6. Let x(t, · ) denote the internal random variable (function in L) defined by setting =
=
x( t, w) =
l \!1!,1
tE
Jr, i?-1 X AW),
* [0, 1],
where X AW) w1 • Here ['7t] denotes the largest element of *N less than or equal to '7t. Thus, for any w (w. , w 2 , , w.), the particle located by x(t, w) starts at the origin at t 0 [i.e. , x(O, w) = 0], and at each time t1 i/'7 (i = 1 , 2, 3, . . . , '7) the particle moves to the right or left a distance 1j.Jr,, depending on whether w1 is + 1 or - 1; at times lying between the t1 the particle remains fixed. The resulting motion is an internal analogue of a standard "symmetric random walk." We now define {J(t, w) = 0x{t, w) for t E [0, 1] and (J) E n. We will show that {J(t. - ) is a Brownian motion on (n, I, P). To do so we need the following results. =
=
•
=
•
•
=
An internal random variable on (0, t!, P) is a function X e L . A collection { X1 : i e / } of internal random variables is •-independent if for every •-finite internal subcollection {X 1 , , X,.} (m e * N) and every internal
6.9 Definition
•
•
•
Applications to Stochastic Processes
IV.6
(6.8) =
'"
215
P( { w E n : X l(w) < (Xl } ).
n l
= l
{X; : i e l} i s S- independen t if, for every finite subcollection { X � o . . . , X '" } replaced and every m-tuple <(X t o , (X '" ) e R'", (6.8) holds with
by
..
.
(m e N) �-
=
Suppose {X; : i e I } is S-independent. Then {0X1 : i e 1 } is an independent collection of random variables on (Cl, i, P).
6. 10 Lemma
Proof" Suppose m e N, ((X1, P({w : 0X;, (w) < =
=
(X l •
.
.
.
, (X '" ) e R'". Then
•
•
.
• 0 X ;m(w) < (X
lim op ({w : X;,(w) <
n - oo
lim
n - oo
a( .Ii
J= l
(X l -
...
!, n
}) .
.
.
P ({w : X ;1(W) < (Xi - ! n
•
X ;m < (X III - ! n
}))
})
Let { X" : n e N} be an internal sequence of • -independent random variables on (Cl, S, P). Assume that there is a standard distribution function F such that *F is the distribution of X" , E(X") 0, and E(X� ) 1 for each n e • N. Let 1/J denote the standard normal distribution. Then for any m e • N N and any (X e • R
6. 1 1 Theorem
=
-
Proof: See Theorem 21 in [2]. 6. 1 2 Theorem
=
0
If '1 e • N - N, then fJ(t, · ) is a Brownian motion on (Cl, i, P).
IV.
216
Nonstandard Integration Theory
Proof: (i) Given t e [0, 1 ], X(t, · ) is tS-measurable, and so {J(t, · ) is j-measur able by Proposition 2.3 1 . (ii) B y transfer from the case o f finite coin tossing we see that the X 1 have identical distributions. Also if St = IJ= t X1 for any k e • N then the St have independent increments by the transfer of Exercise 1. Thus if s 1 < t 1 s2 < t2 � � s. < t. are points in [0, 1], then { x(t 1 , · ) - x(s 1 , ), , x(t., · ) x(s. , are •-independent and so S-independent, and condition (6.6) follows by Lemma 6. 10. (iii) Give n s < t in [0, 1 ], A. = ['7 t] - [17s], and e R,
·)} •
·
•
·
•
•
•
�
IX
� IX}) � IX}) P( {ro : 0X(t, ro) - 0 0 p ({ro · ( ! ) � IX}) ..;r, lim ({ � ! � "'>/�� (IX + �)}) n !�� 0(•1/t) ([1 ( + �)) (by Theorem !� "'C ( fi (IX + �))) IX J IX + ] [lim [ � ..... "" � This establishes condition In general, by a of a stochastic process 1}, we mean a function f(t) for some particular The last result of this section shows P( {w e O. : fJ(t, w) - fJ(s, w)
X( S, W )
=
.
=
=
. .... ""
P
t
Wt
=
w:
Ill•I
ro t � I.,•J t
v.A.
IX
=
6. 1 1)
=
=
1/t
1/n
(6.7)
=
path
X,(w),
= 1/t
0
w e 0..
{X, : t
e
that almost all of the paths of Brownian motion are continuous.
There is a set Q' e 8 with P(Q') = 1 such that fJ( · , w) is a con tinuous and finite function on [0, 1 ] for all w e Q'.
6. 13 Theorem
Proof: For each m, n e N, let Cl..., be the internal set given, using the inter nal extensions of sup and inf, by
{
n.... = w e Cl :
sup
I E (I/n, (i + 1 )/n)
}
for some i < n .
x(t, w) -
inf
I E (Ijn, (l + 1 )/n)
x(t, w) >
.!.. m
IV.6
Applications to
Then for A. P(0111.)
=
'1/n,
({ sup inf ..!_}) ({ max 1± 1 .Jr,}) ({ max ± .Jr,2"}) + ({ min ± ({ � i: }) + ({ � i:}) ({ � t �}) 4 (1 (�)) 4 ( 1 (i:)) nP
w:
nP
w:
S nP
w:
S
S
S
=
�
1 s t s .1. 1
I :S t :S .\ 1
2nP
w:
4nP
w:
n
-
X(t, w) -
r e [O, I/n)
w, >
2m
�
w:
1 :S t :S .\
w1 <
I
w1 <
-
.Jr, 2m
})
-
n
-
�
- rzi2 dt.
For .Jn/2m > 1
2n ./n 2 111 e - '1 2 dt / 0 - U:= l n:- 1 olllft ' Then P(O11111)
1
w:
nP
m
2n P
·�
4n "'
f(O') �
m
w, >
$ J./i/2111 e
=
w1 >
x(t, w) >
r e [O, I /n)
w1 >
=-
Let 0'
217
Stochastic Processes
-
f�
s
sup inf f(0111. ) � 1 ,.
n
-
=
4ne- ./ii'4111 •
sup inf 4ne- ./i/4"' = 1 . ,.
n
Fix (J) E 0. If for some t E *[0, 1] we have 0,l(t, w) = + 00 or 0,l(t, w) = - 00 , then w e 0111 , • for all standard m and n e N, whence w j 0'. If for some s and t e * [0, 1] with s � t we have olx(s, w) - x(t, w) j = a > 0, then for m > 2/a we have w e 0111 • for all n e N (exercise), whence w j 0'. Now suppose w e 0'. By the preceding paragraph, P(t, w) is finite for all t e [0, 1]. Fix e > O in R. Then the set {n e *N : jt - sl < 1/n ==- !x(t, w) x(s, w)l < e/2} is internal and contains all infinite n. Hence it contains a finite n by 11.7.2(ii). Thus if It - sl < 1/n, lx(t, w) - x(s, w)l < e/2 and hence ! P(t , w) p(s, w)l < e. It follows that PL w) is continuous on [0, 1]. 0 Exercise I Jl.6 1. (Standard) Let Xi be defined on the space o. of Example 6.6 by X �w) = e1 if w = (e� o e 2 , e.). Show that the random variables St = L�= 1 X1, •
•
.
,
218
IV.
Nonstanda rd I ntegration Theory
n, have independent increments, i.e., if 1 � k 1 < k 2 < k 3 < k 4 < < k, � n then s.l - s• . s• • - sh • • • , s• . - s., _ I are independent. 2. In Example 6.7, check that 1 �k�
,
,
•
o
o
and
Lro < t P(C,J
1.
�
3 . (a) Show that the process N,(m) defined on (0., i, P ) i n Example 6.7 is right continuous. That is, show that, for each fixed w e 0., the function f: R + --+ Z defined by /(t) N,(w) satisfies lim, ... , s > r f(s) f( t ). (b) Show that the process N,(w) satisfies Propert ies (6.3) and (6.4). 4. (Inter-arrival times) Define the process { f. : n e N} on the space (0., 1, P) of Example 6.7 as follows. For m e 0., f.(m) is the time between the (n - 1 )st and the nth jump of N,( w). =
=
(a) Define the internal analogues { T.,: n e •N} of { f., : n e N} on (0, 8, P). , f. _ 1 (b) Show that P{ f1 > t t } = e - ;. , , and P{ f., > t., l f1 t 1 , '• - d
= e
- Ar
=
.
•
•
•
=
, f., > t.,} = show that P{ f 1 > t 1 , f2 > t2 , e e 41", showing that the f., are independent, identically dis tributed random variables. 5. Prove the result tagged as an exercise in the proof of Theorem 6. 1 3. (Note that we may have s < i/n < t for same values of n e N). (c) Use (b) -
4 " - 41 1 e
•
•
•
to
-
•
•
•
APPENDIX
Ultrafilters
In this appendix we present the essential facts concerning ultrafilters which are needed in the text. In the following, I will be an arbitrary set. A. I Definition
A nonempty collection !/' of subsets of I is a filter on I if
(i) 0 j !/', (ii) A, B e !/' implies A n B e !/', (iii) A e !/' and B 2 A implies B e !/'. A filter !/' on I is an ultrafilter if it is maximal; i.e., whenever C§ is a filter on I and !/' £ C§ then !/' = C§.
The following result shows that this definition of ultrafilter is equivalent to that of Definition 1 . 1 in Chapter I. A.2 Proposition A filter !/' on I is an ultrafilter iff, for every subset A of I, either A e !F or A' = 1 - A e !F.
Proof: Suppose that !/' is a filter such that for every A c I either A e !/' or A' e !/'. Let C§ be a filter with C§ 2 !/' and suppose that B e C§ and B j !/'. But then B' e !/' £ C§, and so 0 = B n B' e C§, contradicting A. l(i) for a filter. Th us there is no filter C§ properly containing !/', and so !/' is an ultrafilter. Conversely, su ppose that !/' is an ultrafil ter and A j !/'. Let C§ be the set {X £ I : A n F £ X for some F e !/' } . Then !/' £ C§ and !/' "=/; C§ (since, for example, A e C§), and so C§ is not a filter since !/' is maximal. But C§ is not empty, and if B, C E C§ and D 2 B then B n C E C§ and D e C§. Thus C§ can fail to be a fil ter only if 0 E ��. That is, we have A n F = 0 for some F e !/' for which we then must have F £ A'. It follows that A' e !/' by A. l(iii). 0 219
220
Appendix
Ultrafilters
We now want to prove the ultrafilter axiom, 1 .2 of Chapter I. To do so we need Zorn's lemma, which is a variant of the axiom of choice. The state ment of Zorn's lemma involves the idea of a partially ordered set and related concepts. A.3 Definition A partially ordered set is a pair (X, S ), where X is a non empty set and s is a binary relation on X which is
(i) reflexive, i.e., x s x for all x e X, (ii) antisymmetric, i.e., if x s y and y S x then x (iii) transitive, i.e., if x s y and y s z then x s z.
=
y,
A subset C of X is a chain if for all x, y e C either x s y or y s x. The element x is an upper bound for a subset B £ X if b s x for all b e B. An element m e X is maximal if, for any x e X, m s x implies x = m. Let (X, S ) be a partially ordered set. If each chain in X has an upper bound then X has at least one maximal element.
A.4 Zorn's Lemma
Zorn's lemma is equivalent to the axiom of choice. A.5 Axiom of Choice For any set A of nonempty sets, there is a function f with domain A such that f(x) e x for each x e A. The function f is called a choice function for A.
We now use Zorn's lemma to prove the ultrafilter axiom. A.6
Ultrafilter Axiom If � is a filter on I then there is an ultrafilter tfl on I
containing �.
Proof: Let j; be the set of all filters which contain �. j; is nonempty since � e j;_ We partially order j; by inclusion; i.e., if .JJ/ , 1M e j; then we say that .JJI s 1M if A e .JJI implies A e IM. It is easy to check that s is a partial ordering on j;. Now let ti be a chain in J;. To show that ti has an upper bound consider § = Uct (CC e ti). Then CG s § for all CG e ti. Also § is a filter. For if A, B e § then A e � 1 and B e CG 2 for some CG 1 and CG 2 in ti. Since ti is a chain, we may assume without loss of generality that CG 1 S � 2 , and so A , B e CG 2 and A n B e CG 2 £ ; . Similarly we check conditions (i) and (iii) of A. l . We deduce from Zorn's lemma that j; contains a maximal element which is then an ultrafilter containing �. D
Append ix
22 1
Ultrafilters
There are some ultrafilters on I which, for our purposes, are quite trivial. Consider, for example, the collection Cfla = {A � I : a e A} for some a e I. It is easy to see that Cfl a is an ultrafilter. A.7 Definition An ultrafilter Cfl is principal or fixed if there is some a e I so that Cfl = {A � I : a e A } . If the ultrafilter Cfl is not principal it is called free.
A.8
Theorem Free ultrafilters exist on any infinite set I.
Proof: The collection !F1 { F � I : I - F is finite} is a filter (check) called the cofinite or Frechet filter. Let Cfl be an ultrafilter containing !F1 • Then Cfl cannot be principal. For if Cfl = {A � I : a e A} and Cfl => !F1 , then the set F { a } ' e Cfl (contradiction). 0 =
=
References
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.
IS Gonshor, H. Enlargements contain various kinds of completions. In "Victoria Symposium on Nonstandard Analysis," (A. E. Hurd and Loeb, P. A., eds.), Lecture Notes in Mathematics, Vol. 369. Springer, Berlin, 1 974. 1 6 Henson, C. W., and Moore, L. Nonstandard analysis and the theory of Banach spaces. In "Nonstandard Analysis-Recent Developments," (Hurd A. E., ed.), Lecture Notes in Mathematics, Vol. 983. Springer, Berlin, 1983. 17 Hewitt, E. Rings of real-valued continuous functions, Vol. I. Trans. Amer. Math. Soc. 64 ( 1948), 45 -99. 18 Hirschfeld, J. "A Non-Standard Smorgasbord" (unpublished notes). Tel-Aviv University.
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1 9 H urd, A. E . (eel.) �Nonstandard Analysis-Rece n t Developmen ts," Lecture Notes i n M athematics, Vol. 983. Springer, Berlin, 1 983. 20 Kelley, J . L. �General Topology." Van Nostrand, N ew York , 1 955. 2 1 Keisler, H . Jerome. Good ideals in fields of sets. A nn . Math. 79 ( 1 964), 338 - 3 59. 22 Keisler, H . Jerome. Ultraproducts and saturated models, Prot:. Kon. Nederl. Akad. v. Werensch., Ser. A, 67 ( 1 964� 1 78 - 1 86. 23 Keisler, H. Jerome. "Elementary Calculus." Prindle, Weber and Schmidt, Boston, 1 976. 24 K eisler, H . Jerome. �Foundation s of I nfinitesimal Calculus." Prindle, Weber and Schmidt, Boston, 1 976. 25 Keisler, H . Jerome. "An infinitesimal approach to stochastic analysis," Memoirs Amer. Math.
Soc., No. 297, America n Mathema t ical society, Pro vidence, Rhode I sland, Vol. 48, 1 984. 26 Lebesgue, H . Sur une generalization de l'integrale defi nie, Comptes Rendus Acad. Sci. Paris, 1 32 ( 1 90 1 ), 1 02 5 - 28. 27 Loeb, P. A . Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Amer. Math. Soc. 2 1 1 ( 1 975), 1 1 3 - 1 22. 28 Loeb, P. A. �An introduction to nonstandard analysis and hypcrfinite probability t heory. In "Probabilistic Analysis and Related Topics, Vol. 2, (edited by A. T. Bharucha-Reid), Academic Press, New York , 1 979. 29 Loeb, P. A. Weak limits of measures and the standard part map. Proc. Amer. Math. Soc. 77 ( 1 979) , 1 28 - 1 35. 30 Loeb, P. A. Review o f [23] a n d [24]. J. Symbolic Logic, 46 ( 1 98 1 ), 673 -676. 3 1 Loeb, P. A. M easure spaces in nonstandard models underlying standard stochastic processes. "Proceedings of the International Congress of Mathematicians." Wanaw, 1 983. 32 Loeb, P. A. A functional approach to nonstandard measure theory. Contemporary Math., 26 ( 1 984� 25 1 -26 1 . 3 3 Loeb, P . A . A nonstandard functional approach t o Fu bini's theorem. Prot:. Amer. Math. Soc. 93 ( 1 985), 343- 346. 34 Loomis, L. H. "An Introduction to Abstract Harmo nic Analysis." Van Nostrand, New York , 1 953.
35 Luxemburg, W. A. J . A remark on a paper by N . G. De Bruijn and P. Erdos, Prot:. Kon . Kederl. Akad. 11. Wetensch., Ser. A, 65 ( 1 962), 343 - 345. 36 Luxem burg, W. A. J . A general theory of monads, In "Applications of M odel Theory to
Algebra, Analysis, and Probability, (W. A. J . Luxem burg, eel.). H olt, Rinehart, and Winston, New York , 1 969.
37 Machover, M . and H irschfeld, J. "Lectures on Non-standard Analysis," Lecture Notes in Mathematics, Vol. 94. Springer, Be rlin, 1 969. 38 Perkins, E. A. A global intrinsic characterization of Brownian local time. Ann. Probability 9 ( 1 98 1 ), 800-8 1 7. 39 Perkins, E. A. Stochastic processes and nonstandard analysis. In "Nonstandard Analysis Recent Developments," ( H u rd A. E., eel.), Lecture Notes in M athematics, Vol. 983. Springer,
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List of Symbols
R, 2 !Jl , 2 +, 2 ·, 2
<, 2
N, 3 R, 3 $, 3 0, 3
( r1 ) , 3
0. 3
"' · 3 "' •• 3 F1 , 3
=. 4
a.e. 4 R, 5
[r] , 5 r, 5
+ . ·. <, 5 at, 5 lrl, 6 •• 7, 86 • r, 7 (A) • • 7 s·. 8
( a 1 , . . . , a"), 8, 72 P ( a 1 , . . . , a"), 8 P', 8 dom P, 9, 72 range P, 9, 72 /(a', . . . , a"), 9 •p, 9 Z, I O XI' • 10 •R, l O
c= , £ , 7 1
•[Jl, 1 0 f/, I I
C 1 X C l X • • · X C0,
L_y , 1 2 11 , 1 2, 74 .... . 1 2, 74 V, 1 2, 74 �. 12 P, /, 1 3
c" , 72 P[b] , r 1 [b], 72 f(x� 72 1 - 1 , 72 gja, 72
fi'x . 74
I, A ,
i\ 1 3 l
A -r�. 1 3
1• 1
<( , ::1- . 26, '�- · 26, ....... , ..... , �.
18
44, I l l , 1 24 124 26
m(x), 26, 44, I l l , G(x), 26 st(p), 27, I l l 0p, 27 •A .., , 28 •z, • N , 2 9
124
Q, ·Q. 30
( s.) , 32 R + , 34 l i m sup, lim inf, 37 .4, .4, 4 1 , 1 1 3 st, 41, 1 07, I l l
', 76 R0 , 76 Rri , 77
V,
-+ , +-+,
Ill, • Ill, 79
n s. 83 •• 84 =
n.s. n�S, 84 [a] , 84
•x, 84 e., 84 M, 85 • , 86 9',( A), 89
A + 8, 44
r: /(x) dx, 57
card A , 105 N � o 1 07
T, , 68
9'(X), 7 1 V.(X)V.,(X), 11 e, j, 7 1 , 7 5 22 5
74
(Vx � o . . . , x. e c), 77
JA(A), 93 , 106 JA(F), 93 • v(X), 96 /.., , 102, 1 23
Ax, Ay, dy, 53 s:(J, P), £). J, P), etc. 56, 57
72
st(B), 107, 1 1 7, 1 22, 1 3 1 (X , .1"), 1 1 0
.r, 1 1 0
.¥. , .r•. fJI,, 1 1 0
226
L1st of Symbols
X", etc.,
I l l , 1 24 "' · I l l , 1 24 st(y), I l l ns(• X), I l l ,'T X !/', 1 1 2
( ' ), 145 12 , 146
.'T X•
e. ,
m(x),
.4, .4, 1 1 3 T0 • T 1 , T2 , 1 1 4
1 15
0X; (i E I), 1 1 6
st(B), 08, A 0 , iJA, 1
1 1 7, 1 22, 1 3 1 18 (X, d ), 1 23 .rd. 1 23 B.( x), 1 23 1,., , 1 23 "' · f: , 1 24 m( x), 1 24 fin(• X), 1 24, ! 54 (i, JJ. 1 28 0, 1 29 st(A), I 3 I + . 1 33 II 11. 1 33 II II .,. 1 34 1 , . 11 11 " 1 34 R "' , 1 34 c0, 1 35 B(S), 1 35 C(S), 1 35 L(X, Y), 1 35 B(X, Y ), 1 35 II T il. 1 35 N(T), 1 36 · ,
X',
.Si . 1 4 1
m.,, m,.. ,
1 40
1 43
X 1 y, ! 46 S1, 1 46
G,
1 49 1 54
fin(* X), ! 54 x. J. 1 54 X0, 1 56
- . 157
{JX, 1 58 1/JA , 1 60
M, 1 60 kA(j), 1 6(} rri.. f ), 160 �. 1 60 p(f), 1 60 '6, 1 60 c( f), 1 6 1 C(X, Y), 1 63 I, 0, 1 66 X£• 16 6
p.l.f., 1 67
S( R), $. 1 68
B( Y). r . 1 68 Ba( Y), Lo• 1 68 B.,(X), Lw• 1 68 L0, 1 69 L. 1 69 i, 1 70
"1/J , 1 72
&, 1 75 R., 1 76 L · . i1 + , 1 76 i, M, 1 76 L. L t . 1 78 !1', 1 79 ""· 1 80 .it, jJ., 1 8 1 M , 1 82 I h dfl, 1 84 C,(X ), 1 89 io(f), 189
M x • . ll x . L x , l x , 1 90 :¥", :Y, 1 9 1 K
-<. f, f -<. V , 1 9 1
a.e.,
1 96 0, 206
(L, 1 ), 1 67
�. P, 207
r . r . 1 67 C[a, b], 1 67 C.(R), 1 67, 1 89
P(A I B), 208
max(f, g), min(/, g), 1 67
suppf, 1 67 I. 1 68
E(X), 207 Px, F X•
n •. 209
207
X(t,w), 2 1 4 {J(t, w), 214
Index
A
Bounded set, 44, 1 29
Brownian motion, 2 1 3 -2 1 7
Abian, A., 1 22 Absolute value, 6 Accumulation point, 4 1 , 1 1 3
c
Alaoglu's theorem, 143
Cauchy sequence in measure, 1 99-200
Alexander's theorem, 1 22
Cauchy-Peano existence theorem, 63 -65
Almost everywhere (with respect to an ultrafilter), 4
Cauchy's principle, 100
Almost everywhere validity, 196
Chain i n an ordered set, 220 Chain rule, 53
Almost unifonn convergence, 198-200
Anderson, R. M ., 1 86, 205, 2 14
Characteristic function, 10
Archimedean property, 19
Choice function, 220
Archimedes, i x
Closed map, 1 2 1
40, 1 10, 1 1 2
Arzela -Ascoli theorem, 62-63
Closed set,
Ascoli theorem, 162
Closed subspace, 1 3 3 Closure o f a set, 4 1 , 1 1 3
Axiom of choice, 220
Cofinite, 3 Cofinite filter, 22 1 Compact metric space, 1 29
8
Compact operator, 1 37- 140, 1 52- 1 53
Banach limit, 102
Compact set, 42, 48, 1 20- 1 22
Banach space, 1 33
Compactification, 1 56- 1 59
Banach space ultrapower, 1 54
Compactness theorem, x
Base, 1 10
Complete integration structure, 1 75
Behrens, M., 56
Complete measure, 1 80
Berkeley, G., ix
Complete metric space, 1 27- 1 29
Bernstein, A. R., 92, 1 52
Complete orthononnal se t (basis), 1 49,
1 51 - 1 52
Bessel's inequality, 1 50- 1 5 1
Comprehensive monomorphism, 98-99,
Best approximation theorem, 1 50 Bilinear fonn, 1 54
106- 107 Con.::u rrent relation, 9 1 , 105
Bliss's theorem, 60
Condi tional probability, 208
Bolzano - Weierstrass theorem, 3 5
Constantinescu, C., 1 59
Borel set, 207 Boundary point, 1 1 8 - 1 1 9
Continuity,
46, 48, 77, 1 1 5, 1 1 9, 1 3 1
Continuous linear operator, 1 36
Bounded linear operator, 1 3 5 - 1 40
227
228
I ndex
Convergence in measure, 199-200 Convergence, •, 104 Convergence, S, 104 Convex set in a vector space, 147 Cornea, A., 1 59 Countable additivity, 1 80 D
Dacunha-Castelle D., ! 54 Daniel, P., 1 65 Darboux's theorem, 55 De B ruij n, N. G., 92 Differential, 53 total 54 Denumerably comprehensive monomorphism, 98-99, 1 06- 107 Derivative, 5 1 -52 Dini's theorem, 61 Discrete metric, 1 23 Distribution function of a random variable, 207 Dual space, 1 40- 1 44 DuBois-Reymond, 1 03 E
Eberlein-Smulian theorem, 143 Egoroff's theorem, 1 99 - 200 Element, 7 1 Enlargement, 90-92 Entity, 7 1 Equality almost everywhere (a.e.), 84 Equicontinuous family of functions, 162 - 1 63 Equiprobability model, 207 Equivalence with respect to an ultrafilter, 84 Equivalent metrics, 1 32 Equivalent norms, 1 3 2 Erdos, P., 92 Euler, L., ix Evenly continuous family of functions, 162- 1 63 Event in a probability space, 207 Expected value of a random variable, 207 Exponential function, 48-49, 52 Extended real number system, 1 76 External entity, 95, 97 -98 External sentence, 95 Extreme value theorem, 46, 50
F
Family distinguishing points and closed sets, ! 57 Fatou's lemma, 1 97 Filter, 3, 2 1 9 cofinite, 3 countable subbasis, 103 Frechet, 3, 22 1 Finite intersection property (f.i.p.), 93 Finite point, 1 24, 1 35 Finitely additive measure, 1 75 First axiom of countability, 1 30- 1 3 1 Formula, 75 • -transform, 79 atomic, 7 5 Fourier coefficient, 149 Frechet filter, 3, 22 1 Fubini, G., 201 Fubini property, 201 in temal, 20 I strong, 201 Fubini theorem nonstandard, 202 -203 standard, 203 -204 Function, 9, 72 continuous, 46, 48, 77, 1 1 5, 1 1 9, 1 3 1 • -continuous, 8 1 , 95-96, 1 3 1 - 1 32 differentiable, 5 1 -56 domain, 72 extension, 72 increment, 53 - 54 injective, 9, 72 n variables, 72 one-to-one, 9, 72 onto, 72 range, 72 restriction, 72 Riemann integrable, 57 S-continuous, l 3 1 - 1 32 surjective, 72 uniformly differentiable, 56 Fundamental theorem of calculus, 59
G
Galaxy, 26, 1 24 principal, 1 24, 1 54 Gonshor, H., ! 57
229
Index
Internal sentence, 95 Internal set, 79 Intersection monad, 93, 1 03, 1 06 Inverse function theorem, 54 Inverse image under a relation, 72
Graph, 92 edge, 92 infinite, 92 k-colorable, 92 vertex, 92
K
H
Hahn- Banach theorem, 1 4 1 - 1 42 Halmos, P. R., 92 Hausdorff space, 1 1 4, 1 1 7 Heine -Bore! theorem, 43 Henson, C. W., 109, 144, 1 54 Hewitt, E., x Hilbert cube, 1 54 Hilbert space, 1 45 - 1 54 Hirschfeld J., 1 22, 1 57, 1 60 Homeomorphism, 1 1 5 Hyperfinite integration structure, 1 68 - 1 69,
1 88 - 1 89 Hyperfinite set, 89 Hyperintegers, 29 Hypernatural numbers, 29 Hyperrational numbers, 30 Hyperreal number system, 5
K-saturation, 70 Keisler, H. J., x-xi, 1 4, 49, 52, 56, 59, 60, 96,
104-106, 1 1 8, 205 Keisler's internal definition principle, 96 Kelly, J. L., xi, 1 60, 1 62 Kolmogoroff, A., 209 Konig's lemma, 94 Krivine, J -L 1 54 Kunen, K., 105 .
.,
L
Language for superstructures, 74-78 Lattice hyperreal, 1 66- 1 67 real, 1 66 - 167 Lebesgue, H., 1 64, 1 80 Lebesgue covering lemma, 1 32 Lebesgue dominated convergence theorem,
1 97 - 198 Image under a function, 9, 72 Image under a relation, 72 Independent random variables, 208 Individual, 7 1 Infinite coin tossing, 208 - 2 1 0 Infinite sum theorem, 5 9 Initial segment o f N , 89 Inner product, 145 Inner product space, 145- 1 54 Inner regular measure, 192 Integral for a standardization, 1 70 with respect to a measure, 1 83 Integral operator, 1 38 Integration structure, 1 66- 1 67 hyperreal, 1 66- 1 67 internal, 1 66 - 1 67 real, 1 66 - 1 67 Interior point, 1 1 8 - 1 1 9 Intermediate value theorem, 46, 50 Internal entity, 95 -97 Internal formula, 95
Lebesgue measurable set, 1 9 1 Lebesgue measure, 1 9 1 Lebesgue monotone convergence theorem,
197 Leibniz, W. G., ix, 1, 53 Lifting, 1 86, 200 Lightstone, A. H., 100 Limit, 45 Linear functional, 1 35, 1 40 - 1 44 Linear operator, 1 3 5 - 1 40 Linear subspace, 1 33 Locally compact space, 1 58 - 1 59 Loeb measure, xii Loeb space, xii ..l:os theorem, 68-69, 86-87 Luxemburg, W. A. J., 70, 92, 93, 100, 104,
105, 1 06, 1 08, 1 1 8, 1 22, 1 3 1 , 1 54, 1 57 M
Machover, M., 1 57 Mapping •. 7 Maximal element in an ordered set, 220
230
Index
Maximal orthonormal set (basis), 149, 1 5 1 - 1 52 Maximum of functions, 1 67 Mean value theorem, 55 Measurable function, 175- 1 78, 1 82 - 1 83 Measurable set, 1 76, 1 80- 1 8 1 Measurable space, 180 Measure, 1 80 Measure space, 1 80 Metric space, 1 23 - 1 32 completion, 128 Minimum of functions, 1 67 Monad, 26, 106, I l l , 124, 1 35 Monomorphism, 79, 86-88 strictness, 79 Monotone convergence theorem, 1 64, 1 7 1 - 1 72, 1 78, 1 97 Moore, L. C., Jr., 109, 144, 1 54 Mostowski collapsing function, 84-88 N
Near point, I l l, 1 24 Near-standard point, I l l , 1 27 Negative part of function, 1 67 Neighborhood, 1 10 Neighborhood base, 1 10 Neighborhood subbase, 1 10- 1 1 1 Neighborhood system, 1 10 Newton, 1., ix Nonmeasurable set, 1 88 Non-standard entities, 90 Nonstandard hull, 144 Nonstandard hull of a metric space, 1 54 - 1 56 Nonstandard hull of a normed space, 1 55- 1 5 6 Nonstandard number system, 5 Nonstandard summation operation, 36-37 Norm of a linear operator, 135 Norm on a vector space, 1 33 Normal distribution, 2 1 4 Normal space, 1 1 8 Null function, 1 69 Null space of a linear operator, 1 36 Number finite, 25 infinite, 25, 28 infinitesimal, 25 non-standard, 25 standard, 25
Numbers finitely close, 26 infinitesimally close, 26 near, 26 0
One point compactification, 1 59 Open ball, 1 23 Open covering, 42, 1 20 Open map, 1 2 1 Open set, 40 , 1 10, 1 1 2 Open subcovering, 42, 1 20 Operator of finite rank, 1 52 - 1 53 Ordered n-tuple, 72 Ordered pair, 72 Ordering partial, 94 total, 94 Orthogonality, 1 46 Orthonormal basis, 1 49 Orthonormal sequence, 149 - 1 52 Orthonormal vectors, 1 49 Outer regular measure, 1 92 p
Parallelogram law, 147 Parameter, 1 2 Parameter set of a stochastic process, 207 Parentheses, 74 Parseval's identities, 1 5 1 - 1 52 Partially ordered set, 220 Partition, 56-57 refinement, 57 Path of a stochastic process, 2 1 6 Perkins, E . A., 205 Permanence principle, I 00- 1 04 Poisson process, 2 10-2 1 3 Polynomially compact operator, 1 52 Positive linear functional, 1 66- 167 Positive part of function, 1 67 Power set, 7 1 cumulative, 7 1 Pre-near-standard point, 1 27 Probability measure, 207 Probability of an event, 206 Probability space, 207 Projection in a Hilbert space, 1 48 Pseudomonad, 1 14 - 1 1 5, 1 1 8
23 1
Index
Q Q-compactification, 1 56 - 1 59 R Rado's selection lemma, 94 Random variable, 207 i nternal, 2 1 4 •- i ndependent, 2 1 4 S-independent, 2 1 5 Rank, 7 1 Rate o f a Poisson process, 2 1 0 Reflexive normed space, 1 43 Regular measure, 1 92 Regular space, 1 1 8 Relation •-transform, 9 binary, 9, 72 complement, 8 concurrent, 9 1 domain, 9 , 7 2 finitely satisfiable, 9 1 n-ary, 8 , 72 range, 9, 72 unary, 8 in V(X), 72 Relational system, 1 1 Remote po i nt, I l l , 1 90 Riemann integral, 57-60 Riemann integration, 56-60 Riemann sum, 56-51 Riesz representation theorem, 1 48, 194 Riesz-Fischer t h eorem , 1 5 1 Ring. 1 75 Robi nson, A., i x - x, 32, 42, 63, 70, 78, 9 1 , 92, 100, 102, 1 03, 109, 1 1 8, 1 20, 1 3 1 , 1 36, 1 38, 1 52, 1 57 Robinson's sequential lemma, 1 0 1 Robinson's theorem, 42, 1 20 - 1 22, 1 32 s Sam ple space, 206 - 207 Saturation, 1 04 - 1 08, 1 1 7 Scalar multiplication, 1 3 3 Schwarz ineq uality, 1 45 Sco pe of a quanti fier, 75 Second dual space, 1 40, 142 Sentence, 75 a tomic, 1 3
compound, 1 3 simple, 1 3 transfer, 2 1 •-transform, 20, 79 truth, 1 6, 7 5 - 76 Separable Hilbert space, 1 49 - 1 50 Separation properties, 1 1 3 - 1 1 4 Sequence, 32-.36 bo u nded, 3 3 - 35 Cauchy, 33-34, 126- 1 2 7 con vergence, 3 2 , 34 -35, 1 1 9, 1 26 double, 35 - 36 limit. 32. 34 - 36, 1 2 6 - 1 2 7 limit inferior, 37-38 limit point, 34-35, 1 26 limit superior, 37-38 Sequence of function s , 60-63 con vergence, 60-6 1 equicontinuou s , 62, 6 3
uniform con vergence, 36, 60-62, 1 2 5 - 1 26 u n i formly bou nded, 62 Sequentially compact space, 1 30 Series, 36- 38
absolute con vergence, 37 -38 convergence, 3 6 - 37 ratio test, 38 Se t dense, 1 1 3 e xh austing. 9 1 -92 •-finite, 89 •-open, 1 1 1 Sigma-algebra of sets, 1 80 Simple function, 1 8 2 reduced representation, 182 S-integrability, 1 86- 1 88 Skolem function, l 8 Smith, K. T., 92 Spillover principle, 1 0 1 Standard enti ty, 90 Standard formula, 95 Standard numbers, 7 Standard part, 27, I l l Standard part map, 27, 4 1 , 1 1 4 Standard part of a function, 1 72 Standard part of a set, 4 1 , I 07, 1 1 7, 1 22, 1 32 Stand ard sentence, 95 Standardization of an int egration structure, 1 70 Stochastic process, 207 Stone, M. H., 1 79
232
Index
Stone-tech compactification, 1 5 8 - 1 59 Stonian integration structure, 1 79 Sronian lattice, 1 79 Stroyan, K., 1 00, 105 Subbase, 1 10 Subgraph, 92 sup function, 1 25 Superstructure, 7 1 Support of a function, 1 67 Symbol connective, 12, 74 constant, 1 2, 74 equality, 74 function, 1 3 logical, 1 2, 74-75 predicate, 7 5 quantifier, 1 2, 74 relation, 1 3 variable, 1 2, 74 T
Term, 1 3 constant, 1 3 interpretable, 1 5 •-transform, 20 Tonelli's theorem, 204 Topological space, 1 10 Topological vector space, 145 Topology, 1 10 compact convergence, 1 63 compact-open, 1 60- 1 63 discrete, I l l finite complement, 1 1 2 half-open interval, I l l Hausdorff, 1 14 jointly continuous, 1 60- 1 62 normal, 1 1 8 pointwise convergence, 160- 1 62 product, 1 1 2, 1 16 - 1 19, 1 2 1 - 1 22 regular, 1 1 8 relative, 1 1 5 - 1 1 6 stronger, I l l trivial, I l l uniform convergence on compact sets, 1 6 3 weak, 1 1 6, 1 1 8, 142 - 1 44 weaker, I l l
Totally bounded metric space, 1 29 Transfer principle, 2 1 , 67-69, 78-83 downward, 79, 82-83 Triangle inequality, 1 23, 1 3 3 Trichotomy law, 6 Tychonotr product theorem, 1 2 1 - 1 22
u
Ultrafilter, 3, 2 1 9 fixed, 3, 88, 22 1 free, 3, 8, 22 1 principal, 22 1 Ultrafilter axiom, 4, 220 Ultrapower, x, 5, 84 bounded, 84 Uniform boundedness theorem, 1 3 7 Uniform continuity, 47 -48, 1 25 - 1 26 Uniform convergence, 36, 60-62, 125, 1 26 Upper bound in an ordered set, 220
v
Variable, bound, 7 5 Variable, free, 75 Vector addition, 1 3 3 Vector space, 1 3 3 Volume o f revolution, 60 w
Wave equation, 65 Weak compactness, 143 Weak operator topology, 154 Weak sequential compactness, 143 Weak topology on normed space, 142- 143 Weak • sequential compactness, 1 44 WeaP topology on normed space, 142 - 1 44
z
Zakon, E., 78, 79 Zero vector, 1 3 3 Zorn's lemma, 220
Pure end Applied Methemetlce A Series of Monographs and Textbooks
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