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(0,1] such that ip(j, •) is upper semicontinuous for j e N and ¥>(j + 1, t) <
V(j,*) = 0 } € J V for j € N ,
V»C? + M ) < ^CM) for J G N , * G / , Mv,= | J { i G / ; V 0 ' , 0 = 0}; obviously M^ £ Af and M$ is Borel measurable. Put D = {8 £ D*; 8(j, •) is Borel measurable for j £ N}.
32
HENSTOCK-KURZWEIL INTEGRATION
2.15 Theorem. Let 6 e D*. Then there exists rj e D with the following property: If G G A,g : I -»• R, M G M and if (2.13) and (2.14) are fulfilled, then
(2.26)
X>(ff)l<2-'+E;gj
for j e N, G = {(s, K)} G S(I, I, V(j, •)) and (2.27) there exists M* G M such that '£\G(K)-g(a)\K\\<2-' G
forjeN,Q
=
{(s,K)}eS(IJ\M*,v(j,-)).
Proof. Put V(J,t) = min{^(j,t),
for j eN,teI\
M^,
n(j, t) =
Y,\G{K)-g{s)\K\\<2-i e provided j G N, 0 = {(a, # ) } G 5 ( 7 , 1 \ M*,MJ, •))• Since 7j(j,<) < ¥>«(ji') f o r j € N, £ G I and r){j,t) < i>s(j,t) for j G N, t G I \ M*, it follows that (2.26) and (2.27) hold. The proof is complete. D
2. GAUGES AND BOREL MEASURABILITY
33
2.16. Denote by P the set of F G A such that F is the primitive of some integrable / : I —> R. Theorem. F G P if and only if there exist f : I —> R, 77 G D such that (2.26) and (2.27) hold. The "if part is a direct consequence of Theorems 1.20, the "only if part follows by Theorems 1.20 and 2.15. 2.17 Theorem. P is a vector space. If G{ : Iv(J) —» R, gi : I —► R, a G R and if G; is the primitive of gi for i = 1, 2, then G\ + G2 is the primitive of g\ + g2 and aGi is the primitive ofagi. Theorem 2.17 can be deduced from Theorem 2.16 or from Definition 1.4 and Lemma 1.8 in a standard way.
3. CONVERGENCE An elementary convergence result is presented in its usual form. 3.1 Theorem. Let I = [a,b] C R, 6 G D*, / , - : / - » R, 7; G R for z G N, / : 7 —► R. Assume that |7i-^/l(0l^ll<2-J
(3-1)
A
provided i, j e N, A = {(*, J ) } G S(7, / , <5(j, •)), A being a partition of I, (uniformity condition) (3.2) fi(t) —> f(t) for t G I, i —>• oo. (convergence condition) Then there exists lim 7; = 7 € R and (3.1) hoids if(~fi,fi) i—>oo
is replaced by ( 7 , / ) . 77mt If z, j G N and A = {(*, J ) } , A' = {(*', J')} are <5(j, •)fine partitions of 7 , then
l£/;W|./|-£/;WII<2-' +1 A
A'
and this inequality holds if fi is replaced by / .
□
Note 1. Let us mention that if fi : 7 —► R, 7; G R, i G N satisfy the uniformity condition (3.1) then the sequence of functions /j : 7 —► R is called equiintegrable. 34
3. CONVERGENCE
35
Note 2. Theorem 3.1 was published in [Kurzweil J., 1980], Theorem 5.2, where we have to put Ui(J, t) = ft(t)\J\, U(J, t) = f(t)\J\, and in [Kurzweil J. and Jarnfk J., 1991 - 92], Theorem 1.3. 3.2. A is the set of H : Iv(7) —> R which are additive and continuous (see Definition 1.6). Put||77|| = sup{|77(J)|;JGlv(7)}. The map 77 H-» \\H\\ is a norm on A and A becomes a Banach space if it is endowed with this norm (this can be proved directly or by the correspondence between additive functions on Iv(7) and real functions defined on 7, cf.Note 1.7). 3.3. Functions /;, i G N from Theorem 3.1 are integrable, 7; = fj fidt and it follows by Theorem 3.1 that / is integrable. Let Fi be the primitive of /; for i G N and let F be the primitive of / . By Theorem 3.1 we have Fi(I) — ► F(I) for i —> oo and analogously (3.3)
Fi(J)->F(J)
for every J G Iv(7) i f i - » o o .
It follows from the continuity of Fi, i G N and of F that (3.4)
Fi-^F
in A.
Moreover, Lemma 1.10 implies that (3.5)
^|Fl(J)-/i(0|J||<2-^1 A
provided i,j G N, A = {(t,J)} G S(I,I,6(j,-)) and it follows from (3.4) and (3.2) that (3.5) is fulfilled, if (Fiji) is replaced by(F,/). 3.4. Let / : 7 — ► R be integrable, F being its primitive and let g : I —> R. It follows from Lemma 1.15 and Theorem 1.17
36
HENSTOCK-KURZWEIL INTEGRATION
that g is integrable and F is the primitive of g if and only if g = f a.e. in 7. However, there cannot exist 6 G D* corresponding by Def inition 1.2 to every g whose primitive is F. Therefore it is convenient to transfer the accent from integrable functions to their primitives. Let P = P(I) be the set of primitives of functions integrable on I. P is a vector subspace of A (cf. Theorem 2.17) and the concept of jB-convergent sequences in P can be introduced in a natural way as follows: 3.5 Definition. Let F{, F G P for i G N. The sequence F{ is called E-convergent to F, Fi —> F, if there exist 6 G D* and integrable fi : I —*■ R, / : 7 —> R such that Fi is the primitive of fi, i e N, F is the primitive of / and (3.1) and (3.2) hold with 7, = Fi(I), i e N. Condition (3.2) is inconvenient and will be replaced by an other one. The following lemma is a preparatory result. 3.6 Lemma. Let rj E D, I = [a,b] F{ : lv(I) -► R, F : Iv(7) -»• R, fi : I -»• R, Mi € N for i e N. Assume that
(3.6)
52\Fi(J)-Mt)\J\\<2-i A
for i,jeN,A (3.7)
= {(t, J)} G 5(7,7 \ Mi, r,(j, •)), Fi(J) -* F( J) for J € Iv(7), i -^ oo.
Then (3.8) there exists / : 7 —> R such that fi —> f in measure for i —► oo,
(3.9) there exists M* G M and a subsequence i(k), k G N such that
fi{k)(t)^
f(t) forte I\M*,
k^oo
3. CONVERGENCE
37
and (3.10)
£|F(J)-/(*)|J||<2-' A
forj € N, A = {(*, J)} G
S(I,I\M*,6(j,-)).
(cf. [Kurzweil J. and Jarnfk J., 1997], Theorem 3.3). Proof. In order to prove (3.8) it is sufficient to show that the sequence / ; is Cauchy in measure. Let us suppose the opposite, i.e. that there exists a e R + such that for every r € N there are p = p(r), q = q(r) such that p,q > r and (3.11)
|{<eJ;|/p(0-/,(0l>*}l>2<7.
Let h € N be chosen such that 2~h+2 < a2.
(3.12) For fc e N put
Xk =
{xel;v{h,x)>-^}.
Xk is measurable, Xi D Xk for / > k and UfcgN -^fc = -^ so that lim \Xk\ — b — a. fc—»oo
Find k G N such that |AT*| >
b-a-a.
Denote
^ M = {*e/;l/p(0-/,(0l>^}-
38
HENSTOCK-KURZWEIL INTEGRATION
By Lemma 1.11 fi is measurable for i G N , hence Ep>q is mea surable as well. Consequently \Ep>q\ > 2a by (3.11),
\xk n EPtq\ + \xk \ EPiq\ = \xk\, \XknEp,q\>\Xk\-\I\Ep,q\, so that (3.13)
\Xk n Ep>q\ > a for all p = p(r),q = q(r), r € N.
Put a = a + £(6 - a) for / € { 0 , 1 , . . . , k}, (3.14)
Ji = [ci-Uct],
for / e { l , 2 , . . . , A ; } .
Let Cp>q be the set of / such that
;,nifcn£Mn(/\M+)/0, where M+ = \JieN Mi and let ti = titPtq be an element of Ji D
x fc n £Piq. Let us set &p,q =
{(ti,Ji);l£CP,g}.
Obviously Ap
£|Fi(J)-/i(0|J||<2-'1
for i G N, p = p(r), q = <j(r), r 6 N. Since evidently (fp(t) - fq(t))\J\
=
3. CONVERGENCE
39
= Fp(J) - Fq(J) - (FP(J) - fP(t))\J\) + Fq(J) -
fq(t)\J\,
(3.15) implies that
E I/PW - fS)\\A < 2~h+1 + E i w A
A
P,
- F<(J)'-
P.?
If (£, J) G AP)(?, then J is one of the intervals given in (3.14) and by (3.7) there exists r G N so large that \Fp(r)(J)-Fq(r)(J)\<2-h.
E A
P(T-),q(T")
This inequality together with (3.15) implies that (3.16)
E A
l/p(r)(0-/,(r)(0ll^l<2- h + 2 .
p(»-),«(r)
On the other hand, if (t, J) G Ap(r)j(7(r), then t G Ep^r^q^ and l/p(r)(0 ~"/g(r)(*)l ^ a - Moreover, AP)<7 was constructed in such a way that Xfc n EPtq n (/ \ M+) c ( J P ; («, ^) e AP)
Ei/p(«)-/«(oim>^EiJi^ff2' which contradicts (3.16) and (3.12).
HENSTOCK-KURZWEIL INTEGRATION
40
Hence the sequence ft is Cauchy in measure and it follows that there exist / : / —» R such that (3.8) holds. Moreover, there exist a subsequence i(k), k G N and M* G A/", M* D [Jien Mi such that fi(k){t) -> f(t)
for
k-^oo,teI\M\
The validity of (3.10) follows from (3.6), (3.7) and (3.9). The proof is complete. □ Theorem 3.1 is a starting point to 3.7 Theorem. Let rj G D, F{ G A, ft : I -~* R, Mi G Af for i G N, F G A. Assume that
(3.17)
£ \Fi{K)\ < 2-^ + ^2 e
forij
e
G N, 9 = {(5,/f)} G
(3.18)
\K\ ^'s)
S(I,I,r,(j,')),
^|Fi(A')-/i(«)|A'||<2^ e
for «,j G N, O = {(s,K)}
G
S(I,I\Mi,V(j,-)) (uniformity conditions)
and (3.19) Fi —» F in A.
(convergence condition)
Then (3.20) ft is integrable and Fi is its primitive for i G N, (3.21) there exists f : / —> R such that ft —* f in measure for i —► oo,
(3.22) there exist M* G A/" and a subsequence i(k), k G N such that M* D M; for i £ N and /*(*)(*) -+ /(*) for
t G I\M*,
k^oo,
3. CONVERGENCE
(3.23)
41
EW)l<2-+E^fL
for j G N, 6 = {(s,K)}
(3.24)
e «S(/,/,7,(j,-)),
52\F(K)-f(s)\K\\<2-1 e
for j G N, 0 = {(s,K)} G S(I,I\M*,ri(3,-)), (3.25) / is integrable and F is its primitive. Proof. (3.20) follows by Theorem 1.20 while (3.21) and (3.22) follow by Lemma 3.6. (3.23) is a consequence of (3.17) and (3.19) while (3.24) is a consequence of (3.18), (3.19) and (3.22). Moreover, (3.25) follows by Theorem 1.20 from (3.23) and (3.24). The proof is complete. D 3.8 Definition. Denote by CxBlCp(i4) the set of convex balanced compact subsets of A. For 77 G D denote by Q(r}) the set of G G A such that there exist g : I -»• R, M £ Af and
(3.26) forjGN, e = (3.27)
£l*WI^' + E f i {(s,K)}eS(I,I,V(j,-)), £|G(ff)-0(*)|ff||<2->" e
for j € N, 0 = {( S , K)} e S(I, I \ M, V(j, •))• 3.9 Theorem. Q : D -► CxBlCp(A). Proof. Let 77 G D. It follows directly from the definition of Q(r}) that Q(t]) is a convex balanced subset of A. For G G Q(r))
42
HENSTOCK-KURZWEIL INTEGRATION
put HG(a) = 0, HG{t) = G([a,i\), t G (a,ft]. It follows from (2.26) that {HQ; G G Q(v)} ls a s e t of uniformly bounded and uniformly continuous functions. Let Fi G Q(rj) for i G N. By the Arzela - Ascoli theorem there exist F G A and a subsequence i(k), k € N such that ^i(fc) —► -f1 i n ^4 for k —► oo. By Theorem 3.7 there exist f : I -+ R and M* <E Af such that (3.23) and (3.24) hold. It follows that F e C?(»7) and Q{-q) is compact. The proof is complete. □ 3.10 Theorem. (i) P = Ur,eDQ(v). (ii) Let 771,772 G £>. Then there exists TJ3 E D such that
Q(vi) + Q(v2)cQ(v3). Proof, (i) is a consequence of Theorem 2.16 and it can be verified that (ii) holds for 773 defined by rfy(j,t) = -min{77i(j-f l,t),T]2(j + l,t)}, The proof is complete.
j G N, t G / . □
Theorem 3.7 is a basis for a new concept of convergence in the space of primitives P. 3.11 Definition. Let F{, F G P for i G N. The sequence F{ is said to be Q-convergent to F, F{ —*• F, if (3.28) there exists 77 G D such that F{ G Q(r}) for i G N, (uniformity condition) (3.29) Fi —> F in A. (convergence condition) Comparison of F-convergence and Q-convergence is made in the following theorem.
3. CONVERGENCE
43
3.12 Theorem. Let Fi, F e P for i G N. (i) HFt -^
F, thenFl
- ^ F.
(ii) If Fi —> F, then there exists a subsequence i(k), k G N IT
such that F{(k) — > F for k —> oo. Proof. Let Fi —> F. There exist 6 e D*, integrable / ; : / — > R, / : / -» R for i € N such that F; is the primitive of fi, i G N, F is the primitive of / and (3.1) and (3.2) hold with 7* = Fj(7). It follows by Lemma 1.10 that
'£\Fi(J)-Mt)\J\\<2-* A
for i,j G N, A = {(t, J)} G 5(/,/,«5i(j,-)), where ^(j,*)) = (3.2) implies that there exists /i : / —> R + such that |/i(<)| < MO for i G N,
*G /
so that
J2\Fi(J)\<2-J + ^\J\h(t) A
fovi,jeN,A
=
A
{(t,J)}eS(i,i,61(j,-)).
Let <52 G D* fulfil /i(0 < — \ — , <52(j, *) < 61 (j, 0 for j G N, 02(1,0 f €/. It follows that (2.13) and (2.14) are fulfilled if G = F , g = fi, i G N, M = 0, <5 = 62By Theorem 2.15 there exists n e D such that (2.26) and (2.27) hold if G = Fi, g = fi, i € N, which implies that F; G Q(T/) for i G N. Moreover, F; -► F in A by (3.4), so that Ft -^ holds.
F and (i)
44
HENSTOCK-KURZWEIL INTEGRATION
Let Ft - ^ F. By Definition 3.11 Fi -> F in A for i -» oo and there exists rj e D such that F{ € Q(f?) for i G N. Since Q(r7) is compact (cf. Theorem 3.9) we have F G <3(?7). By Definition 3.8 there exist /,- : 7 -» R, / : 7 -» R, A/,-, M G A/" such that (3.30)
provided j , j 6 N , 0 = {(5, K)} 6 5(7, 7, ^(7, •), (3.31)
Y,\WK)-h{s)\K\\<2-i e provided i,j G N, 0 = {(5, tf)} G 5(7, 7 \ Mu v(j, •)»
£|F(*)-/(*)I*II<2-' e provided j € N, 9 = {(s, # ) } G 5(7, 7 \ M, 77C7, •)• Theorem 1.20 implies that (3.32) Fi is the primitive of fi for i G N and F is the primitive of/. It follows by Theorem 3.7 (cf. (3.21) and (3.25)) that the sequence fi converges in measure to a function the primitive of which is F so that fi —► / in measure for i —* 00. By (3.22) there exist a subsequence i(k), k G N and M* G M, M* D Mi for i G N such that /<(*)(*) -* /(*) for : ^ G 7 \ 717*, & — 00. Put 9i(k)(t) = / i ( f c ) (0, 5(0 = /(<) for * G 7 \ M\k
G N.
3. CONVERGENCE
45
9i(k)(t) = 0, g(t) = 0 for t G M*, k G N. Obviously (3.33) Fj(fc) is the primitive of &(fc) for k G N, F is the primitive of £ and (3.34)
9i(k)(t) ^ g(t) for * e / , * - oo.
Let j , fc G N, A' = {(*, J)} G 5(7, 7, V(j + 2, •))• Put A'2 = {(«, J) £ A ' ; t £ M*}, A[ = A' \ A'2. Then (cf. (3.30) and (3.31)) (3.35)
(3.36)
^
^2-^
l F l { k ) { J )
J ] \Fm(J)
+
^
-
^
r
- 9i(k)(t)\J\\ < 2- J - 2 .
*;
By Lemma 1.15 we conclude that there exists <5i G D* such that
(3.37)
Y
,)K[
if j G N, 0 = {(*,*)} G Put
, <2-^~ 1
S(I,M*MJ,.)).
<5(j, s) = min{77(j + 2,t), S^j, t)} for j G N,i G 7. Let now j,A: e N and let A = {(t, J)} G S(I,I,6(j,-) partition of 7.
be a
46
HENSTOCK-KURZWEIL INTEGRATION
Put A 2 = {(*, J) 6 A; t e M*}, Ai = A \ A 2 , 7i(fc) = *i(fc)(-0 for A;€ N. Then (cf. (3.35), (3.36), (3.37))
(3.38)
| 7i(fc) - £>(*)(<)|./Il < J Z l*i(fc)(J) - 9i(k)(t)\J\\ < A
£
A
\FHk)(J) - 9m(t)\J\\
+ ^
\FiW(J)\
<
A,
< 2~ J '- 2 + 2 _ J ' - 2 + ^
I ' l < 2~j~2 + 2~J~2 + 2~j~1 < 2~j. -*-*—
(3.38), (3.33), (3.34) imply that Fi{k) Theorem 3.12 is complete.
- ^ F.
The proof of □
The next theorem shows that ^-convergence and <5-convergence are equivalent from the point of view of topology. 3.13 Definition. Let W b e a topology on P. U is said to be tolerant to E-convergence if F{ —► F implies that F{ —► F in (P,U). Analogously U is said to be tolerant to Q-convergence if Fi -£+ F implies that F{ -» F in (P,W). 3.14 Theorem. Let U be a topology on P. Then U is tolerant to E-convergence if and only if it is tolerant to Qconvergence. Proof. The " i f part follows directly from Theorem 3.12 (i). Assume that U is tolerant to E-convergence but is not tolerant to Q-convergence. It follows that there exists a sequence F,-,
3. CONVERGENCE
47
i G N and F such that F; —► F and that there exist U e U and a subsequence i(k), k G N such that F e U and (3.39)
Fi(Jfc) $U
Obviously F;(fc) —> F.
for i e N .
By Theorem 3.12 (ii) there exists a
subsequence k(l), I G N such that F^k(i)) —* F. Therefore there exists m G N such that F^k(i)) G £/ for / > m, which contradicts (3.39). The "only if part is valid and the proof is complete. □
4. A N A B S T R A C T SETTING
4.1. In this chapter the concept of (^-convergence from Chap ter 3 will be studied from a more general point of view. Let X be a Banach space, ||rc|| being the norm of x G X, E C X. For x e X, p G R+ put B(x, p) = {ye X; \\y-x\\ < p} and B{p) = {y G X; \\y\\ < p}, dist(x,£) = inf{||:c-y||;
yEE}.
By U{\\ • ||) we denote the topology on X which is induced by the norm. If Y c X, let U(\\ ■ \\)\Y be the restriction of U(\\ ■ ||) to Y, i.e.
«(IHI)lv = {£/ny;tf€W(|H|)}. We will write {Y,U(\\ ■ ||)) instead of (Y,U(\\ ■ \\)\Y). By CxBlCl(X) we denote the set of convex balanced closed subsets of X. Let D be a set, D ^ 0, Q : D -> CxBlCl(X") and assume that (4.1) for any di,d,2 G D there exists d.3 G D such that Q(di) +
Q(d2)cQ(d3). 48
4. AN ABSTRACT SETTING
49
Put (4.2)
P = U{Q(d);deD}.
It follows from (4.1) that P is a vector space. Let £; G P for i G N, x e P . The sequence X{ is called Q-convergent to x, Xi —► x, if there exists d e D such that X{ G Q(d) for i G N and if X; —► x in X for i —> oo. Observe that all the above conditions are fulfilled, if we put (*)
X = A,P=P,D
= D,Q = Q.
4.2 L e m m a . If Xi —► x, then (XJ — x) —► 0 for i —> oo. Proof. There exists d £ D such that X{ G Q(d) for i G N and ||x* — a;|| —>■ 0 for i —* oo. Moreover, x G Q(d), since Q(d) is closed. By (4.1) there exists c/3 G D such that 2Q(d) C Q(ds) and therefore G Q ( ^ ) for i G N. Consequently, (x; — x) —► 0 for i —► 00 and the proof is complete. 4.3 Definition. Let U be a topology on P. U is called Q
tolerant to Q-convergence, if x, —► x implies that the sequence X{ is convergent to x in the topological space (P,U). 4.4 T h e o r e m . Let U be a topology on P. Then the follow ing conditions are equivaient: (4.3) U is tolerant to Q-convergence, (4.4) for every d G D the identity map
i(d):(Q(d)M\\-\\))-(PM) is continuous, (4.5) if di G D, x G t/ G W, then there exists ( 6 K + such that
x + Q(di)nB(c)ct/.
□
50
HENSTOCK-KURZWEIL INTEGRATION
Proof, (i): (4.3) implies (4.4). Assume that (4.4) does not hold. Then there exist d G D and x G Q(d) such that the identity map is not continuous at x. There exist Xi G Q(d) for i G N such that Xi —> x in X but the sequence X{ is not convergent to x in (P,U). It follows that (4.3) does not hold. (ii): (4.4) implies (4.5). Assume that (4.4) holds. Let dx G D, x G U G U. There exist diAz G -D such that a; G Q(d 2 ), Q{d\) + Q(d 2 ) C Q(rf3). Since the identity map i{dz) is continuous and x G Q(d3), there exists ( 6 R + such that Q(d3) D B(x,C) C £/. Moreover, x + Q(d 1 )DS(C) C (ar + Q(d!))nB(a:,C) C Q{d3)nB(x,0
C £/
and (4.5) holds. (iii): (4.5) implies (4.3). Assume that (4.5) holds. Let Xi —► x so that there exists d G D such that a?; G Q(d) for i G N and X{ -> x in X . Since Q(d) is closed, we have x G Q(d)- By (4.1) there exists d\ e D such that 2Q(d) CQ(di). It follows that Xi = x + (xi - x) G x + Q(di) n B(\\xi - rc|| + -). i
If x E U £ U, then Xi E U provided i is sufficiently large. Therefore (4.3) holds. The proof is complete. □ 4.5 T h e o r e m . Let U be a topology on P. Assume that ( P , U) is a complete topological vector space and that U is finer thanW(||-||)|p. Then (4.6) if 3 = {Z} is a Cauchy filter in (P,U) and if 3 —► x in X, then x G P, (4.7) if 3 = {Z} is a Cauchy filter in (P,U) and if 3 ->• 0 in X, then3->0 in(P,U).
4. AN ABSTRACT SETTING
51
Proof. Let 3 = {Z} be a Cauchy filter in (P,U). Since (P,U) is complete, there exists y G P such that 3 —► y in (P,U). The assumption that ZY is finer than U(\\ • \\)\p implies that 3 -» y in X . Hence both (4.6) and (4.7) follow. □ 4.6 Theorem. Let U be a topology on P. Assume that (P,U) ^ a topological vector space, thatU is finer than U(\\-\\)\p and that (4.6) and (4.7) hold. Then (P,U) is complete. Proof. Let 3 = {Z} be a Cauchy filter in (P,W). By as sumption U is finer than U{\\ ■ \\)\P SO that 3 is a Cauchy filter in X and there exists x G X such that 3 —► x in X. It follows from (4.6) that x G P . Put 3 - x = {Z - x\ Z G 3} so that 3 - x -* 0 in X and 3 - x is a Cauchy filter in ( P , W). By (4.7) we get 3 — x —*■ 0 in (P,U) a n d consequently 3 -* £ in {PM)-
□
4.7. For p : D -»• R + put
Obviously i?(p) C P . Theorem. Let U be a topology on P Then (i) W is tolerant to Q-convergence if and only if + (ii) for x e U eU there exists p : D - R such that
x + R(p) C U.
The proof follows immediately from Theorem 4.4. 4.8. For Z C X denote the convex hull of Z by convZ.
HENSTOCK-KURZWEIL
52
INTEGRATION
Definition. Let Wlc
= W*LC(P,Q)
= {convR(p);p:
D - M+}.
Let U1C be the set of U c P with the property if y G £/, then there exists W G 2H/,c s u c n * n a t 2/ + W U. 4.9 Lemma. 2tf£ c (A.21).
/ui
c
^ s (A-13)> (A.15), (A.16), (A.20) and
Proof. It is obvious that (A. 13) and (A.20) are fulfilled. Q(d)D B(p(d)) is balanced for d G D so that R(p) is balanced and ab sorbing for p : D -> R+ and (A.21) holds. Let pi : D -» R+ for i G {1, 2}. Put p 3 (d) = min{pi(d),p 2 (d)} for d e D. Then R(p3) C #(pi) n #(p 2 ) and (A.15) holds. Let us prove that (A.16) holds. By (4.1) there exists A(d) G D for d G -D such that 2Q(d) C Q(A(d)). For p : D -» R+ put A(d) = |p(A(d)). If x,y G convi?(A), then a: = E i = i <2iW;, 2/ = Y?j=i0jzj w h e r e w * e WO n 5 ( A ( rf t))> *i e Qid'j) n B(A(d$)), "»,)% G R + U { 0 } , di,dj G Z> for i G {1,2,...,*:}, i G {1, 2 , . . . , /}, Zti
a» = 1, E j = i & = 1. We have
fc
J
-
4. AN ABSTRACT SETTING
53
Since Wi G Q(di), \\wi\\ < A(rf,), it follows that 2w{ G Q(A(di)), \\2wi\\ < p(A(di)) for i G {1,2,...,/:} and analogously 2ZJ G Q(A(d$)), \\2Zj\\ < p ( A « ) ) for j G { 1 , 2 , . . . , / } . Hence fc
i=l
*
„
J=l
conv U {Q(A(d)) n fl(p(A(d));d e £>} C convfl(p) and (A. 16) holds. The proof is complete.
□
4.10. Put WLC = ZV(2U2C), cf- Corollary A.5. Theorem, i / j ^ has the following properties. (4.8) W£c is a topology on P and (P, U*LC) is a locally convex space, (4.9) U*LC is finer than W(|| • ||), (4.10) WLC\Q{d)=U{\\-\\)\Q{d)fordzD, (4.11) U*LC is tolerant to Q-convergence, (4.12) if V is a topology on P such that (P, V) is a locally convex space and V is tolerant to Q-convergence, then V is coarser than U*LC. Proof. Since W*LC fulfils (A.13), (A.15), (A.16), (A.20) and (A.21), Ulc is a topology on Y and (Y,Ulc) is a locally convex space by Theorem A. 12 and (4.8) is valid. (4.9) follows directly from Definition 4.8. Let d E D, x G Q(d), x G U G W£ c . Definition 4.8 implies that there exists p : D —*• R + such that x + convR(p) C U. By (4.1) there exists d^ G D such that 2Q(d) C Q(d3).
HENSTOCK-KURZWEIL INTEGRATION
54
Consequently B{x, p{d3)) n Q(d) = x + B(p(d3)) n (Q(d)
-x)c
Cx + B(p(d3)) n Q(d 3 ) c x + R(p) C U, which together with (4.9) implies (4.10). Moreover, (4.11) is a consequence of (4.10). If y G V G V, then there exists a convex neighbourhood of zero V\ in V such that y + Vi C V. Since V is tolerant to Q-convergence, Theorem 4.4 implies that there exists p : D —> R + such that R{p) C V\. It follows that conv i?(/>) C Vi, since V\ is convex. Therefore y+convi?(/9) c V so that V is coarser than U*LC. (4.12) holds and the proof is complete. Note. U*LC is the finest of all topologies U on P such that (i) (P,U) is a locally convex space, (ii) the identity map i(d) : Q(d),U(\\ • ||)) -» (P,W) is con tinuous iov d £ D (or equivalently, ZY is tolerant to Qconvergence, cf. Theorem 4.4).
□ 4.11 T h e o r e m . Let E : P —* M be a linear map fulfilling (4.13)
£; —► rr implies E(x{) —> H(x).
TAen (4.14)
H is continuous in
(P,Ulc).
Proof. Let e > 0. It follows from (4.13) that for every d € D there exists p(d) € R + such that |S(x)| < e for a: G Q(d) n B(p(d)). Since E is linear, we have \E(y)\ < e for y G conv#(p) G 20^(7• The proof is complete. □
5. A N A B S T R A C T SETTING WITH D
COUNTABLE
5.1. The setting from Chapter 4 is continued and the space (P,Utc) ls studied (cf. Theorem 4.10) with the additional as sumptions
(5.1)
D =N
and (5.2)
2Q(i) C Q(i + 1) for i e N.
Obviously (4.1) is fulfilled and therefore definitions and results from Chapter 4 can be used. Of course, Q(i) is convex, balanced and closed for i 6 N . In the present notation we have (5.3)
/t>:N-R
+
,
R(p) = (j Q(i) n B(p(i))
and (5.4) {convR(p);p : N —>• M + } is a 0-neighbourhood base in The main result of this chapter are Theorems 5.2 and 5.7. 55
56
HENSTOCK-KURZWEIL INTEGRATION
5.2 Theorem. Ulc has properties (4.8) - (4.12) and, more over, (P,Ulc) is a complete space. WLC h a s properties (4.8) - (4.12) by Theorem 4.10. For showing that (P, U*lc) is also complete the following lem mas are needed: 5.3 Lemma. Let 3 be a Cauchy filter in (P,Ulc), 3 —* x in X. Then x G P.
x G X,
Proof. Assume that x G X\P. Put a(i) = dist(x, Q(i)) so that a{i) > a(i + 1) > 0 for i G N since Q(i) is closed in X, Q(i) C Q(i + 1) and P = U{Q((»); « € N } . Let 0 : N -> R+ fulfil (5.5)
0(i) < \a(i),
0(i + l ) < 0 ( i )
for i G N. There exists Z G 3 such that Z - Z C conv#(0). Let z G Z. There exists / G N such that z G Q(/). Since 3 —► £ in X, there exists y £ Z such that
||i/-x||
= Y^ctiWi, i=i
where a , G l + U {0}, £ * l i Qi = 1, and w{ G Q(i) n 5(0(t)) for i = 1,2,..., A:. It may be assumed that k > I + 1 so that 1
(5.6)
k
z - Y^aiWi - x = y- x+ ^ i=l
i=l+l
a tWi.
5. ABSTRACT SETTING WITH D COUNTABLE
We have 5Z i = 1 aiW{ G Q(l), z - Y!>i=i aiwi (5.7)
\\z-^2aiWi-x\\>cr(l
e
57
Q(l + 1) s o that
+ l).
On the other hand, (5.6) implies that i
k
\\Z -^QLiWi i=\
- X\\ < \\y -Z\\+
< l
^2 i=l+l
a
i\\wi\\
<
l
The contradiction resulting from (5.7) proves the lemma.
□
5.4 Lemma. Let p, a : N —> R + where p(j + 1) < p(j) and
conv | J Q(j) n 5(p(j)). J'=3
Proof. Assume that i
u G conv[|J Q(j) n B((r(j))] + Q{1 + 1) D B(2<x(/ + 1)). Then there exist Wj G Q(j') n B(a(j)), aj G K + U {0} for j G { 1 , 2 , . . . , / } , £)J = 1 ^ = 1 and v G Q(l + 1) n B(2cr(/ + 1)) such that a i t w j + t ; = 2 ^ -f*Wj + 3=1
i=i
--v.
HENSTOCK-KURZWEIL INTEGRATION
58
We have (cf. (5.2)) 4itfj e Q(j + 2) n B(4a(j)),
^v e Q(l + 3) n B(4a(Z + 1))
so that u G conv[|J Q ( j + 2) D 5(4cr(i)) U Q(/ + 3) D B(4a(l + 1))] C j=\
1+3
conv[\J
Q(j)nB(pU))]
3=3
and the lemma holds.
□
5.5 Lemma. Let 3 = {Z} be a Cauchy filter in and let 3 -»• 0 in X. Then 3 -» 0 in (P,W£ C ).
(P,Ulc)
Proof. Assume that 3 is not convergent to zero in (P,U£C). Hence there exists p : N —> M+ with p(i) > p(i + 1) for i G N such that Z \ convi?(p) ^ 0 for every Z G 3Let a : N -► R+, a(j + 1) <
2 G Zi \ conv#(p).
There exists I G N such that 2 G Q(/) and there exists y G Zi n B(a(l + 1)). Since z — y G Zi — Z\ C conv.R(cr), there exist /c G N, to,- G Q(j) n B(a(i)), a,- G R+ U {0} for j = 1, 2 , . . . , fc, ^ , = 1 ckj = 1 such that fc
z~y = J2aJwJ-
5. ABSTRACT SETTING WITH D COUNTABLE
59
It may be assumed that k > I + 1 so that by (5.2) we get k
i
j=i+i
i=i
It follows that k
/
a w
\\z- Y, j i\\<\\v\\+ E " J K H k
+ \),
z - E aj-Wj G Q(l + 1) n 5(2a(/ + 1)) and z € conv[|J Q(j) n B(cr(j))\ + Q{1 + 1) n £(2
By Lemma 5.4 we get
z G conv[(J Q(j) n B(p(j)) C conv#(p). J=3
The contradiction with (5.8) makes the proof complete.
D
5.6. Proof of Theorem 5.2. U*LC has properties (4.8) - (4.12) by Theorem 4.10 and {P,WLC) is complete by Theorem 4.6 and Lemmas 5.3 and 5.5. The proof is complete.
60
HENSTOCK-KURZWEIL
INTEGRATION
5.7 Theorem. Let xt -► 0 in (P,WLC).
Then xt - ^ 0.
Proof. Let Xi —► 0 in (P,Ulc) and let the theorem be false. Ulc is finer than U(\\ ■ \\)\p and this implies that ||xi|| —> 0. Therefore there exists i^') G N for j G N such that x^ $. Q(j) and i(j) > i(j — 1) for j > 1. Moreover, for j e N there exists A;(j) € N such that *,-(,■) e Q ( * ( j ) ) \ Q ( * ( j ) - i ) ; obviously we have k(j) > j . If necessary, the sequences i(j), k(j), j G N may be replaced by subsequences and without loss of generality it may be as sumed that k(j + 1) > k(j) for j G N. Let a : N -► R + fulfil
xi(l) G conv [j {Q(j) n £(cr(j))}
for some / G N. Then there exist m G N, Wj G Q(j), ||WJ|| <
i=i fc(0-l
m
i=i
j=fc(0
The norm of the right hand side of the second equality does not exceed a(k(l)). Since k(i)-i
Zi(i) e Q(*(0) \ Q(*(0 - 1), X
<W
e
G(*(0 - *)>
5. ABSTRACT SETTING WITH D COUNTABLE
61
the norm of the left hand side exceeds a(k(l)). This contradic tion implies that the assumption (5.9) is false and the proof is complete. □ Theorems 5.2 and 5.7 may be applied in the following exam ples. 5.8 Examples. (i) Let X = Co be the space of sequences x = (xi, X2, • • •) of reals Xj such that Xj —► 0 for j —* 00, ||x|| = max{|xj|; j G N}. Put Q(i) = {x C X; Xj = 0 for j > i}. Then
is the vector space of x G CQ such that the set {j\Xj ^ 0} is finite. (ii) X = A,6i€D, 6i+1(j,t) < \8i{j + 1,0, Q(i) = Q(6i) for i, j e N, t e I. Then
P = |J Q(t) c P. Since the elements of P are primitives of integrable functions, they will be denoted by F;, F as in the Chapters 1 and 2. Let Fi,F G P for i G N. If F{ - ^ F , then F< - ^ F , but Q-convergence does not imply Q-convergence (cf. Definition 3.11). U*LC is a topology in P ; it is tolerant to Q-convergence but it need not be tolerant to Q-convergence. (iii) Let X = C = C(/,R), the space of continuous func tions from / to E with the maximum norm. Let us define Y = {xeC;
\x(t) - x(s)\ <\t-s\
for t,sE
I}.
Put Q(i) = 2lY for i G N. Then P = |J{Q(?);iGN}is the subset of Lipschitz functions from C.
62
HENSTOCK-KURZWEIL INTEGRATION
5.9 Note. There is a link between the concept of ^-con vergence and the concept of two norms convergence, shortly 7-convergence, as it was introduced in [Alexiewicz A., 1950]. Let Y be a vector space over R. A functional x i-» \\x\\ (x E Y, \\x\\ E R) is called an F*-norm, if 1° ||x|| > 0, 2° \\x\\ = 0 if and only if x = 0, 3° ||* + y | | < | M | + ||y||, 4° A„ -»• 0, An E R implies ||A n x|| -> 0, 5° ||x n || -»• 0 implies ||Ax n || -► 0 for every A E R. Let Y be equipped with F*-norms || • ||i, || • H2 such that (m) || - ||i — 0 implies || - ||2 — 0. A sequence {xn; n E N} is said to be 7-convergent to y ( xn —^ y) if there exists k E N such that Iknlli < 2k for n E N, ll^n - vh —> 0 for n -+ 00. Let the following conditions hold. (5.10) Y is a vector space equipped with norms || • ||i, || • H2 (i.e. ||Aar||i = |A|||a:||t- for A E R, x E Y, i = 1,2) such that (ni) holds (i.e. there exists K e R + such that lk|| 2 < « N | i for xEY). (5.11) {x E Y; \\x\\i < 1} is closed in (Y, || • || 2 ). For Z C Y denote by compl(Z, || • H2) the completion of Z with respect to || • H2. Put (5.12)
D = N, Q(i) = {x E Y- \\x\\x < 2~1} for i E N, P = | J Q(t), X = compl(Y, || • || 2 )
5. ABSTRACT SETTING WITH D COUNTABLE
63
and let || • || be the continuous extension of || • IJ2 from Y to X (so that (X, || • ||) is a Banach space). Then P = Y and the Q-convergence coincides with the 7convergence. Example. For x : [0,1] -> R put ,. .
, \x(t) - x(s)\
1>(x) = suP{'
:t_)
.
.
;
1
; *, s e [0,1], t / s},
Y = {x : [0,1] -> R; x(0) = 0, ip(x) < 00}. Put ||x||i = 1>(x), \\x\\2 = sup{|x(0|; t G [0,1]}. for x e y , D = N, Q(z) = {x e K; |lrc|ji < 2'} for t G N. Consequently ||x|| 2 < ||x||i for
xeY'
X = compl(y,||.|| 2 ) = C 0 ([0,l],R), where C 0 ([0,1],R) = {x e C([0,1],R); x(0) = 0} is equipped with the supremum norm. Then X{ —> x if and only if X{ —► x. 5.10 Note. There is a link between the topology Ki^v anc * the mixed topology 7 [ T I , T 2 ] which was introduced in [Wiweger A., 1961]. Let y be a vector space, which is equipped with topologies Ti,T2 such that (y,TI), (Y,T2) are Hausdorff topological vector spaces and T\ is finer than r 2 . A. Wiweger introduced the mixed topology 7(7-1, r 2 ] by defin ing a corresponding 0-neighbourhood base and proved the fol lowing result (cf. (Wiweger A., 1961], (2.2.1) and (2.2.2)):
64
HENSTOCK-KURZWEIL
INTEGRATION
Let there exist an F*-norm || • ||i o n Y such that T\ is the topology of (Y, || • ||i). Then 7[n, r 2 ] is the finest of all topologies r on Y such that (5.13) (Y,T) is a topological vector space, (5.14) T2\v is finer than r | v , where for every ri-bounded V
CY.
Assume now that the conditions (5.10) and (5.11) hold and make use of the notation (5.12). Then P = Y and the topologies Uj>v and 7[TI,T2] coincide. 5.11. Results from this chapter can be extended to more general situations. Let X be a vector space, T = {T} a topology on X such that (X, T) is a complete locally convex vector space. Denote by Sj a 0-neighbourhood base of (X, T) consisting of convex balanced neighbourhoods of zero. Various concepts and results from Sections 4 and 5 can be transferred to this situation if the balls B((), C € R + are replaced by the sets H G -ft. By CxBlCl(JC) denote the set of convex balanced subsets of X which are closed in the topological space (X, T). Let Q : N -> CxBlCl(X),
2Q(i) C Q(i + 1) for i e N,
p = U «(«)■ Then P is a vector space. Let X{ G P for i e N, x G X. The sequence Xi is called Q-convergent to x, X{ —> x, if there exists d £ D such that Xi G Q(d) for i G N and if ^ —* a; in (X,T). Of course, x G Q(rf) and —► 0. The concept of a topology U on P which is tolerant to Q-convergence is introduced in the same way as in Section 4.3.
5. ABSTRACT SETTING WITH D COUNTABLE Theorem 4.4 can be transferred if we write (4.5') if di e D, x G U G U, then there exists Heft that x + Q{di) H H C U
65
such
instead of (4.5). Let us prove that (4.5') implies (4.3). Let x G U G U, There exist j G N, H G ft and k € N such that x{ E Q(j) for i € N, x{ - x G H for i > k. Then
x e Q(j), Xi-xe
Q(j + 1) for i e N,
Xi = x + (xi - x) e x + Q(j + l)nH
for i > k
and (4.3) holds. The remaining part of the transferred Theorem is proved in an analogous manner. Theorems 4.5 and 4.6 can be transferred as well. For p : N - ft put R(p) = (J i e N Q(i) n p(i), W*LC = {conv R(p); p : N -» ft}, WLC =
U(WLC).
Then Theorems 4.5, 4.6, 4.7, Lemma 4.9, Theorems 4.10, 5.2, Lemmas 5.3, 5.4, 5.5 and Theorem 5.7 can be transferred. For instance, the proof of Lemma 5.3 can be adapted as fol lows: Assume that x G X \ P. Since Q(i) is closed, there exists a : N -»■ ft such that (x + a(i)) n Q(i) = 0 for i G N. It can be assumed without loss of generality that a(i + 1) C a(i) for i G N. Let 9 : N -> iD fulfil 0(i + i ) c 0 ( t ) c i * ( t )
66
HENSTOCK-KURZWEIL INTEGRATION
for i e N . After this the proof can be continued with obvious changes. We conclude that Ulc is the finest of all topologies U such that U is tolerant to Q-convergence and (P, U) is a locally con vex vector space. Moreover, if and only if X{ —> x in For example we may put X = V = {x : R -^ R;x e
C{oo)},
the topology T on X being generated by the system of seminorms HI • Hl^jfe, where |||x|||;,fc = max{|x^)(*)|; t e H , i]J = 0 , 1 , . . . , A;} for i, k G N. Define Q(i) = {xeV;
x(t) = 0 for \t\ >i},ie
N.
Obviously Q(i) e CxBlCl(X) for i € N and
p = U w = p° is the set of functions x from £> with a compact support.
6. LOCALLY C O N V E X TOPOLOGIES T O L E R A N T TO Q-CONVERGENCE 6.1. In this chapter the space P together with Q-convergen ce from Chapter 3 are studied. Therefore U^c = U(W*LC(P, Q)), cf. Definition 4.8. The main result is Theorem. U(\\ • \\)\p is the finest of all topologies U on P such that U is tolerant to Q-convergence and (P,U) IS a locally convex topological vector space. Note 1. It is an immediate consequence of Theorems 6.1 and 4.10 that W £ c = W ( | H | ) | p . Note 2. Theorem 6.1 remains valid if Q-convergence is re placed by ^-convergence (cf. Theorem 3.14). Note 3. The above theorem was proved in [Kurzweil J. and Jarnfk J., 1998]. There are slight differences in the proofs; Q in this book denotes a mapping which is different from the map ping which is also denoted by Q in the quoted paper, but both mappings play the same role. 6.2. For K G Iv(J) let PK be the set of F e P such that
F(J) = o if J n K = 0, J e lv(J). If Fe P, put j FK(J)=<
F(J n K) for J n IntK ^ 0, o for J n Int/iT = 0.
67
68
HENSTOCK-KURZWEIL INTEGRATION
T h e o r e m . Let K(i) = [d,di] G Iv(J), F{ G PK^ re I. Assume that (6.1)
K(i + 1) C tf(i) for t € N,
(6.2)
{r}=f]
for i G N,
K(i),
||Fi|| - > 0 for z ^ o o .
Then there exists <5 G 7J) such that ^ G Q((5) for i G N. Note. It follows that Ft - ^ 0. Proof of the theorem. By assumption there exist <5; G Z?, /i : 7 —► R and Mj G A/" for i G N such that
,,3)
£i*<'>i * * - ' + ! : &
provided j G N and A = {(t, J)} G S(1,1,6i(j, (6.4)
•)),
^|Fi(J)-/i(t)|J||<2^ A
provided j G N and A = {(t, J)} G 5(7,7 \ Mit 6{(j, •))• Without loss of generality we may assume that (6.5)
6k(j, t) < 6i(j, t) for i,j,keN,k>i,te
Put (6.6)
0i(t) = dist(t, K{i)) for i G N, t G 7
I.
6. LC TOPOLOGIES TOLERANT TO Q-CONVERGENCE
69
and define 6 as follows: (6.7)
6(j,t) = e1{t) for
j£N,teI\K(l),
6(j, t) = min{0 i+ i(0,6i(j + i + l,t)} for i,j e N , ( £ K(i)\K(i
+ l).
Let 6(j, r) be so small that \Fi(J)\ < 2-J'- 2 if \\Fi\\ > 2-i~2, reJE
Iv(J), J C / n (r - <5(j, r), r + 6(j, r))
and let 0 < 6(j + 1, r) < £(j, r) for j G N. Obviously <5 G £>. Finally put
M = {r} U (J M/, where M/ G AT for i G N and M[ D M;U{* G 7\K(z); /<(*) # 0}. We have to prove that
(6.8)
£| F , ( 7 ) |< 2 - J
+
^_li
provided i j ' e N and A = {(*, J)} G 5(7, J, 6(j, •)), and (6.9)
^|Fi(J)-/t(0|J||<2^ A
provided i,j G N and A = {(*, J)} G 5(7,7 \ M, «5(j, •))• LetjGN, A = {(t,J)}eS(I,I,6(j,-)). Put A0 = { ( « , J ) G A ; f G / \ / < ( ! ) } ,
70
HENSTOCK-KURZWEIL INTEGRATION
Afc = {(*, J) G A; t G K{k) \ K(k + 1)} for k G N, A00
= {(t,J)G
A;t = r}.
Systems A^ (k G N U {0,oo}) are pairwise disjoint and their union is A. Let i e N. Since F; G PK{i), we get (cf. (6.7)) (6.10)
^2 \Fi(J)\ = °
for
A: = 0 , 1 , . . . , i - 1.
A*
Let ife G N, jfc > «. T h e n (cf. (6.3))
(Ml)
glfiWI<2->->-+gw+'^+M)
since Afc G 5(7, tf(fc) \ AT (A; + 1), 4 ( j + k + 1, •))• Moreover, (6.12)
^|Fi(J)|<2-^1
since at most two couples (£, J) are elements of AQQ. It follows from the definitions of 8 and A^ (k G N U {0, oo}) and from (6.10), (6.11), (6.12) that (6.8) holds. (6.9) holds by an analogous argument. The proof of Theorem 6.2 is complete.
□ 6.3 Lemma. Let p : D —> R + , r G I. m = ra(r) G N such that
Then there exists
PL{rn) n 5(2—) c (J(W); 6 e A P(<5) > 2—}, where L(m) = I D [r - 2 ~ m , r + 2 - m ] . Proof. If the lemma does not hold, then for k G N there exists
Fk G pL{k) n z?(2-fc) \ |J{ W);<* e AP(<5) > 2-fc}, which contradicts Theorem 6.2.
D
6. LC TOPOLOGIES TOLERANT TO Q-CONVERGENCE
71
6.4 Lemma. For p : D -» R + there exist £ G R + , r G N, a = Co < Ci < • • • < Cr = b such that r
(6.13)
| J Jfo_ltCi, n B(0 C | J Q(6) n B(p(«)). i=l
6&D
Proof. Put T/(T) = 2 - m ( T ) for r G /, m(r) being given in Lemma 6.3. By Lemma 1.2 there exists A = {(r i) [Ct-i,Ct]);i = l , 2 , . . . , r } e 5 ( / I / , 7 7 ) which is a partition of I, a = Co < Ci < ' ' • < Cr = b. Put f = min{2 _ m ( T i ) ; i = 1, 2 , . . . , r } . Then (6.13) holds by Lemma 6.3. The proof is complete.
□
6.5 Lemma. For p : D —> R + there exists A G R + such that P n B(A) C conv (J Q(6) n 5(p(d)). 6€D
Proof. Let p : D —> R + and let f and Ci, i = 1,2,..., r be defined as in Lemma 6.4. Put A = £. If F G P n B(A) then (6-14)
^ = £^F[c<_1;Ct], i=i
'%-!,<«] e % - i , C i ] n B ( 0 for i = l , 2 , . . . , r . It follows by (6.14) and (6.13) that r
F G conv | J P [Ct _ 1>G] n£(C) C c o n v ( J { W ) nfl(/>(6)); <5 G D}, which completes the proof.
□
72
HENSTOCK-KURZWEIL INTEGRATION
6.6. Proof of Theorem 6.1. Let U be a topology on P which is tolerant to Q- convergence. Theorem 4.4 implies that (4.5) holds so that (6.15) if F 6 U e U then there exists C : D -> R+ such that F + R(C) = F+{J
Q{d) n B(((d)) C U.
d£D
Assume in addition that (P,U) is a locally convex vector space. Let V be a convex neighbourhood of zero in {P,U). By (6.15) there exists p : D —► E + such that R(P)
c v
and, since F is convex, conv R(p) C V. By Lemma 6.5 there exists A G 1R"1" such that P n £(A) C conv R(p) c V. The proof is complete. 6.7 Theorem. Let E : P —► R be a iinear map fulfilling (6.16) Fi-^F
implies E(F{) -> E(F).
Then (6.17) there exists 7/ : i" —► R of bounded variation such that E{F) = I FdH
for F e P.
6. LC TOPOLOGIES TOLERANT TO Q-CONVERGENCE
73
Proof. S is continuous in (P,i/£ c ), where
wLC = u(<mic{p,Q)) by Theorem 4.11. Since U^c = U(\\ ■ ||)|p, E can be extended uniquely to a continuous linear map from A to R. By Section 3.2 and Note 1.7 A is isomorphic to the Banach space C 0 (/,R) of continuous functions G : I —> R, G(a) = 0, so that (the extended) E can be interpreted as a continuous linear functional on C 0 (/,R). For F e C(J,R) put EX(F) = S(G) where G(t) = F(t) — F(a), G E Co. Si is a continuous linear functional on C(I, R) and by the Riesz theorem there exists H : I —> R of bounded variation such that =!(F) = / FdH for F € C{I,R) Ja
and S(G) = S i ( G ) = / GdH for
GeC0(I,}
Ja
The proof is complete.
□
Note. Let M : I —* R. M is called a multiplier of integrable functions if / M is integrable for every integrable / : / —* R. (6.18) M is a multiplier if and only if there exists Z : I —► R of bounded variation such that M = Z almost everywhere. ( For a more general result see [Sargent W. L. C , 1948]. (6.18) was extended to integration in two-dimensional inter vals in [Lee T. Y, Chew T. S. and Lee P. Y., 1996].) Observe that H is unique if it is required that H(t) = \(H(t+)
+ H(t-))
for * G IntI, H(b) = 0
74
HENSTOCK-KURZWEIL INTEGRATION
and Z is unique if it is required that Z{t) = -(Z(t+)
+ Z(t-))
2
for t e IntJ.
If f, T): I -> R, then £ {dr] can be defined in an analogous way as 6„ fdt was defined in Section 1.4, where ^ = i / ( ^ ) l ^ t l 1S replaced by EiUfC'OOK^t) ~ vfai-i)) and A = {(U, [c2i-i, CM]); i = 1,2,..., fc} is a partition of if = [c,rf](see also [Kurzweil J., 1957] Definition 1.2.1 with U(r,t) = S(T)ri(t).) It is not difficult to prove that rb
rb
I Zfdt = f Ja
ZdG
Ja
and it can be deduced from the Theorem in [Kurzweil J., 1958] that
I ZdG = Z(b)G(b) - I GdZ. J a,
Ja
If H(t) = Z(b) - Z(t) for t e I, then rb
rb
/ Zfdt = f Ja
GdH
Ja
for every integrable / and G(t) = / /ds, G(a) = 0. Ja
7. TOPOLOGICAL VECTOR SPACES T O L E R A N T TO Q-CONVERGENCE
7.1. Notation and concepts from Chapter 4 and from Ap pendix will be used in this chapter. The goal is in establishing conditions sufficient for (P, U) to be a complete topological vec tor space, U being a topology on P tolerant to Q-convergence. 7.2. Let ft* be the set of u : N x D -»• R + such that (7.1)
u(j + 1, d) < u(j, d)
for
j e N, d e D.
Put T
(7.2)
S(u) = £ R(u(j, •)) = U E *("(* ')). jen
reNy=i
where R was introduced in Section 4.7. For flcfi* put
2rj = 2rj(fi) = {S(cj);u;eft}. Lemma. Let ft C ft* fuihl (7.3) if u; e ft then there exists a G ft such that crQ, d) < w(2i,d) forj eJS,deD, 75
76
HENSTOCK-KURZWEIL INTEGRATION
(7.4) ifui,u2
G ft, then there exists ^ e H such that
U3(j,d) < min{uj1(j,d),uJ2(j,d)}
(7.5) ifu(j,d)
for j G N,d G D,
= 2'j forj G N, d G D, then u G ft.
Then 2U(ft) futfis (A. 13) - (A. 16). Proof. It is obvious that (A.13) is valid and that (A.15) follows from (7.4). Since R(£) is balanced and absorbing for £ : D -»• R + , it is obvious that (A. 14) is fulfilled. If v, o G ft* and if a(j, d) < w(2j, d) for j G N, d G £>, then ^)c^i?(u;(2j,.))
and also (cf. (7.1))
S(a)cJ^R(u(2j-lr)) jen so that S(
{P,U) is a topological vector space, U is finer than U{\\-\\)\P, {P,U) is a Hausdorff space, U\Q{d)=U{\\-\\)\Q{d)fordtD, U is tolerant to Q-convergence.
7. TV SPACES TOLERANT TO Q-CONVERGENCE
77
Proof. Lemma 7.2 implies that 20(ft) fulfils (A.13) - (A.16). It follows from Corollary A.5 that U is a topology on Y and that (7.6) holds. If w is defined by (7.5), then R(u>(i,-)) C B{2~i) for i e N, 5(w) C 5(1) and if a fulfils (7.3), then S(a) c B(2 _ 1 ), etc. Hence (7.7) holds and (7.8) holds as well. Let d € D, x e Q(d), x e U € U. The definition of U implies that there exists u 6 ft* such that x + S{u) c U. By (4.1) there exists d3 e D such that 2Q(d) C Q(d3). Consequently B(x, w(l, d 3 )) n Q(rf) = re + 5(0, w(l, d 3 )) n (Q(d) - x) c C x + S(0, w(l, d 3 )) n Q(d3) Cx + R(u(l, ■)) C x + S(u) C U, which together with (7.7) implies (7.9). Moreover, (7.10) is a consequence of (7.9). The proof is complete. □ 7.4 Theorem. Put U£v = W(20(ft*)). Then U^v is a to pology on P fulfilling (7.6) - (7.10) and (7.11) if V is a topology on P such that (P, V) is a topological vector space and if V is tolerant to Q-convergence then V is coarser than U^v. Proof. Since ft* fulfils (7.3), (7.4), (7.5), it follows by Theo rem 7.3 that U^v is a topology on P fulfilling (7.6) - (7.10). Let 20 be a 0-filterbase in ( P , V), W <E 20. By (A.19) there exists a sequence Wj such that Wj € 20 for j E N and oo 76N
k
fc=lj=l
78
HENSTOCK-KURZWEIL
INTEGRATION
Since V is tolerant to Q-convergence, Theorem 4.4 implies that for j G N there exists pj : D —> R + such that
R(pj) = U W ) n
B
(PM))
C wt.
Put u(j,d) = min{pfc(d); k = 1, 2 , . . . ,j} for j € N, d € D . Thenu; G ft*, 5(u;) = ^ i 2 ( u ; 0 - . ) ) c H / J'GN
and (7.11) is valid. The proof is complete.
□
Note. U^y is the finest of all topologies U on P such that (i) (P,U) is a topological vector space, (ii) the identity map i{d) : {Q(d),U(\\ • ||)) -► (P,W) is con tinuous for d G D (or equivalently £Y is tolerant to Qconvergence, cf. Theorem 4.4). 7.5 Definition. V = {Vitj; i € N u { 0 } , j G N} is called a structure on D (with respect to Q) if (7.12)
£><,,• c D,
V0J = 0,
ViJ+1 C P i j C Z> i+1J (7.13)
(7.14)
for i,j e N,
\JViJ=DiorjeN,
if z,j € N, di,d2 G X>jj +1 , then there exists d 3 G Vitj such that Q(di) + Q(d2) C Q(d 3 )-
Denote by D* the set of all structures V on £>.
7. TV SPACES TOLERANT TO Q-CONVERGENCE
79
7.6. Let A = {Xitj;i, j G N} where (7.15)
Xij G (0,1], Xij > \i+ij,
Xij > A i j + 1 for i , j e N
and let A* be the set of all such A's. Definition. Let X> G D>*. By Q(V) denote the set of u G ft* such that there exists A G A* and fceN such that (7.16)
u(j, d) = ^
if d G P ii2 *-iy \ Z>i-i, 2 *-ij, *', J e N.
7.7 Lemma. Let V G D*. Then ft(2>) fuifils (7.3), (7.4) and (7.5). Proo/. It is obvious that (7.4) and (7.5) hold for ft(£>). To verify (7.3) assume that u is defined by (7.16) and define a : N x D -> (0,1] by
Vi_lt2kj,
where X'{j = A^- for i, j G N. Obviously A'' = {X'i:j;i,j G N} G A*, a G ft(£>). If j G N and d e D, then there exists e G N such that d G P<>2*j \ ^i-i,2*i = ^t,2*-»2j \ ^ i - i ^ - ^ r It follows that
w(2j, d) = Aii2j
and (7.3) holds. The proof is complete.
□
7.8 Lemma. For V G D* put ZY[2>] = W(2U(fi(X>))) (c/. Corollary A.5J. Tien ( P , U[V}) is a Hausdorff topological vector space, U[V] is finer than U(\\ • \\)\p, U[V]\Q{d)=U{\\-\\)\Q(d)
for
deD
and U\V] is tolerant to Q-convergence. Lemma 7.8 follows immediately from Lemma 7.7 and Theo rem 7.3.
80
HENSTOCK-KURZWEIL INTEGRATION
7.9 Lemma. Let V G D* and let 3 = {Z} be a Cauchy filter in (P,U[V}). Assume that 3 -» 0 in X. Then3^0in(P,U[D}). Proof. Assume that Lemma 7.9 is false. Then there exists u G il(V) such that (7.17)
Z \S(u)^
for Z e 3-
Let u be defined by (7.16), let A' = {A^-; J , j ' e N } e A fulfil
(7.18)
X'ij < Ai>4j, 5 3 X^j
< \Ki for i,j G N
and let o : N x D —► (0,1] be defined by (7.19)
It follows from (7.16), (7.18), (7.19) that v(j, d) < w(4j, d) for j G N, d G L> so that /J(o-(i,-))ci2K4;-,0)forieN. Since 3 is a Cauchy filter in (P,U), that Z2 -
there exists Zi G 3 such
Z 2 G S(
By (7.17) there exists (7.20)
zeZi\
S(w).
There exist d G D , / G N such that (7.21)
z G Q(d), where d G P / | 2 * \ Z>i_i,2k-
7. TV SPACES TOLERANT TO Q-CONVERGENCE
Since 3 —> 0 in X, there exists x G Z\ with (7-22)
||x|| < ^ A u .
Moreover, z - x G Zx - Zx C S(a) = Y, RW, ■)) jew and there exist r 6 N and (7.23)
G R(a(j, •)) for j G { 1 , 2 , . . . , r } ,
Wj
r
(7.24)
Z-X = ^2WJ.
For every j 6 {1, 2 , . . . , r} there exists
WjGQ(cgn£(0-(j,^))
and /(j) G N such that d
3
e V
i(j),2k+1i \^0')-i.2 f c + 1 i-
(7.24) can be rewritten as
(7.26)
z- ^
w3=x+
J2 WJ-
By (7.26), (7.22), (7.25), (7.19) and (7.18) we get
(7.27)
||*- J2 «»ill
81
82
HENSTOCK-KURZWEIL INTEGRATION
JGN
Moreover, if l(J) < I, then (cf. (7.12)) (7.28)
WJ G Q(dj), dj G VlU)i2k+ij C X>i,2*+i;
and (7.14) leads (by induction from larger indices j to smaller one's) to E wj e Q(d') where d! G Pj ) 2 *. 'Ci)<* Since z G Q(d) with d G X>/>2*: by (7.21), it follows from (7.8) that there exists d" G T^i^-1 C "P^i such that
It follows by (7.27) that (7.29)
z - ^2 WjE Q(d") D B(u(l, d")) c i2(w(l, •)),
and by (7.25) we get that Wj G Q(dj) D B(a(j, dj)) C R(a(j, •)) C i2(o;(4j, •)) for j G N. It follows from (7.29) that
z = (z- Y, WJ)+ J2 wi e e/2(w(l,-))+ E
^(4i,-))c5'(a;),
which contradicts (7.20). The proof is complete.
□
7. TV SPACES TOLERANT TO Q-CONVERGENCE
83
7.10 Lemma. Let V e D*, y G X \ P, v G X. Let 3 = {Z} be a Cauchy filter in (P,U[D\), 3 -> v in X. Assume that there exist Q e R + such that (7.30)
dist(y, Q(d)) > Q for d G P M , i G N.
Then v ^ y. Proof. Assume that Lemma 7.10 is false, i.e. that 3 —* y in X. Let A = {Xid; i,j G N} G A* fulfil (7.31)
$ > ' + U < k u
for / G N .
Define w G ft(2?) by w(j, rf) = Xitj if d G 2?i)4j \ Vi-i^j,i,j
GN
(cf. (7.16) with k = 3). There exists Z G 3 such that Z-ZcS(w). Let z e Z . There exist d e D such that (7.32)
z G Q(rf)
and / G N such that d G D/,2 \ A-i,2- Since 3 —* V in X, there exists i G Z such that (7-33)
||x_j,||
Moreover, z-xeZ-ZcS(u>)
=
52R(u(j,-)) J6N
HENSTOCK-KURZWEIL INTEGRATION
84
so that there exist r G N and
WJ e R(u(j, •)) = | J Q(d) n B(u>(j, d)) for j e { 1 , 2 , . . . , r} such that r
(7.34)
Z-X = ^2WJ. i=i
Therefore there exist dj £ £> such that WJ € Q(d,) n B(u(j,dj)),
j G {1,2,...,r}
and there exist /(j) G N such that d
o
e
Vi(j)Aj \^(j)-i,4i-
It follows from (7.34) that 2 -
^
Wj-y
'(?')<'
(7.35)
= x~y+
^
Wj,
*0')>'
||* - Y,WJ- y\\ < II* - y\\ + E A<+^ < Ci,*
(cf. (7.33) and (7.31)). If /(j) < I, then d, G T^i(j),4j C A,4j and it follows from (7.14) by induction (from large indices j to smaller ones) that there exists d' G T>it2 such that
J2 WJ G Q(d'), and by (7.14) and (7.32) there exists d" G £>;,i such that
Therefore
II* - E
w
i ~ 2/11 ^ dist(j/,Q(d")) > CM,
iU)
which contradicts (7.35). The proof is complete.
D
7. TV SPACES TOLERANT TO Q-CONVERGENCE
85
7.11 Theorem. LetD C O*, D / 0, T(D) = sup{U[D];V G D} (cf. Section A2 ) Then (i) T(D) is a topology on P which is finer than U{\\ ■ ||) and (P, T(D)) is a Hausdorff topological vector space, (ii) T(B)\Q(d)=U(\\.\\)\Q{d)fordeD, (iii) T(D) is tolerant to Q-convergence. Assume that for every y G X\P with i e N such that (7.36)
there exists V e D anci ^ 6 1R+
dist(y, Q(d)) >Q for ieN,
de Vitl e V.
Then (iv) ( P , T ( B ) ) is a complete space. Proof, (i) and (ii) follow from Lemma 7.8 and Theorem A.8. (iii) is a consequence of (ii). Theorem 4.6 will be employed to prove (iv). First, let 3 = {Z} be a Cauchy filter in T(D), 3 -» 0 in X. Then 3 is a Cauchy filter in U[D] for every V € D so that 3 -> 0 in {P,U[D]) for V G D by Lemma 7.9. Consequently 3 -> 0 in (P,T(D)). Second, let (7.36) be fulfilled. Let 3 be a Cauchy filter in (P,T(D)), v G -X", 3 -► u in X . Then 3 is a Cauchy filter in \p,U[D}) for every V G D and by (7.36) and by Lemma 7.10 we get that v G P . Now (iv) follows from Theorem 4.6 and the proof is complete. D
8. P AS A COMPLETE TOPOLOGICAL V E C T O R SPACE 8.1 Notation. Theorem 7.11 will be applied in the special case X = A, P = P, D = D,Q = Q. A structure on D will be denoted by V = {T>ij; i,j G N} and D* is the set of structures onD. Theorem (Main Result). (i) T(D*) is a topology on P which is finer than U(\\ ■ \\) and (P, 7"(D*)) is a Hausdorff topological vector space, (ii) T(B*)\Q{s)=U(\\.\\)\Q{6)for6eD, (iii) T(D*) is tolerant to Q-convergence, (iv) (P,T(D*)) is complete. Note. Since F{ —► F implies that F{ -?-> F (cf. Theorem 3.12), T(D*) is tolerant to ^-convergence as well. Theorem 8.1 is a consequence of Theorem 7.11 and the fol lowing 8.2 Theorem. Assume that (8.1)
HeA\P.
Then (8.2) there exist
DGD*
and & G R+ for i 6 N such that
dist(H,Q(6))>Si for 6 G ViA G V, i G N. 86
8. P AS A COMPLETE TVS
87
8.3. The rest of this chapter is devoted to the proof of The orem 8.2, which is split into cases A, B, C, D. We introduce the following notation which is connected with 77 EA: (8.3)
~5{t) = limsup ¥$■, |JHo \J\
H(t) = liminf I-/H0
^ ^ \J\
where t G J G Iv(7),
(8.4)
r = /\{te/;I(t)=i(<)6R},
(8.5)
7/:Iv(/)U{0}->[O,oo], 7/(0) = 0, 7/(L) =
inf{sup{^|i/(J)|;A={(f,J)}G5(L,T,C)};C:/-K+} for L G Iv(7). For J G Iv(7) denote by Int( J, 7) the interior of J relative to 7 (i.e. Int( J, 7) = 7 \ Cl(7 \ J) and put (8.6)
W = |J{Int(J,7); J G Iv(/),r/(J) = 0}.
Obviously (8.7) (8.8)
if Ji, J 2 G Iv(7), Ji C J 2 , then T/( Ji) < T/( J 2 ), W is open in 7.
Properties of r\ and W are described in Lemmas 8.4 - 8.13. If 77 G P, then \T\ — 0 and 77 fulfils the strong Luzin condition by Theorem 1.17. Therefore r\{I) = 0 and W = 7.
88
HENSTOCK-KURZWEIL INTEGRATION
8.4 Lemma. Let m e N, L{,K G Iv(7) fori G { 1 , 2 , . . . , m}, Li pairwise nonoverlapping, Ul^i Li C K. Then rn
(8.9)
5 > W <»?(*).
Proof. Assume that T](K) < oo, so that rj(Li) < oo for i G { l , 2 , . . . , r a } . Let ( : I -> R+, e e R + . There exist A* =
{(t, J)} e S(Li,T, 0 such that ^\H(J)\>V(Li)-e for i e {1,2,.. . , r a } . Put A = (J™ i A f . It follows that A G S(K, T, C), m
m
H J
Y,\ ( )\ = T,Y,\ ( )\*T,r>(LJ-rne A
H J
i=l
i=l
A{
and (8.9) holds. The proof is complete.
□
8.5 Lemma. Let Li,K G Iv(7) fori G {1,2, . . . , m } , if C
( X ^ . Then m
(8.10)
»;(#)< J > ( £ i ) . i=l
Proof. Assume that r/(Li) < oo for i G {l, 2 , . . . ,?n}. Let e G R + . Since /f is continuous, there exist Q : / —► R + such that
22\H(J)\
8. P AS A COMPLETE TVS
89
Let C(<) = min{Ci(«); i =
A=
l,2,...,k},
{(t,J)}eS{K,T,Q.
Put i-i
Ai =
{(t,J)eA;teLi\\jLj} i=i
for iG {1,2,.. . , r a } . Then A; G 5(7, T D L if C;) and m
m
me. A
i = l A;
i=l
Hence (8.10) holds. The proof is complete. 8.6 Lemma. Let K G Iv(7), K CW.
D Then rj(K) = 0.
Proof. (8.6) implies that there exist m £ N, I ; e Iv(J), r)(Li) = 0 for i = 1,2,..., m, tf C (J£Li Int(Li, / ) . Lemma 8.5 implies that r)(K) = 0. The proof is complete. □ 8.7 Lemma. Let m G N, Lt, if G Iv(7) for i = 1, 2 , . . . , m, Li pairwise nonoverlapping, K — U£Li A- Then m
(8.11)
frtff) = £ > ( I i ) . i=l
Lemma 8.7 follows immediately from Lemmas 8.4 and 8.5. 8.8 Lemma. Let Li,K e Iv(7), Li C L; + i for i G N, oo
(J Li = IntK,
90
HENSTOCK-KURZWEIL INTEGRATION
a = lim rj(Li) < oo. Then i—KX>
(8.12)
V(K)
= a.
Proof. By (8.7) we have (8.13)
n(K) > a.
Without loss of generality it can be assumed that L\ ^ 0, Li^ 0 for i G N. Then
Li+i\
cl(L i + 1 \ Li) = J2i U J2i+i with J2i G Iv(7) and J 2 i+i € Iv(J) or J 2 ;+i = 0 for i G N. Obviously 77(^1) < a, v( Jk) < a for fc G N \ {1}. Since H is continuous, there exist £t : / —> R + for i G N such that £ | f f ( J ) | < i7(Lx) + I for A = {(*, J)} G
S(K,TnLltCi),
A
J2\H(J)\
for A = {(t, J)} G S(K,Tn
Jk,Ck),k G N \ {1},
and there exists A G R + such that \H(L)\ < § if L G Iv(A"), |L| < A. Let A" = [c,rfJ. Put C(0 = 1 for t G / \ K, C(c) = C(d) = A, C(t) = Ci(0 for t G L l 5 C(t) =
(2i(t)iorteJ2i\Li,
C(t) = C2i+i(<) for * e ^2i+i \ £»•
Let e =
{(s,V)}eS(K,T,Q.
8. P AS A COMPLETE TVS
91
Put
el={(s,V)eQ]seLl}, ®2i={(s,V)ee;seJ2l\Li}, @2i+i ={(s, V)eB;se
&'
J 2 ;+i \ Li},
={(s,V)ee;se{c,d}}.
Since @ is a finite set, there exists / G N such that
e = e'u|j0 f c , /t=i
X; mv)\ = E E 1^)1+E itf(y)i ^ e
fc=i
efc
9'
i £
< viL,) + - + £ > ( J f c ) + e2- 2 - f c ) +£-
+ e.
fc=2
It follows that r)(K) < a, which together with (8.13) implies (8.12), and the proof is complete. □ 8.9 Lemma. Let K G Iv(7), rj(K) < oo. Then rj is additive and continuous on Iv(K). The proof follows by Lemmas 8.7 and 8.8. 8.10 Lemma. Let K G Iv(/), IntK C W. Then n(K) = 0. The proof follows by Lemma 8.8, since r](L) = 0 for L G lv(K), L G IntK by Lemma 8.6. 8.11 Lemma. Let K G Iv(I). ifI\Wr\IntK^Q.
Then r)(K) > 0, if and only
Proof. If n(K) = 0, then IntK C W by (8.6). If IntK C W, then n(K) = 0 by Lemma 8.10. The proof is complete. D
92
HENSTOCK-KURZWEIL
8.12 Lemma. I\W
INTEGRATION
is a perfect set.
Proof. Let s G Int7 be an isolated point of I\W. There exist c,d £ I such that c < s < d, [c,d]r\W = {s}. If ci G (c, s), then T?([C, ci]) = 0 by Lemma 8.6, so that r)([c, s]) = 0 by Lemma 8.10. Similarly v([s,d[) = 0. By Lemma 8.5 we get r]([c, d]) = 0 so that 5 G W, which contradicts the assumption that s G I\W. Analogously neither a nor b can be an isolated point of 7 \ W. The proof is complete.
□
8.13 Lemma. Let e G R + . Then there exists ( : I -* R+ suc/2 fcfiat A
provided A = {(t, J)} G 5(7, T, C) a D d J c W /or (*, J ) G A. Proof. Since W is open in 7 (cf. (8.8)), there exist L t G Iv(7) for % G N such that (J i e N L, = W. Let e G R+. Since 77 is continuous and t](Li) = 0 for i G N by Lemma 8.10, there exist Ci : 7 -♦ R + such that
£|jy(ir)|< e 2-* A
for A = {(*, AT)} G 5(7, T n Li, Ci), i e N. Put C(<) = 1 for t G 7 \ iy, C(<) = Ci(0 ^ < e L< \ Uj-=1 L,-. Let 9 = {(<, J ) } G 5(7, T, (),JcW for (f, J ) G 0 . Put 0^ = {(*, J ) € 6;« € Lf \ Uj=l M > « e N. Then
E i^(J)i = E E wj)i ^ E^2_t ^£e
The proof is complete.
t6N 0 i
t€N
□
8. P AS A COMPLETE TVS
93
8.14 Lemma. Let N C /, \N\ > 0, a G R+. Assume that for t G N there exists J(t) = [u(t),v(t)] c / such that t G J(t), \J(t)\ >aforteN. Then there exist r e N , t i , i 2 ) . . . l t r e i V such that J(ti) are pairwise disjoint and
(8-14)
J2\J(U)\>^.
Proo/. For t G Af put K(t) = [u(t)-2\J(t)lv(t)
+ 2\J(t)\).
Obviously \K(t)\ = 5\J(t)\ for t G AT. Let ti G A^ fulfil
\J(t1)\>±8ap{\J(t)\;teN}. UN C K(h), put r = 1; obviously |TV| < |/f(*i)| and | J ( t i ) | > Lj4. Otherwise there exists t2 € N \ K(ti) such that |J(t2)|>^sup{|J(t)|;
teN\K(t1)}.
It follows that J(*i) n J(t2) = 0. If A/" C tf (
\N\<\K(h)\
+ \K(t2)\,
|j(tl)| + | j ( « 2 ) | > M .
Otherwise there exists £3 G N \ {K(ti) U K{t2)) such that IJ(t 3 )\ > l-sup{|J(t)\; teN\
(K(h) U K(t2))},
...
Since \J(t)\ > a for t e N, this process stops after a finite number r of steps and (8.14) holds. □
94
HENSTOCK-KURZWEIL INTEGRATION
8.15 Lemma. Let (8.1) hold. Then either (8.15) \T\ > 0 or
case A
(8.16) |T| = 0 and there exists K G Iv(I) such that 0 < n(K) < 00, case B or (8.17) \T\ = 0, 7}(J) G {0, oo} for J G Iv(I), W is not dense 1, case or (8.18) \T\ = 0, n(J) G {0, oo} for J G Iv(I), W is dense in I\W # 0. case
in C I, D
Proof. Let none of (8.15) - (8.18) hold. Then \T\ = 0,W = I so that T](I) = 0 by Lemma 8.10. By (8.1) H G A so that H : lv(I) -» R is additive and continuous. Since n{I) = 0, there exists 9 G D* such that
£lW)l<2-; A
provided j G N, A = {(«, J ) } G S(I,T,0(j,-)). Moreover, //(*) exists and H{t) G R for t G I \T, \T\ = 0. Therefore Theorem 1.19 implies that H G P, which contra dicts (8.1). The proof is complete. □ It follows from Lemma 8.15 that Theorem 8.2 is valid pro vided the following theorems hold: 8.16 Theorem (case A). Let (8.15) hold. Then (8.2) holds. 8.17 Theorem (case B). Let (8.16) hold. Then (8.2) holds. 8.18 Theorem (case C). Let (8.17) iioid. Then (8.2) holds. 8.19 Theorem (case D). Let (8.18) hold. Then (8.2) holds.
8. P AS A COMPLETE TVS
95
8.20 (case A), proof of Theorem 8.16. Let (8.15) hold. Since |{t € I; H(t) = - c o or Hit) = oo}| = 0 (see [Saks S., 1937], Chapter IX, Theorem 4.40, [Bruckner A., 1994], Chapter 4, Theorem 4.4 (Denjoy, Young, Saks), it follows from (8.4) and (8.15) that \{teI;H(t)>H(t)}\>0. H and H_ are measurable and therefore there exist (3 € (0, \) and a measurable set E c I such that |E| > 0, H(t) > H(t) + 40/5 for t e E. Moreover, there exists h : E —> R measurable such that H{t) - 20^ > h(t) > H,(t) + 20(3 for t e E. For 0 < a < r let ETy0 be the set of t G E such that there exist L+(t),L-{t) G Iv(J) fulfilling teL+(t)nL-(t),
I
^<|L + (0|
\H{L+(t))-h(t)\L+(t)\>2D0\L+(t)\, k //(L-(0)
- M0l^"(«)l < -2O0|L-(O|.
£T)<7 is measurable and for every r G R + we have
E = \J{ETy, ae(0,r)} so that there exists
|£ T , CT(T) | > 1 | ! .
96
HENSTOCK-KURZWEIL INTEGRATION
There exists JH G N such that 2~JH < P\E\.
(8.21)
For i,j G N p u t T>oj = 0, (8.22)
> 2"f+'}| >
Vij = {6e D- \{t G I;6(j+jH,t) >\I\-2-*-i»}, V =
{Vitj;i,jeX}.
Then (8.23)
V is a structure on D with respect to Q.
To prove (8.23) observe that (7.13) holds and that (7.12) holds, since 6(j,t)>6(j + l,t) for 6 G D, j G N, t G / . Moreover, if i, j G N, 6X,62 G Vitj+i and if
63{j,t) = ^min^C? + l,t),62(j + 1,0}, then \{t G I;6k(j + l+jHtt)
> 2-i+l+1}\
> \I\ -
2->-1-i»
for k = l,2, \{t G /;min{<5fc(j + l+jH,t);k
= 1,2} > 2~i+j+1}\
> \I\ - 2~j~JH so that 63 G T>itj.
>
8. P AS A COMPLETE TVS
97
Let Fk G Q(Sk), k = 1,2. By Definition 3.8 there exist fk • I -* R, Mfc G A/" such that
for i e N , 9 = { ( a ) } e 5(7, /, 4 ( j + 1, •))> * = 1, 2 and ^|FfcW-/fc(S)|^||<2-^1 e for i G N , 0 = {( 5 , A-)} G 5(7,7 \ Affc> Sk(j + 1, •)), A:= 1,2. It follows that
£ |*<*)+ft(*)l < 2->+X W ( _ l I _ J + _ L _ ) <
f o r j G N , 0 = {(5, AT)} G 5(7,7,8 3 (j,-))C 5(7,7,<51(j + l,-))n 5(7,7,<52(7 + l,-)) and that Y, \Fx(K) + F2(K) - (h(s) + f2(s))\K\\ e
< 2->
for j G N, 9 = {(5, K)} G 5(7,7 \ (Mx U M 2 ), <53(j, •)) C C 5(7,7\M 1 ,<5 1 (i + l , - ) ) n 5 ( 7 , 7 \ M 2 , < 5 2 0 + l,-)). Therefore Fx + F2 G Q(63), Q(h) + Q{h) C Q(63)
HENSTOCK-KURZWEIL INTEGRATION
98
so that (7.14) is valid and (8.23) holds as well. Let i e N, 6 e Viti, F e Q(S). By Definition 3.8 there exist / : I -> R, M e M such that
Y,\F{J)-f{t)\J\\<2-^» provided A e S(I,1 \ M, 6(1 + j H , •))• Put r = 2 " \ TV = £ T / 7 ( r ) p | { * € 7; 6(1 + j > , 0 > 2~i+1} \ M. Since (3e(Q,\),8 that
£ Vitj, it follows from (8.20), (8.21), (8.22)
1*1 > If!. For t € N put J(t) =
/
L+
\
L"(0
W
if
if
'(«) ^ MO, f(t)>h(t).
We have r = 2 " i , < r ( r ) < | J ( 0 | < r , (5(l + i / / , * ) > 2 - i + 1 for < G N. By Lemma 8.14 there exist r £ N, ti,t2,- ■ ■ ,tr £ N such that J(£;) are pairwise disjoint and
Put A = {(ti,J(ti));i
=
1,2,...,r}.
8. P AS A COMPLETE TVS
It follows that A £ S(I, I\M,
99
8(1 + j H , •)) so that
On the other hand, it follows from (8.19) that r
r
] T \H(J(U)) - f(U)\J(U)\\
> 20/3^) |J(*i)l > P\E\ > 0,
i=l
i=l
so that r
\H(J(U)) - F(J(U))\ > 0\E\ - 2 " 1 - " > 0.
£ i=i
Since r < a,2-i) > ^ follows that
where
0 = OS|£|-2-» ) J L and the proof of Theorem 8.16 is complete. 8.21 Lemma. Let (8.16) hold. Let e G (0, \). Then there exist m e N \ {1}, K\,K2,...,Km G I v ^ ) pairwise disjoint and such that (8.24)
(8.25)
\H(K{)\ > (1 - eHKi)
> 0 for i G {1, 2 , . . . , m},
^»y(/f<)>(l-4e)i7(A'), i=l
100
HENSTOCK-KURZWEIL INTEGRATION m
(8.26)
\K\
£ l ^ l < X -
Proof. Since 77 is continuous on lv(K) (cf. Lemma 8.9), there exists a G R + such that (8.27)
v(L) < (1 - Ae)ri{K) for L G Iv(X), |£| < 2a.
Since |T| = 0, there exist an open set U G R and C : I —> (0, a) such that (8.28)
TcU,
\U\ < 1^1, (* - £(*),« + C(0) C U for « € T,
X>( J )l^( 1 +
(8.29)
£2
)^)
A
provided A = {(*, J)} G S(K, T, Q. It follows from (8.28) that
(8.30)
E l J l < 'f' f ° r A = {(*'J)} G 5 (*' T '0A
Moreover, there exists A* = {(i, J)} G S(K,T, () such that (8.31)
El^(J)|>(l-e2)77(^). A*
We have A-
iej"
where J C N is a finite set, Vi C K are pairwise disjoint inter vals relatively open in K.
8. P AS A COMPLETE TVS
101
Analogously to (8.31) there exists A; = {(t, J)} G S(clVt, T, Q for i G J such that
E \H{J)\ > (1 - e2)i7(clV-). Put 0! = A* U | J A*. Obviously 0 i G S(K,T,().
It follows that
(1 + e^V(K) > E \H(J)\ > E l# (-01 + E E l^ J )l ^ ei
i£j
A*
A;
^ ( l - ^ M ^ + a-^E^KK), y
2e2
T,^( i) < ^-rfWSince r\ is additive on Iv (K) by Lemma 8.9, we have
!/(*) = J>(./) + $>(cl(V-) < A*
i£j
2e2
E^)+(i^2)^(n A*
(8.32)
l-3e
2
E^)>(^5)^)-
Now put (8.33)
0 + = {(*, J) G A*; \H(J)\ > (1 - e ) ^ ( J ) } ,
102
HENSTOCK-KURZWEIL INTEGRATION
0 - = A*\0+. By the definition of 77 there exists
A(t,J) =
{(t,J')}eS(J,T,Q
for (t, J) e 0 ~ such that
\H(J')\>(l-e2)V(J),
E so that E
\H(J')\ - \H(J)\ > (1 - e*)r,{J) - (1 - e)V(J) >
(8.34)
>e(l-e)77(J)
for (t,J)eS-.
Put
0 2 = 0 + u|jA ( f ) J ) G-
and write for a moment 0 2 = {(s, V)}. Obviously 0 2 G S(K,T,QIt follows (cf. (8.33) and (8.34)) that
(8.29), (8.31),
(l + £ 2 h(/O>EW0l = = E i*( J )i+E i^( J )i+E( E \ff(J,)\ - \H(J)i) = e+
e-
e - A (tiJ)
E ^ i + D E 1^(^)1-IH(J)D> A-
e-
A (t ,j)
8. P AS A COMPLETE TVS
(l-e2)V(K)
103
e(l-s2)J2v(J). e-
+
Hence e-
and by (8.32)
e+
A*
e-
e+
(8.35)
E ^(J) ^ *
6
\ * 2
26
\(K)
> (1 - 4 £ )r?(K).
e+
Put (8.36)
{Klt K2,...,
Kn} = {J; («, J ) G 0 + , r/( J) > 0}.
Then (8.24) holds by (8.33) and (8.36), (8.25) follows from (8.35) and (8.26) follows from (8.30) since A* C S(K,T,(). Moreover, (8.25) and (8.27) imply that m > 1, since 0 < C(t) < a for t e I and \K{\ < 2\ for i = 1,2,..., m. The proof of Lemma 8.21 is complete. D 8.22. Lemma 8.21 will be used repeatedly. A system of in tervals Yi j will be constructed in such a way that various relations involving these intervals, their Lebesgue measure and functions H and 77 are fulfilled. Let (8.16) hold. In the first step let e G R + , 1 - 4e > e x p ( - 2 - 1 ) .
HENSTOCK-KURZWEIL INTEGRATION
104
By Lemma 8.21 there exist m e N \ {1}, K1,K2,..., Km e lv(K) pairwise disjoint and such that (8.26), (8.25) and (8.24) hold. Put y(°) = K, r(°) = m, Y}1] = AT,- for j = 1,2,..., r<0). Rewriting the above relations we get (8.37)
Y™ e Iv(y(°)) for h = 1,2,..., r<°\ yj
being pairwise disjoint,
2>i<*p.
(8.38)
i=i r(0)
(8.39)
E^ (1) )> ex p(- 2_1 M y(0) )' i=i
(8.40) |tf ( i f >)| > exp(-2- 1 )r 7 (y/ 1 ) ) > 0 for i = 1, 2 , . . . , r<°>. In the second step let e e M + , (1 - 4e) > e x p ( - 2 - 2 ) and for /i = 1,2,...,r<°> apply Lemma 8.21 with if = Y^K There exist m G N \ { l } , KUK^...,Km e Iv(/C) pairwise disjoint and such that (8.26), (8.25) and (8.24) hold. Put r^(h) = m, r/^J = ^ for j = 1, 2 , . . . , r( 1 )(/ 1 ). Rewriting the above relations we get (8.41)
^Glvfr'1')
for Za = l,2,...,r< 1 >(/ 1 ) >
Y^ • being pairwise disjoint for j = 1,2,..., r ^ ^ / i ) ,
8. P AS A COMPLETE TVS r ( 1 ) (<0
8 42
(- )
105
|V(l).
£ i^i
(8.43)
£
7 ? (y/ i ^)>ex P (-2- 2 )r ? (y/ i 1 )),
i=i
(8.44)
IHCyfj)! > exp(-2- 2 )7 7 (r/ i 2 ]) > 0 forj = l,2,...,rW(Zi).
By induction we find that (8.45) there exist r<°> € N \ {1}, r( 1 )(/ 1 ) G N \ {1} for h G N, k < r(°>, rW{lu...,lk) € N \ { 1 } for h G N, Z* < 1 r ^ " ) ^ ! , . . . , h-i), i = 1,2,..., jfc and jfc e N. Let us call the fc-tuple (h,..-
,lk) G Nfc admissible, if
/t^r^-1^/!,...,/*-!)
for i = 1,2,..., k. The empty set is also called admissible. (8.46) There exist
r<°>€iv(/),
Y^tivfxtlJ
for (li,..., lk) G Nfc admissible, k G N. Moreover, the following conditions are satisfied: (8.47) the intervals Y^ ' ik_1j
are
pairwise disjoint for
j = i,2,...Mk-1)(lu...,lk-l), provided (l\,...,
Z/t-i) are admissible,
fceN
106
HENSTOCK-KURZWEIL INTEGRATION
rV-VOu.-.lk-i)
ly(k-l)
|
for A; e N, (/i,..., /fc-i) admissible, rf*-1)^,...,**-!)
(8.49)
iVSli^j) * <*P(-*-kMYtlj
£
for k e N, (/i,..., /fc_i) admissible and (8.50)
I*Wi*!.,J| > exp(-2-*)»7(y/;*>.iJJ > 0
for fc e N, (Zi,..., Ik) admissible. The next lemma represents the common part of the proofs of Theorems 8.17, 8.18 and 8.19. 8.23 Lemma. For k e N let there exist r(°\r^(lu...,lk),Y^,Yl[kllk such that (8.45), (8.46), (8.47) and (8.48) hold. Put Y = {0, y<°>, Y^
lk;
k 6 N, (/ l 5 ..., lk) admissible}
and let there exist \i: Y -♦ R + U {0} such that (8.51) //(0) = 0, 0 < n(U) < /i(V) for U, V e Y, 0 ^ U C V,
8. P AS A COMPLETE TVS
107
and
(8.52)
exp^-'MY^lj < r(fc-1)(Ji>...,/fc-i)
forfcG N, (Zi,..., /fc-i) admissible. Assume that l^(^.,,JI>exp(-2-fc)/i(yW.|It)
(8.53)
forfce N, ( / i , . . . , /fc) admissible. Then (8.2) holds. Proof. For m G N put Y m = {Y^lk;
keN,fe>m,(!i
/fc) admissible} U {0},
-co *=niKiv m=lYm
For £ C 2 put i/(E) = inf { ^ n{Vi); V- G Y, ( J K D £ } . Then ^ is an outer measure. If E C Z, then
for A; € N, ( / i , . . . , /&) admissible and it follows that
i/(E) = inf { £ /x(K); Vi G Ym, | J K D £}. igN
i€N
for m e N . Therefore v has the following property: if EUE2 C Z, inffli - s|;< G £ i , s G £ 2 } > 0, then u{Ei U £ 2 ) = i/(Ei) + i/(E 2 ),
HENSTOCK-KURZWEIL INTEGRATION
108
which is characteristic for the so called metric measures (see [Lukes J. and Maly J., 1995],Theorem 36.4, [Rogers C.A., 1970], Theorem 19) and which implies that every Borel measurable subset of Z is ^-measurable. If Vi, Vj e Y, then either V{ n V3; = 0 or V- C Vj or Vj C Vt. It follows that (8.54) v(E) - inf{^2i€N n(Vi);Vi e Y m , V{ pairwise disjoint, (J i e N Vi D E} for m G N, £ c Z. Let £ C Z be closed. Then ^ = P I \JlY™..,U! Ci» • • • »'*) admissible, Y ^ ^ n £ ^ 0}. fceN
Since the sets V C\ Z where I ' e f are both closed and open in Z, only finite sums may be used in (8.54) to obtain v(E) and with respect to (8.52) and (8.46) we have (8.55)
v{E)=
lim y^{/i(Y, (fc) i j ; (h, ■ ■ ■ ,h)
admissible,
If E = Z n Y , | p ) / p , p G N, ( / i , . . . , / p ) admissible, then
i/(Zn Y,(p) . ) = lim
y
{n(Y,(k) ,
m p + i ,...,T>H
( / i , . . . , / p , r a p + i , . . .,mfc) admissible}. By (8.52) and (8.55) we get
m
);
8. P AS A COMPLETE TVS
Define V = {Vitj;i,j Let j H e N,
109
e N} as follows:
2~J" < 4 ^ -
(8.56)
4e
For i,j eN put V0j = 0, Vij = {Se D;u({t 6 Z;<5(j +jH,t)
> 2"^'})
>»/(Z)(l-2-J'-J'")}.
(8.57) Then
X> is a structure on D with respect to Q. The proof is omitted since it is analogous to that of (8.23). Let i E N, 6 e £>i,i. By (8.57) we get !/({« e Z; 6(1 + ;*,*) > 2 - + 1 } ) >
(8.58)
^ -
Let k € N fulfil 2-2fc|y0|
(8-59)
<2-
i
-J'w)
... . , \Yi[ ..,ik I < 2 " ' + for ( / i , . . . , /fc) admissible .
Let us denote by C the set of admissible ( / i , . . . , Zfc) such that Yt[k)
lk
n{te
Z;6{l+jH,t)
> 2 - t + 1 } jk 0.
For every ( / i , . . . , Ik) € £ there exists (8.60)
*,,,...,,, 6 Y j ^ . . ^ n { t G Z;6(l + jH,t)
> 2^+1}.
110
HENSTOCK-KURZWEIL INTEGRATION
Put Since
UYiuU
D
(«e Z;S(1 +jH,t)\ > 2-+ 1 }
A
it follows by (8.52) and (8.58) that
E^(C..,Ji> e_1 EMC..,J>
(8.6i)
A
A
> e ->2>(2ny«.,„)> A
> e-xv({t
> 2~i+'})
e Z;6(j + jH,t)\
>
^ . 2e
Since <5(l+i//,^,...,/J>2-i+1 for (Z 1 } ..., lk) e C, it follows from (8.59) that
AeS(IJ,6(l+jH,.)). Let F EQ(6).
(8.62)
Then
EIF(C.,JI
^
2 l jH 2 _1
~ ~ + ' EiC,J
A
A
It follows from (8.48) that
E{l y i ( i*Ul5d'•■"'*) a d m i s s i b l e } < 2-2fc|F0| and we get by (8.62), (8.59), (8.56)
(8.63)
E A
^(^...JJI
^ 2~l~j" + 2l~1-2fc |y°| <
8. P AS A COMPLETE TVS
< 2~l-JH
+ 2~l~jli
< 2~JH <
111
u{Z) 4e
It follows from (8.61), (8.63) that
E i ^ i l « J - ^ l . J i ^4e . \H-F\\>
4eq
where q is the number of all admissible fc-tuples ( / i , . . . , lk) and (8.2) holds for & = 4^r • The proof of Lemma 8.23 is complete.
□ 8.24 (case B), proof of Theorem 8.17. By Section 8.22 there exist A°\ r^(h,..., lk), Y(0], Yt[k) A for k e N such that (8.45) - (8.50) hold. Putting n = 77, we get that (8.51) holds. Moreover, the sec ond inequahty in (8.52) is vahd since 77 is nonnegative and ad ditive on Iv(7) and the first follows from (8.49). Finally, (8.53) holds (cf. (8.50)). Lemma 8.23 implies that (8.2) holds. The proof of Theorem 8.17 is complete. The following lemma is a direct step to the proof of Theorem 8.18. 8.25 Lemma. Let (8.17) hold. Then (8.64) there exists L G Iv(7) such that r){J) = 00 for J 6 Lv(L). Moreover, if K e Lv(L), \H(K)\ > 0, then there exist m G N \ {1}, Kj G Iv(tf) for j e {1,2,..., m} such that (8.65)
(8.66)
Kj G Iv(A'), j G { 1 , 2 , . . . , m} are pairwise disjoint ,
\H(Kj)\ > 0 for j G { 1 , 2 , . . . , m},
112
HENSTOCK-KURZWEIL INTEGRATION
(8-67)
£|#.|<]*l,
(8.68)
Yl WKi)\ = UW|. 3= 1
Proof. By (8.17) there exists L e Iv(7), L c I\W. Since rj(J) = 0 or r)(J) = oo for J e Iv(7), Lemma 8.11 implies that (8.64) holds. Let K € Iv(L), | ^ ( ^ ) | > 0. Since ff is continuous, there exists a e R + such that \H(J)\ < \H(K)\ for J e Iv(A"), \J\ < 2a. Let M € Iv(tf), |M| < 1£1. Put C(0 = a for * C / . Since T?(M) = oo by (8.64), there exist m <E N, A = {(*,-, Jj);j = l , 2 , . . . , m } e S(M,T,Q, cf. (8.4), such that m
£
|ff (4)1 > 2|ff (ff)l, |ff (Jj)\
> 0 for i € {1, 2 , . . . , m } .
i=i
We have |J,| < 2a so that m > 1. Since ff is continuous, there exist Kj G Iv(Jj) such that (8.65), (8.66) and (8.68) hold. (8.67) holds since Uj=i Kj c M. The proof is complete. D 8.26 (case C), proof of Theorem 8.18. Let (8.17) hold. It follows that (8.64) holds. Let K € Iv(L), \H{K)\ > 0. By Lemma 8.25 there exist m e N \ {1}, Kj E Iv(K) such that (8.65) - (8.68) hold. Put r<°) = K, r(°) = m,M0) = 0, p(Y(°>) = \H(K)\, YJ1} = Kj, niY}") = \H(Kj)\ for j e {1,2,.. .,r(°)}. It follows that (8.45) - (8.48), (8.52), (8.53) hold for k = 1.
8. P AS A COMPLETE TVS
113
For h G {l,2,...,r<°)} apply Lemma 8.25 with K = Y^K There exist m G N \ {1} and Kj G I v ^ 1 * ) such that (8.65) (8.68) hold. Analogously as above put rW(l\) = m, Y{fj = Kj, M(VJpj) = \H(Kj)\ioTJ€{l,2,...,rW(l1)}. It follows that (8.45) - (8.48), (8.52), (8.53) hold for k = 1,2 and by induction we get that (8.45) - (8.48), (8.52), (8.53) hold for k G N. Moreover, (8.51) holds. Now Lemma 8.23 implies that (8.2) holds. The proof of The orem 8.18 is complete. 8.27 Lemma. Let (8.18) hold, let K € Iv(J), (I \ W) n IntK ^ 0. Then there exist m G N \ {1}, Kt G Iv(iif) for i G { 1 , 2 , . . . , m} such that (8.69)
Ki G Iv(K), i 6 {1, 2 , . . . , m} are pairwise disjoint ,
(8.70)
fj(Ki) = oo, \H(Ki)\ > 0 for * G {l, 2 , . . . , m},
(8-71)
El^l^^T' i=l
(8.72)
£ | J f ( t f , - ) | = H(tf), i=l
where (8.73)
(8.74)
E(K) = \H(K)\
if H(K) ? 0,
E(K) = 1 if H(K) = 0.
114
HENSTOCK-KURZWEIL INTEGRATION
Proof. Since T/(J) € {0,00} for J e Iv(7) and (I\W)n IntK ^ 0, it follows by Lemma 8.11 that r}(K) = 00. There exists M G lv(K) such that \M\ < l-^, (I \ W) n IntM 7^ 0, so that r}(M) = 00. Since H is continuous, there exists a e R + such that |if ( J ) | < E(tf) for J e Iv(J), IJ| < 2a. By Lemma 8.13 there exists C, : / —► (0, Q] such that
(8.75)
EM^W A
provided A = {(*, J ) } <E <S(M, T, C) and J c W for (*, J ) e A. Since T/(M) = 00, there exists 0 = {(*, J)} e S(M,T,Q such that (8.76) Y, \H(J)\ > 3E(K), \H(J)\ > 0, J C IntM for (*, J ) € 0 . e Put 0 i = {(t, J)eG,JcW},Q2
= Q\ 0 X .
By (8.75) we have £ |ff(J)| < 3 ( J 0 , ex so that
Y,mj)\>2z{K). Put {J-,(f, J) G ©2> = {[Cl,C2],[c3,C4],...,[c2m_1,C2m]} where m € N, C2» < C2i+i for i < m.
8. P AS A COMPLETE TVS
115
We have (8.77) (8.78)
[ c 2 i _ i , C 2 i ] n ( J \ H O ^ 0 for z e { l , 2 , . . . , rn}, \H([c2i-i,c2i])\>0
forie{l,2,...,m},
m
(8.79)
X)|ff([C2i_ilc«])|>2S(A').
Since | J | < 2 Q for (t, J ) e 9 , we have \H(J)\ < 3{K) for («, J) G 0 , so that |//([c 2 i_i,C2i])| <2(/if) for i e { l , 2 , . . . , m } and (8.80)
m > 1.
Put J = {i € {1, 2 , . . . , m}; (c 2 i-i, c2i) n (7 \ W) / 0}, /C = { i € { l , 2 , . . . , m } \ J . For i e j there exist c ^ . n ^ £ Z such that (8.81)
(8.82)
c 2 i _! < 4 . ! < c'2i < c2z, ( 4 . ! , 4 ) n (7 \ W) # 0,
|ff([4--i>4])l ^
^([CM-I.C«])|.
If i e K, then (C2i-i,c«) C W, {c2i-i,C2i}
n(I\W)^
Assume that c2i € 7 \ W. Since 7 \ W is perfect, we have (c2i,C2i +
e)n(I\W)^
116
HENSTOCK-KURZWEIL INTEGRATION
for e e M + . It follows that either i = m, or C2i < C2i+l,
or In the last case we have necessarily i + 1 6 ,7, since / \ W is perfect. For i e A:, C2J E I \ W put 4 - 1 = C2i-1, 4 = C2i + £i with £j G K+. An analogous situation occurs if C2i-i € / \ W. Moreover, [c2;-i,C2i] C IntM by (8.76). If all Si, i e K, are sufficiently small, then (8.83)
(8.84)
(8.85)
|H ( [ 4 - 1 . 4 1 ) 1 > \\H([c2i^,c2i})\
for i € /C,
( 4 - 1 . 4 ) n (/ \ WO ?6 0 for i e AC,
[4-1,4] CM
(8.86) intervals [ 4 - i > 4 1
are
forie/C,
pairwise disjoint for
t e {l,2,...,m}.
8. P AS A COMPLETE TVS
117
By (8.79), (8.82) and (8.83) we have m
(8.87)
£
iHdc'^, 4-])|
>E(K)
and (8.88)
| # ( [ 4 - i , 4 ] ) l > 0 fortG{l,2,...,m}.
By (8.81) and (8.84) we get (8.89)
(4_1,4i)n(/\V^)^0
forie{l,2,...,m}
and (8.81) together with (8.85) imply that (8.90)
[ 4 - i > 4 ] C M for z G { 1 , 2 , . . . , m}.
Moreover, m > 1 by (8.80). Therefore there exist ifj G IvQc^-^c^]) such that (I\W)n
IntKi ^ 0, \H(Ki)\ > 0 for i € { 1 , 2 , . . . , m}
and that (8.72) holds. (8.69) holds, since [ c ^ . ^ c y G Iv(M) for i = 1,2, . . . , m and since [ c ^ ^ ! , ^ ^ are pairwise disjoint. Moreover, ri(Ki) = oo for i £ {1,2,... ,m} by Lemma 8.11, so that (8.70) holds. (8.71) follows from (8.90) since \M\ < J—L The proof is complete. D 8.28. Now Theorem 8.19 can be proved in the same way as Theorem 8.18 was proved in Section 8.26, with the minor change that Lemma 8.27 is applied instead of Lemma 8.25 and that we have to put /x(y(°)) = E{K). (If fc G N, then Y,Whl.,iJ5
('!.■•■»'*) admissible} = E(K)
and by (8.55) we obtain v(Z) = 2(K).)
9. O P E N
PROBLEMS
9.1. Let the notation of Section 8 be kept (e.g. Q : D —► CxBlCp(.4) is introduced in Definition 3.8). If the topologies UTv and 7"(B*) are different the following problem is open: (9.1) Is {PMTV) complete? Since (P, T(B*)) is complete, it follows by Theorem 4.5 that (9.2) if 3 = {Z} is a Cauchy filter in (P,T(B*)), F G A and if 3 -> F in A, then F £ P. Since T(D*) is tolerant to Q-conbvergence (cf. Theorem 7.11), Theorem 7.4 implies that UTV is finer than Tip*). There fore (9.3) if 3 = {Z} is a Cauchy filter in (P,Z/f. v ), F 6 i and if 3 ->• F in A, then F G P . Theorems 4.5 and 4.6 lead to the conclusion that (9.4) if 3 = {Z} is a Cauchy filter in {PMTV) a n d if 3 —> 0 in A, then 3 -* 0 in {PMTV) is a necessary and sufficient condition for the affirmative answer to (9.1). 9.2. Do results from Sections 6 and 8 hold for some other integrations, which are based on gauges and integral sums of the type ^ / ( t ) m ? A
If they do, can they be obtained by a common approach? 118
9. OPEN PROBLEMS
119
For suitable integrations (on n-dimensional intervals or more general sets, n G N), see e.g. [Bongiorno B., Pfeffer W. F. and Thomson B. S., 1996], [Buczolich Z. and Pfeffer W. F., 1997], [Jarnfk J. and Kurzweil J., 1985, 1995, 1997 ], [Jurkat W. B. and Nonnenmacher D. J. F., 1994 a, b, c], [Jurkat W. B. and Nonnenmacher D. J. F., 1995 a, b], [Kurzweil J. and Jarnik J., 1990, 1991-92, 1992, 1996, 1997], [Nonnenmacher D. J. F., 1994, 1995], [Pfeffer W. F., 1987, 1991, 1993]. Note. In this note couples {t, J ) are admitted such that t el=
[a,b] C R , J G l v ( J ) .
For E C R put diamE = sup{|t — s\; t, s G E}. Let / : / —► R. The function / is called *integrable if there exists 7 G R and for (3 6 (0,1] there exists 8 G D* such that k
provided j G N, a = CQ < c\ < • • ■ < Ck = b, U G / , Pdia.m({U} U [ci_i, Ci}) < a - (k-i, [ci_i,Cj] C {ti-6{j,ti),U
+ 6(j,U)) for i = 1,2,...,*;.
It is well known that / : / —*■ R is Lebesgue integrable if there exists 7 G R and 6 e D* such that k
120
HENSTOCK-KURZWEIL INTEGRATION
provided j G N, a = c0 < c\ < ••• < Ck = b, U £ I, [ci-itCi] C (U - 6{j,ti),ti
+ S(j,ti)) for i = 1,2,..., A;
(see [McShane E. J., 1969, 1983],[ Kurzweil J., 1980], [McLeod R. M., 1980]). Hence every Lebesgue integrable function / is *integrable and obviously every *integrable / is integrable. Let G : I —>• R and assume that G(t) e R for t e I. It is not difficult to prove that G is * integrable and G is its prim itive. Therefore G G *P, where *P is the set of primitives of * integrable functions. Consequently there exist * integrable functions which are not Lebesgue integrable. Put H(a) = 0, H(t) = exp(-|ln(*-a)|2)sin
for
te(a,b].
t —a Then H £P\*P. Can results from Sections 6 and 8 be extended to "integration? 9.3. Stepfunctions are linear combinations of characteristic functions of intervals. In [Jarnfk J. and Kurzweil J., 1995], Theorem 2.1 it was proved that (9.6) if n E N, if / „ is an interval in R n and if g : In -» R is strongly p-integrable, then there exists a sequence of stepfunctions gj : In —► R, j € N such that gj is strongly /9-equiconvergent to g for j —► oo. Let n = 1 and denote by PST the set of primitives of stepfunctions h : I —► R where / = [a, b] C R. Then strongly p-integrable means integrable and it can be deduced from (9.6) and Theorem 2.15 that (9.7) for every F £ P there exist 6 e D and a sequence Fj G PST, j e N such that Fj e Q(6) for j £ N and \\Fj - F\\ - 0 for j - oo.
9. OPEN PROBLEMS
121
It follows that (9.8) PST is dense in (P,U) if U is a topology on P tolerant to Q-convergence and (9.9) P is the completion of ( P 5 T , T ( D * ) | P s T ) . Can (9.7) be extended to other integrations? Note. In [Jarnfk J. and Kurzweil J., 1997] a result analogous to (9.6) vas proved for * integration on n-dimensional intervals. 9.4. Denote by PPOL the set of primitives of polynomials on I = [ab] C R. Let c G (a,fc), g(t) = 0 for t G [a,c], g(t) = 1 for t E (c, b].
By Weierstrass theorem there exists a polynomial gj such that \9j(t)\ < 2~j for a
2~j,
- 2 " j < gj(t) < 1 + 2~j for c - 2~j < t < c + 2~j, \9j(t) - 1| < 2~j for c + 2~j provided j 6 N, j > jo where a
c + 2~jo
Let Gj be the primitive of gj for j G N and let G be the primitive of g. Obviously (9.10)
HO,- - G|| -> 0 for j -» oo
and it is not difficult to prove that (9.11) there exists 6 e D such that Gj G Q(S) for j G N (cf. Theorem 2.15). Therefore (9.12) PPOL is dense in (P,ZY) provided U is a topology on P tolerant to Q-convergence.
122
HENSTOCK-KURZWEIL INTEGRATION
Let F e P. Do there exist 6 e D and Fj 6 Ppoz, for j e N such that ||Fj - F\\ -> 0 for j -> oo and Fj 6 Q(<5) for j G N?
APPENDIX
A . l Filters. Let Y be a set. Let 3 be a set of subsets Z of Y, shortly 3 = {Z}. 3 is called a filter on Y provided
(A.l) 3 ^ 0 , 0 ^ 3 , (A.2) if Zi 6 3 , 2 i C Z 2 C Y, then Z 2 G 3, (A.3) if ZuZ2e 3, then Zi n Z2 G 3Let 55 be a set of subsets of Y. 05 is called a filter base, if (A.4) 53 ^ 0, 0 i 23, (A.5) if B1,B2e 55, then there exists S 3 € 55, 5 3 C # i n B2. Let 3(55) be the set of Z C Y such that there exists B G 55, £ C Z. It follows that 3(55) is a filter; it is called the filter generated by 55. Obviously 55 C 3(55). A.2 Topology. Let Y be a set. Let U be a system of subsets of Y such that (A.6)
124
HENSTOCK-KURZWEIL INTEGRATION
(Y,U) is called a Hausdorff topological space, if for every couple x,y eY, x ^ y there exist a neighbourhood Vx of x and a neighbourhood Vy of y such that Vx n Vy = 0. Let E C Y. Put Z^l^, = {U C\ E;U C £/}. Obviously ZY^ is a topology on E. U\E is called the topology induced on E. For the sake of simplicity it may be written (E,U) instead of (E,U\E). Let U\,Ui be topologies on Y. Ui is finer (stronger) than Hi (and U\ is coarser (weaker) than £V2), if U\ C ZY2. The coarsest topology on y is the trivial topology, the finest topology is the discrete topology. Let T = {r} be a set of parameters and let UT be a topology on Y for r G T. Let us denote by sup{WT;r G T} the set the elements of which are intersections U\ D U2 n • • • n Ur, where r G N and r(i) G T, t/j G WT(i) for z = 1,2,..., r, and unions of such intersections. It follows that sup{ZYT;r G T] is a topology on y and that it is the coarsest of topologies U on Y which are finer than UT for r £T. A . 3 . Let Y be a vector space, V C Y. Recall that V is called balanced (circled) if AV C V for A G [—1,1] and V is called absorbing (radial), if for every y G F there exists /^ G M+ such that j/ G AF if A G R, |A| > /x. Let U be a topology o n F . If y £ Y, U £ U implies that y + U G U, then £V is called invariant with respect to shifts. A.4 Definition. Let Y be a vector space and let (Y, U) be a Hausdorff topological space. (Y,U) is called a topological vector space if (A.9) the map (x, y) 1—► x + y from Y x F to Y is continuous (A. 10) the map (A, y) \-+ Xy from R x Y to Y is continuous. If (y, £/) is a topological vector space and if A G R, A ^ 0, x G Y, then the map y — i > re + Ay is a homeomorphism of V. A.5 Theorem. Let Y be a vector space. Let U be a topol ogy on Y.
APPENDIX
125
Then (A.ll) (y,U) is a topological vector space if and only if there exists 2D = {W} C 2Y such that the follow ing conditions are fulfilled (A.12) ifyeY,U G U, then y + U eU, (A.13) 0 £ 2 D , 2 D # 0 (A. 14) every W from 2D is balanced and absorbing, (A. 15) ifWuW2 € 2D there exists W3 G 2D such that W3 C
wx n w 2 , (A. 16) for W G 2D there exists Wi G 2D such that Wi + Wi C (A. 17) 2D is a base of the filter of neighbourhoods of zero in (y,W) fshortiy O-hiterbase in {Y,U)). Corollary. Let 2D C 2 y fulfil (A.13), (A.14), (A.15) and (A. 16). Let U = W(2D) be the set ofUe2Y such that (A.18) ifx G U, then there exists W G 2D such that x+W CU. It follows that U is a topology on Y such that (A.12) and (A.17) hold. Therefore (A.ll) holds. Note 1. Theorem A.5 is a detailed version of Theorem 1.2, [Schaeffer H. H., 1980]. Note 2. It follows from (A.16) by induction that (A. 19) for W G 2D there exists a sequence of Wt G 2D, i G N such that oo
oo
i=i
j=i
j
E ^ = UE w * cW t=i
Note 3. (y, ZY) is a Hausdorff topological vector space if and only if (A.20) for every y G Y, y / 0 there exists W G 2D such that y$W.
126
HENSTOCK-KURZWEIL INTEGRATION
A.6 Definition. Let (Y,U) be a topological vector space, let 3 = {Z} be a filter in Y. 3 is called a Cauchy filter (in (Y,U)), if for every neighbourhood W of zero in (Y,U) there exists Z e 3 such that Z - Z cW. A.7 Definition. Let (Y,U) be a topological vector space, y EY and let 3 = {Z} be a filter in Y. For every neighbourhood of zero V in (Y, U) let there exist Z E 3 such that Z C y + V. Then 3 is called convergent to y, 3-y(in(y,W)). A.8 Theorem. Let (Y, UT) be a topological vector space for r ET. Then (Y,sup{UT; r ET}) is a topological vector space. Theorem A.8 can be verified directly from Definition A.4 and the definition of sup{UT; r E T} in Section A.2. A.9 Theorem. Let (Y, UT) be a topological vector space for r E T, let 3 = {Z} be a filter in Y, 3 -+ 0 in (Y,UT) for TET. Then 3 -► 0 in (Y,sup{WT; r E T}). Proof. Let 0 E U E sup{ZVr;r E T}. There exist r E N, r{i) E T, Ui E W(r(«)) for i = l , 2 , . . . , r such that 0 E Ui D U2 H • • • n Ur C U. There exist Z{ E 3 such that Zj C Ui for i = 1,2,..., r. It follows that Z = Zx n Z 2 D • • • D Zr C t/. The proof is complete. D A.10 Definition. Let {Y,U) be a topological vector space. (Y,U) is called complete, if for every Cauchy filter 3 on Y there exists y EY such that 3 —* yA.11 Definition. Let (Y, ZY) be a Hausdorff topological vec tor space. (y,U) is called a locally convex topological vector space, shortly a locally convex space, if the set of convex neigh bourhoods of zero is a 0-neighbourhood base. A. 12 Theorem. Let Y be a vector space and let 2U be a set ofW C Y fulfilling (A.13), (A.15), (A.16), (A.20) and (A.21) every W from 2U is balanced, absorbing and convex.
APPENDIX
127
Let U be the set of U C 2Y which fulfil (A.18). Then U = U(2B) is a topology on Y and (Y,U) is a locally convex topological vector space. Proof. Theorem A. 12 is a consequence of Corollary A.5 and Note 3 in Section A.5. □
List of symbols A 10, 35 B(x,p) 48 B(p) 48 cless£ 22 c\E 21 convZ 51 CxBlCl 48 CxBlCp 41 D 21 D* 9 Z> 48 V 78 2>ij 78 D 85 D* 78 dens.E 22 diam£ 119 dist(x, E) 48 \E\ 9 / k 11 FK 67 Fi-^F Fi^->F H,H,ii J 8 Int(J,J) Inttf 8 Iv(tf) 9 M ^ 27
36 42 12
7[ri,r 2 ] 64 P, P(7) 33, 36 P 49 PK 67 PPOL PST
T(6,K)
121
120 Q 48 g ^ ) 4i R(p) 51 5(7, T,C) 9 sup{UT;reT} S{u) 75 T(D) 85 U\V\ 79 « ( | | . | | ) 48
WflHDIv 48 W£ c 53 W-JV 77 2&LC 5 2 2H(0) 75 X 48 X,- - ^ i 49 (V,W(|| - ||) 48
87
128
21
<5 9 A 8 A 79 A* 79
ft*
75
25
INDEX
absorbing A.3 additive 1.6 balanced A.3 closed set A.2 coarser topology A.2 Cauchy filter A.6 circled A.3 complete topological vector space A. 10 convergent filter A.7 convex hull 4.8 density point 2.4 discrete topology A. 2 ^-convergent 3.5 equiintegrable 3.1 essential closure 2.4 filter A.l filter base A.l filter generated by A.l finer topology A.2 gauge 1.3 Hausdorff topological space A. 2 Henstock - Kurzweil integrable 1.4 induced topology A.2 integrable 1.4 integral 1.4 129
124 10 124 123 124 126 124 126 126 51 22 123 36 34 22 123 123 123 124 9 124 9 124 9 9
130
HENSTOCK-KURZWEIL INTEGRATION
*integrable 9.2 invariant with respect to shifts A.3 locally convex space A. 11 locally convex topological vector space A. 11 Lusin's condition iV 1.14 mixed topology 5.10 multiplier 6.7 neighbourhood A.2 neighbourhood filter A.2 nonoverlapping 2.13 open set A.2 partition of K 1.1 primitive of / 1.8 Q-convergent 3.11 Q-convergent 4.1 radial A.3 set of density points of E 2.4 strong Lusin condition on / 1.13 system of tagged intervals on / 1.1 tag 1.1 tagged interval 1.1 tolerant to ^-convergence 3.13 tolerant to Q-convergence 3.13 tolerant to Q-convergence 4.3 topology A.2 topology induced on E A.2 topological space A.2 topological vector space A.4 trivial topology A.2 T-tagged 1.1 two norms convergence 5.9 C-fine 1.1 7-convergence 5.9 0-filterbase A.5
119 124 126 126 15 63 73 123 123 29 123 8 11 42 49 123 22 15 8 8 8 46 46 49 123 124 123 124 123 8 62 8 62 125
REFERENCES
Alexiewicz A., 1950 On sequences of operations (II). Studia mathematica XI, 200 - 236 Bongiorno B., Pfeffer W. F. and Thomson B. S., 1996 A full descriptive definition of the gage integral. Canadian Math. Bull. 39, 390 - 401 Bruckner A., 1994 Differentiation of Real Functions. American Mathematical Society, Providence Buczolich Z. and Pfeffer W. F., 1997 Variations of additive functions. Czechoslovak Math. J. 47 (122), 525 - 554 Cousin P., 1895 Sur les fonctions de n variables complexes. Acta Math. 19, 1 - 6 2 Henstock R., 1961 Definitions of Riemann type of the variational integrals. Proc. London Math. Soc. 11, 402 - 418 Henstock R., 1963 Theory of Integration. Butterworths, London Jarnik J. and Kurzweil J., 1985 A non-absolutely convergent integral which admits transfor mation and can be used for integration on manifolds. Czechoslovak Math. J. 35 (110), 116 - 139 131
132
HENSTOCK-KURZWEIL INTEGRATION
Jarnik J. and Kurzweil J., 1995 Perron-type integration on n-dimensional intervals and its properties. Czechoslovak Math. J. 45 (120), 116 - 139 Jarnik J. and Kurzweil J., 1997 Another Perron type integration in n dimensions as an ex tension of integration of step functions. Czechoslovak Math. J. 47 (122), 557 - 575 Jurkat W. B. and Nonnenmacher D . J. F., 1994 a A generalized n-dimensional Riemann integral and the diver gence theorem with singularities. Acta Sci. Math. (Szeged) 59, 241 - 256 Jurkat W. B. and Nonnenmacher D . J. F., 1994 b An axiomatic theory of non-absolutely convergent integrals inW1. Fund. Math. 145, 221 - 242 Jurkat W. B. and Nonnenmacher D . J. F., 1994 c A theory of non-absolutely convergent integrals with singular ities on a regular boundary. Fund. Math. 146, 69 - 84 Jurkat W. B. and Nonnenmacher D . J. F., 1995 a The fundamental theorem for the v\ -integral on more general sets and a corresponding divergence theorem with singulari ties. Czechoslovak Math. J. 45 (120), 69 - 77 Jurkat W. B. and Nonnenmacher D. J. F., 1995 b A Hake-type property for the v\-integral and its relation to other integration processes. Czechoslovak Math. J. 45 (120), 465 - 472 Kurzweil J., 1957 Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (82), 418 - 449
REFERENCES
133
Kurzweil J., 1958 On integration by parts. Czechoslovak Math. J. 8 (83), 356 - 359 Kurzweil J., 1980 Nichtabsolut konvergente Integrale. Teubner, Leipzig Kurzweil J. and Jarnfk J., 1991-92 Equiintegrability and controlled convergence of Perron-type integrable functions. Real Analysis Exchange 17, 110 - 139 Kurzweil J. and Jarnfk J., 1990 The PU-integral: its definition and some basic properties. Lecture Notes in Math. 1419, Springer Verlag, Berlin, pp. 66 - 81 (Proceedings of the Henstock conference held in Corelaine, Northern Ireland, August 1988.) Kurzweil J. and Jarnfk J., 1992 Equivalent definitions of regular generalized Perron integral. Czechoslovak Math. J. 42 (117), 365 - 378 Kurzweil J. and Jarnfk J., 1996 Perron type integration on n-dimensional intervals as an ex tension of integration of stepfunctions by strong equiconvergence. Czechoslovak Math. J. 46 (121), 1 - 20 Kurzweil J . and Jarnfk J., 1997 A convergence theorem for Henstock - Kurzweil integral and its relation to topology. Bulletin de la Classe des Sciences, 6e serie, Tome VIII, Academie Royale de Belgique, 217 - 230 Kurzweil J. and Jarnfk J., 1998 Henstock - Kurzweil integration; convergence cannot be re placed by a locally convex topology. Bulletin de la Classe des Sciences, 6e serie, Tome IX, Academie Royale de Belgique, 331 - 347
134
HENSTOCK-KURZWEIL INTEGRATION
Kurzweil J., Mawhin J. and Pfeffer W. F., 1991 An integral defined by approximating BV partitions of unity. Czechoslovak Math. J. 41 (116), 695 - 712 Lee P.-Y. and Chew T. S., 1985 A better convergence theorem for Henstock integrals. Bulletin London Math. Soc. 17, 557 - 565 Lee P.-Y., 1989 Lanzhou Lectures on Henstock Integration World Scientific, Singapore Lee P.-Y. and Vyborny R., 1993 Kurzweil - Henstock integration and the Strong Lusin condi tion. Bolletino U. M. I. (7) 7-B, 761 - 773 Lee P.-Y. and Vyborny R., 1999 The Integral; An Easy Approach after Kurzweil and Hen stock. Cambridge Univ. Press Lee T. Y., Chew T. S. and Lee P.-Y. , 1996 Characterization of multipliers for the double Henstock integ rals. Bull. Austral. Math. Soc. 54, 441 - 449 Lukes J. and Maly J., 1995 Measure and Integral. Matfyzpress, Publ. House of the Faculty of Mathematics and Physics, Charles University Prague McLeod R. M., 1980 The Generalized Riemann Integral. The Mathematical Association of America. The Carus Math ematical Monographs No. 20 McShane E. J., 1969 A Riemann-Type Integral that Includes Lebesgue - Stieltjes, Bochner and Stochastic Integrals. Memoirs of the AMS, Providence No. 88
REFERENCES
135
McShane E. J., 1983 Unified Integration. Academic Press, New York Nonnenmacher D. J. F., 1994 A desriptive, additive modification of Mawhin's integral and the divergence theorem with singularities. Ann. Polon. Math. LIX, 85 - 98 Nonnenmacher D . J. F., 1995 Sets of finite perimeter and the Gauss - Green theorem with singularities. J. London Math. Soc. (2) 52, 335 - 344 Pfeffer W. F., 1987 A Riemann type integration and the fundamental theorem of calculus. Rendicont Circ. Mat. Palermo, Ser. II 36, 482 - 506 Pfeffer W. F., 1991 The Gauss - Green theorem. Adv. in Math. 87, 93 - 147 Pfeffer W. F., 1993 The Riemann Approach to Integration. Cambridge University Press Rogers C. A., 1970 Hausdorff Measures. Cambridge University Press Saks S., 1937 Theory of the Integral. Monografie Matematyczne, Warszawa Sargent W. L. C. 1948 On the integrability of products. J. London Math. Soc. 23, 28 - 34 Schaefer H. H., 1980 Topological Vector Spaces. Springer-Verlag, New York
136
HENSTOCK-KURZWEIL INTEGRATION
Wiweger A., 1961 Linear spaces with mixed topology. Studia Mathematica XX, 47 - 68
