Non-linear operator valued measures and integration

R O CZNIKI P OLSKIEGO TO W A R Z YSTW A M A T E M A T Y C ZN E G O Séria I: PRACE M A T E M A T Y C ZN E XXVII (1988)

To m a s z Sz c z y p i n s k i (Krakôw)

Non-linear operator valued measures and integration

This paper is an attempt to give a representation o f the theory of integration with respect to operator measures with values in the space iV(S, F) o f non-linear transformations.

Integration with respect to such measures was considered by many authors in connection with representation o f orthogonally additive operators, see e.g. Friedman, Tong [9 ] and Batt [2]. In their papers they considered integration o f totally measurably functions only. W e extend the class of integrable functions to the space Jt{n) which contains also some unbounded functions.

If one takes L(S, F), the space o f all bounded linear operators instead of N (S, F), the theory by Dobrakov [5 ] becomes a particular case o f the theory presented here.

The paper is a part o f author’s doctoral thesis [14]; the reader is also referred to [10] and [11] for some partial results.

The paper consists o f five sections. Section 1 is a brief exposition of some basic concepts of the theory of operator measures. Next the definition of integration o f totally measurable functions is presented and discussed.

Section 2 is devoted to extending the class o f integrable functions; we consider also various conditions which are equivalent to integrability.

Section 3 deals with various principal properties of integration in the above sense, e.g. orthogonal additivity and absolute continuity o f integral.

In Section 4 various types of convergence o f measurable functions are discussed. It contains also the Vitali-type theorem.

Section 5 contains examples, which show some relations to other theories o f integration and, on the other side, the possibilities o f applications to investigation o f non-linear operators.

I should like to express my thanks to Professor J. Musielak for his kind, continuing help and encouragement during the work. M y thanks are ex­

tended also to Dr. W. M. Kozlowski for his reasonable remarks and im­

provements of the text.

314 T o m a s z S z c z y p i r ï s k i

1. Preliminaries. Let F be a Banach space and let 5 be a separable Banach space. By N (S , F) we denote the space o f all functions U: S-> F satisfying the following conditions:

(1) U (0) = 0,

(2) U is uniformly continuous on every K a = [reS: ||r|| < a j .

Further, let X be a non-empty set, 0A a (5-ring (a ring closed with respect to the forming o f countable intersections) of subsets of X and <r(^) the ti­

ring generated by Let 0$ denote the а-algebra of all subsets of X which are locally in i.e., 0% = [E с X : E n A e 3P for every A e 0*}.

A set function p: 0* -> N(S, F) is said to be an operator valued measure if p has the following properties:

(1) л ( 0 ) = 0,

(2) for every r e S and for every sequence of pairly disjoint sets (B,) from 00

& such that U B, e 0P we have

i = 1

00 00

p( (J Bt) r = £ p(Bi)r.

i = l i = 1

The operator valued measure p is countably additive in the strong operator topology (in the point-wise sense) but not necessarily countably additive in the uniform operator topology. Let us note that by the O rlicz- Pettis theorem (see IV. 10.1. in [7 ]) the countable additivity in the strong and in the weak topology are equivalent conditions.

For every a > 0 and E с X the semivariation of the operator valued measure p is defined by the formula

sv(pa, E) = sup [|| £ р(Е{)r,-||: (J Е{ с E, E{e0P, ||rf|| < a, X ^ i ^ n } .

• i = i <=i

For every a, <5 > 0 and E с: X the <5-semivariation of the operator valued measure p is defined by the equality

s v ^ , E) = sup {|| £ [p{Edr( —p(£ ,)r-]||: (J £, a E, Е{е& ,

i = 1 «=1

M > 1ИИ < a, l h - ^ l l < à, 1 ^ i ^ n}.

For every E а X the scalar semivariation of vector measure v: 0? - + Z (Z is a Banach space) is defined by the formula

\\v\\{E) = sup (|| £ а; г(£;)||: (J с £, Et e0?, |аг| < 1, 1 < i < л }.

i = 1 «'=1

If p is an operator measure on then, for each re S , ||ji(-)rll is a

countably subadditive set function. For every r eS and E с X the sub­

measure majorant for ц is defined by the formula Ц(Е)г = sup {Ц/2(A )r||: А с E, АеЗР).

A set N e & is called a fi-null set if sv(/ia, N) = 0 for every ot > 0. The term /г-almost everywhere refers to the complement of a /г-null set.

A set N <= X is called а ц-negligible set if N is a subset of a /г-null set.

The symbol denotes the or-algebra of unions E u N, where Ee& and N is a /^-negligible set.

In this paper we assume that /г satisfies the following conditions:

(/d) for each Ee£P, ot > 0, svô(fia, E) tends to 0 as <5 tends to 0, (fi2) for each r e S submeasure majorant for /г is continuous on i.e., if Ene & , En \ 0 , then limЦ(Е„)г = 0.

П

By g we shall denote the space o f all ^-simple functions on X with values in S.

A function / : X -> S is called totally .^-measurable if it vanishes outside a set A e & and if there exists a sequence (s„) of ^-simple functions converging uniformly to / By g 0 we shall denote the space o f all totally measurable functions acting from X to S.

П

1.1. Definition (a) For every serf o f the form s = £ г{ -1А.the integral i = i

of s over Ee$ is defined by

П

I'sdfi = J ц ( Е п Ai)rt.

E i=

1

(b) For every f e g 0, f vanishes outside a set A e the integral of / over E e $ is defined by

j f d f i = f f d f i = lim js„dn,

E EnA и E

where (s„) о fi and sn =$ f. Such an integration is well defined. This fact can be proved using condition (/Л), because || js d /г— Js'd/гЦ < s v ,,^ , £ ) for

A E

every s, s’ efi such that sup||s(x)|| ^ ot, sup||s'(x)|| ^ a and sup||s(x) —s'(x)||

X X X

^<5. The detailed discussion was carried out in [14], Section 1.

The following proposition is easy to check (cf. [14]):

1.2. Proposition. For every f, f u /2e 0 such that supp f x nsupp f 2= 0 we have

f ( / + /i + / j № = J ( / + / i № + I f dll

E E E E

316 T o m a s z S z c z y p i r i s k i

for every E e $ .

1.3. Corollary. I f / i,/ 2e S0 and suppf k n su ppf 2 = 0 , then for every E e 0 t

J(/i + f i)d g = f / i ^ + f / 2^ .

£ E E

1.4. Proposition. For every f , g e S 0 such that sup||/(x)|| ^ a, xeE

sup||g(x)|| and sup||/(x) —0(x)|| ^ <5 we have

xeE xeE

||$ fd n ~ jg d u|| ^ svô(na, E) for Ее 01.

E E

P r o o f. If Е е 01 and f , g e S 0 are such that

||/(х)Ж а, ||0(x)| | ^ a for each x e E and sup||/(x)-0(x)|| ^ <5, xeE

then we can find a natural number n0 such that, for n ^ n0 and x e E ,

||s„(x)|| < a, |K(x)|| ^ a,

sup I |s„ (x ) — s^, (x)|| ^ S, where s„ =* / and s'n=lg.

xeE

From the definition o f the <5-semivariation we immediately obtain that for each n ^ n0

| | Jdp— js'ndfi\\ ^ s\0{ца, E).

£ £

Thus

IlS f d f i - \gdfi\\ < S\ô{tia, E).

E E

1.5. Definition. A sequence (A„) of sets from is said to be ц- convergent to О (Л„^>-0) iff for every r e S Ц(А„)г-> 0 as n-+ oo.

1.6. Th e o r e m (Absolute Continuity Theorem on S0). I f f e â 0, A ne $ and A „ {-^> 0, then j / d ^ - ^ 0 .

Ân

m ^

P r o o f. S te p 1. I f seS , s = then i= 1

m m

II f 5^11 = IIZ МЛ. < Z 1И4.

°»

An i= 1 i = 1

because А и n £ , с : Л и and A n ^ > 0.

S te p 2. Let f e S0, (sk) S, sk=} f and A = supp/e^*. For each

a, e > 0, there exists a ô > 0 such that s\0(ga, A) ^ (cf. condition (/Л)).

Since the sequence (sk) converges uniformly to f it follows that for S there exists a К > 0 such that for every n

sup\\sk- l An( x ) - f - l An(x)\\^S, where k > K .

X

Denoting a = m a x {a b a2}, where acj = sup||/(x)||, a2 — sup||sk(x)||, we ob-

x k,x

tain (cf. Proposition 1.4) for every n and к > К

Il J f d g - J skdg II ^ s vô(ga, s\ô{ga, A) ^ \г.

AnnA AnnA

From step 1 for every к and for sufficiently large n we have

|| j skdg|| ^ ^£. Then AnA„

111/<41 = 11 1 /<41 <11 1 /<*/<- 1 s*<4l + ll 1 s*<4l<£

An AnAn AnAn AnAn AnAn

for к > К and for sufficiently large n. The theorem is proved completely.

1.7. Proposition. For every Ng & the following conditions are equivalent: (a) sv(ga, N) = 0 for every a > 0,

(b ) g ( N ) r = 0 for every reS, (c) § sdg = 0 for every se S.

N

P r o o f, (a) -►(b). If re S , then ||r|| < a for some a > 0 and g ( N ) r

= sup|HB)r|| ^ s\{ga, N) = 0.

В c N

Be0>

n

(b) - » (c). If s e ê , s = Yjг1' !аг then i= 1

Y Wn{Ai n N ) r i\\=0,

N i= 1 i = 1

because A{ n N c. N and g (N )r i = 0 for each i.

( c ) ->(a). s\(ga, N ) = sup {||Jsd/i||: s eS , sup||s(x)|| ^ a } = 0 since §sdg

N X n

= 0 for each se S’.

If A e 3$*, then A is o f the form A = A' и N , where A e 01 and N is g- negligible. W e extend the definition o f integral by the formula § fd g = J f d g

A A’

for every / e S0.

Dinculeanu in [4 ] introduced integration o f totally measurable functions with respect to linear additive set functions g: Ш -> L(S, F). Batt [2 ] and Friedman and Tong [9 ] considered non-linear and finitely additive cases. In

E

their papers, N (S, F) was equipped with the uniform topology, similarly as in [8]. W e assumed ju to be countably additive in the point-wise sense.

2. Integrable functions.

2.1. Definition. A function / : X —> S is called measurable if there exists a sequence o f ^-simple functions (sn) such that sn - * f /i-almost everywhere on X.

Similarly as in [7 ], one can prove that / - 1 ( G ) e ^ * i f / is a measurable function and G is an open subset o f S.

2.2. Theorem. Let a sequence of simple functions (s„) converge ц-almost everywhere to a function f Then there exists a non-decreasing sequence of sets

00

Z kG & {k = 1, 2, ...) such that (J Z k — X \ N , where N = N 0u Z 0, N 0 is fi-

k= 1

null, Z 0 cz {x eX : f ( x ) = 0} and on each set Z k, к ⁼ 1, 2, ..., the sequence (sn) converges uniformly to the function f

P r o o f. This proof employs some ideas from [12] and the reader is referred to this paper for further details. Put Tm — {xeX\ ||sm(x)|| > 0 }; it is

00

obvious that TmeâP for every m. Therefore, T = (J Tm is a member o f <r(^).

m= 1

For every m we may introduce a new <r-algebra by the formula

= \A n Tm: Aeâ#] <= & (cf. [4 ], p. 5, Proposition 8); by цт we denote the restriction o f g. to SAm.

Let us denote by Q = {rx, r2, ...} a countable set such that Q = S. For every rpeQ we define a vector measure gm{ - ) r p: 0>m-> F ; to ftm( ’) r p there corresponds a finite, positive control measure (cf. [7], IV. 10.5). Let us define

318 T o m a s z S z c z y pin s ki

® 2P (W)

Лт{Е)= I 2~_{p= 1} p7171 г Л т{1т)p7FT for еуегУ Ее^гп-

In this way for every natural m we have constructed a new positive measure Xm. Similarly as in the proof of the Egoroff Theorem in [1 2 ], we conclude that s„—> f Am-almost everywhere in Tm.

00

Put X{E) = Yj 2~тЛт{Ттг\Е) for Eeffl. One can check that Я is a m= 1

finite positive measure on 0Ï and sn-> f Я-almost everywhere. By the classic Egoroff Theorem ([7 ], III.6.12) we get a non-decreasing sequence o f sets

00

Hke 01 such that Я( П (-ХДЯ*)) = 0.

к — 1

Let us observe that every A e ^ is a union o f A' and A", where A' = A n Te u{ij/) and A " = A \A' e$, furthermore Я (Л ") = 0, since Xm(Tm n A " ) = Xm( 0 ) = 0 for every m.

By this fact one can get a non-decreasing sequence of sets Z ke & such that Z k c= Hk and À( П (A ^ Z *)) = 0.

k = 1

00

Writing N — П (-V\Zk), we have again N = N ' v N ”, where

k = 1

N' = N глТ e o (.J/) and

N " = N \ N ' c z X \ T = { x : s„(x) = 0 for every n}.

Observe that À(N') = 0 implies Xm(N ' n T J = 0 for every m and conse­

quently the control measure À^{N'r\Tm) = 0 for every m, p. From the definition of control measure ([7 ], IV .10.5) it follows that scalar semivari­

ation ||//m( - ) rPll(N ' n Tm) is equal to 0, which implies f m(N ' n Tm)r p = 0; then by the local uniform continuity of // and the density of Q in S we obtain that N ' r\Tm is fim-null and obviously N ' n T m is //-null as well. Since AT

00 00

= (J (N ' n Tm) and f ( N ' ) r ^ ]T jI(N ' n T m)r — 0 for every r e S by the

m = 1 m = 1

countable subadditivity o f Д, it follows that N ' is //-null set.

Note that N " <= lx: s„(x) / + / (x )} u {x: f ( x ) = 0}, and write N k

= N " n \x: s„(x) f (x)} and Z 0 = N " n {x: / (x) = 0). is //-null, be­

cause s„ - * f //-almost everywhere. Finally,

N = N ' u N " = N ' u N k u Z 0 = No u Z 0,

where W0 = N ' и N j is a //-null and Z 0 <= {* : / (x) = 0}, which completes the proof.

00

R em a rk . In the case where X — (J X n, X ne3P, one can repeat the

n— 1

proof with X„ playing role of Tm and get a non-decreasing sequence o f sets

00

Z keid? such that sn =$f on every Z k and [ j Z k = X \ N 0, where N 0 is //-null (cf. [12]).

2.3. Lemma. Let (v„) be a sequence of countably additive vector measures on a-algebra I f the sequence (v„) is uniformly additive on 9R, then the sequence of their scalar semivariations ||v„|| is uniformly continuous on $01

P r o o f. From the assumptions it follows that, if (Л к) c SJÎ and Ak \ 0 , then ||v„(v4k)|| -* 0 uniformly with respect to n. Because is a tr-algebra therefore ||v„(£fe)|| ->0 uniformly, where (Bk) is a sequence o f disjoint sets of Ж

W e obtain (cf. [6], II, Theorem 5.7) that the family o f submeasure majorants (v„) is uniformly continuous on

But ||v„(£)|| ^ v„(£) (cf. [7], IV .10.4), so the sequence of scalar semivari­

ations is uniformly continuous on $R.

320 T o m a s z S z c z y p i n s k i

During the proof o f the following theorem we use some concepts from [5].

2.4. Theorem. I f a sequence of 0 - simple functions (sn) converges p-almost everywhere to a function f and the integrals j s„ d/i are uniformly countably

(•)

additive on .0, then there exists the limit lim \s„dpeF uniformly with respect

n E

to Eg

This limit is independent on the choice of the sequence of simple functions which satisfy the above assumptions.

P r o o f. From Theorem 2.2 it follows that there exists a sequence o f sets Z ke 0 > such that sn =4/ uniformly on Z k and X \ Z k \ N , where N

= N 0 u Z 0, N 0 is a /i-null set, Z 0 a {x: f ( x ) = 0) . Observe that \s„dp = 0 N

for n = 1, 2, ... For every n, p, к = 1, 2, ... and each set the following inequality holds:

НК**/*- fs„d/i||

E E

*Z\\\S' -l Ztdn- fsp*/Zki/M|| + ||fs„^/i||(A'\iV\Zk) + ||jspd/i|[(X\JV\Zlt).

E к

Since the scalar semivariations | | J ( * ) are uniformly continuous on M (cf.

Lemma 2.3), we can find a number k0 such that for к > k0 we have the following inequalities:

sup||Jsnd/i||(X\iV\Z*) ^ sup HJsp d/r|| (X \ iV \ Z k) ^ .

n n

Because the sequence (s„) converges uniformly to a function / on every Z k, then for every к there exists ock > 0 such that

sup||s„-fZk(x)||

and by Proposition 1.4

| | f v l Zk^ - $Sp-lzkdg\\ < svsin*, Z k)

к к

for every n, p and a > a*, where <5 = sup||s„-lZk(x) — sp- l Zk(x)\\.

X

Hence for sufficiently large к ^ k0

\\$snd p - $spdp\\ ^ s\0{pa, Z k) + & + i e .

E E

Since Z ke 0 , we have svô(ga, Z k) ^ for sufficiently large n, p. Therefore,

\\js„dp- Jspd/i|| < в.

E E

Since F is complete, it follows then that lim j s„ dpt exists and the limit is П E

uniform with respect to E e M .

For the proof of the uniqueness assertion in the theorem observe that for any sequence o f ^-simple functions (r„), which satisfies the above assumptions, the sequence (sl5 t x, s2, t2, ...) converges almost everywhere to the function / and their integrals are uniformly countably additive on $.

From the previous part of the proof we conclude that there exists a limit for this sequence o f integrals. Hence it follows that for each E e $

lim §sndg = lim jtnd/л.

n E n £

The theorem is proved completely.

2.5. Theorem. The following classes of functions are equal :

M i = {/ : X -*S: there exists a sequence (s„) of simple functions which converges to f /л-almost everywhere and such that

lim §s„d/i exists for each Ee0t},

n E

M 2 = {/: X -*S: there exists a sequence (s j of &-simple functions which converges to f /л-almost everywhere and J s„ d/i are

(•)

uniformly countably additive set functions on Щ ,

М ъ = {/ : X —>5' measurable such that, if A n \ N 0, where N 0 is /л-null set, (An) c= $ , e > 0, E e & , then for sufficiently large n I I J / 'M d l < e for every Ae0>, A cz A n such that

f ’ 1A E $o}>

M A = {/: X -*S measurable such that, if E e0t, Bn cz A n \ N 0, where N 0 is /л-null set, (Bn) cz &>, (An) <z® and f - l Bne S 0, then j f ' l B d/л-^О as n -* oo},

E

M$ = {/: X -*S measurable such that there exists a non­

decreasing sequence o f sets (Z k) cz 0>, X \ Z k \ N , where N = Z 0u N o, N 0 is a /i-null set, Z 0 a {x: / (x) = 0}, /• l Zfc e<f0 for each к = 1,2, ... and lim \ f - l Zkd/i exists for every

к E

Е е 0t}.

P r o o f. W e shall prove the following sequence of inclusions:

M i cz M 2 <zz М ъ cz M A cz M 5 cz M x.

M i cz M 2. Since lim j s„ d/л exists for every E e (%, it follows by the n E

Nikodym theorem ([7 ], Theorem IV.10.6) that the sequence o f measures j s„ d/i is uniformly countably additive on 01.

<•)

322 T o m a s z S z c z y p i r ï s k i

M 2 c= ,МЪ. Let f g'M2 • Then there exists the sequence o f ^-simple functions such that s„ -> f //-almost everywhere and the sequence o f sub­

measures v „(-) = || |'s„////|| is uniformly countably subadditive on Consider (’•)

the family of submeasures majorants

v„{E) = sup||j\d//||.

В<=Е В

Let Ak \ N 0, where N 0 is //-null. Since / e M 2, vn{Ak\No) - * 0 , k - + oо

uniformly with respect to n ([6 ], II, Theorem 5.7). Thus for sufficiently large к and for Ae0> such that A \ N 0 c= Ak \ N 0 we have sup || J s„d//||^^e.

» a\n q

Since N 0 is //-null, j sndpi = 0 and consequently, No

(i) sup||js„d//||

n A

Recall that f - l Ae S0; hence there exists a sequence (t„) o f ^-simple functions such that t „ - l A =$ f 1A. W e get two sequences o f simple functions sn- 1a~> f ‘ 1a /^-almost everywhere and tn- l A = i f - l A uniformly. Take the sequence (sj •1A, t x- l A, s2- l A, ...); observe that it converges to f - l A almost everywhere. Note that J sn- 1A dji is uniformly countably additive. On the

(•)

other hand, there exists lim § t„ -lAdfi for every E e $ (cf. Definition 1.1 o f the

» E

integral on S0). By the Vitali-Hahn-Saks theorem this sequence is uniformly countably additive on .# as well. Therefore the sequence of integrals (\ s x ' l Adfi, ft x- l Adn, ...) is uniformly countably additive. Using Theorem

('■) <•)

2.4, we conclude that there exists the limit o f this sequence.

Thus lim \sn- l A d/u. = lim\t„-lAdn for every Ee.tf.

n £ Л £

Finally, for sufficiently large n we have

(ii) \\$tn- l Adpi— js n-l^d//|| < for each E e & .

E E

Compute

llï/^.4^11 ^ I I . f U n- l Adfi\\ + \\Un-iAd f i - j;S„-lAdfi\\ +

E E E E E

+ ||JV-M//|| ^ £.

£ _

The above inequality is true, because: \\i f ‘ l Ad/i— \tn- l Adju\\ ^ je from

£ £

the uniform convergence o f tn- l A to f - l A;

\\$tn- l Ad n - $sH- l Adn\\ by (ii) for n sufficiently large;

E E

\\§s„ ' l Ad/j\\ ^ y£ by (i) for A a Ak and к sufficiently large.

E

М ъ c= It is obvious.

с М ъ. Let / e Since / is a measurable function there exists a sequence o f ^-simple functions (s„) such that sn-> f ^-almost everywhere. It follows then by Theorem 2.2 that there exists a sequence of sets Z k£0>, к

00

= 1, 2, ..., such that f - l z e S0 and {J Z k = X \N , where N = iV0 u Z 0, N 0

k k = 1

is ji-null, Z 0 cz {x: f ( x) = 0 }. T o complete the proof we have to show that lim§ f - l Zkdn exists for every E e $ . W e shall check the Cauchy condition.

к E Let n ^ m.

Then

| | = II

||

E E E E E

= 1 \ if - iz n^ „ M \ ^ e

E

for sufficiently large m since f e J l A, /•fz„\zme ^o and Z „\ Z m cz ( X \ Z 0) \ Z m \ N 0.

,,#5 Let f e . / / 5. Consider the sequence o f sets Z ke0>, к 00

= 1 ,2 ,. . ., such that (J Z k = X \ N , where N = N 0kjZ 0, N 0 is ^u-null, Z 0 с {x: f ( x ) = 0} and f - l Zke £ 0(к = 1, 2, ...).

To every Z k, к ^ 1, there exists a sequence o f simple functions (s*) such that skn= t f - l Zk as n-> oo and s*(x) = 0 for every x £ Z k, n = 1, 2, ...

Fix Z k, к ^ 1. Let ak = sup ||/(x)|| + 1 . Let us choose a subsequence (s jj such that

I |s*k ( * ) - / ( * ) ! I < s < l/ k for x e Z k, where Ô is chosen such that s\0(/iak, Zk) < l//c.

Put sk = s*k- W e have thus constructed the sequence o f simple functions (sk). Observe that sk- * f ^-almost everywhere, because

l k ( * ) - / ( * ) l l < V * for x e Z k, к = 1,2, ..., . 00

sk(x) = f ( x ) = 0 for x e Z 0\ (J Z k, fc = 1 oo

and if x$ У Z k, then x e N 0, and N 0 is ju-null.

k =о

324 T o m a s z S z c z y p i n s k i

Since/g there exists limit o f the sequence { § f ' l z kdg}kL1 for every E

E Write \im \ f - l Zkdn = v(E).

к E

Fix к ^ 1 and compute:

|Jv (E) - fsk dpi\\ = 11 v (£ ) - f sk • l Zk dfi\\ (since sk = sk • l Zk)

E E

< \\$sk-lz kd n - j f ' l Zkdfi\\ + \\v{E)- S f - l Zkdn\\

E E E

< z k) + h for к > k 0

^ 1/k + ^e ^ 2fi + i e = £ for к > ki > k0.

Theorem 2.5 is proved completely.

2.6. Definition. A measurable function / is said to be integrable iff / is a member of Jf(yi) = = Л г — Л ъ — Л А = Л 5. The integral of / with respect to g, is defined by the formula

\fdg = lim \sndg, Eeffl,

E n E

where (s„) is a sequence o f ^-simple functions converging to / ^-almost everywhere such that lim §sndg exists for every E e M. Independence o f choice

n E

of (s„) is a consequence o f Theorem 2.4.

2.7. Theorem. I f f e Л (ц), then for every Ee£%

J f df i = lim $ f - l Zkdn,

E к E

where (Zk) is an arbitrary non-decreasing sequence of sets from such that Û Z k = X \ N (JV = JV0 u Z „ , N 0 is ii-null, Z 0 a {x: f { x ) = 0}), f - l z. e S 0 fc= 1

and lim\ f ' l Zkdg exists for each E from

к E

P r o o f. From Theorem 2.5 it follows that to f e Л ( ц ) there corresponds a sequence o f sets (Zk) with the above properties. Similarly as in the proof of Theorem 2.5 ( Л 5 cz J t x), we can construct a sequence o f ^-simple functions (sk) convergent /^-almost everywhere to / such that

lim \ f ’ l zkdii = lim j sk dg. for every Eg01. •

к E к e

Let us take another sequence o f sets ( Wk) with the same properties as (Z k).

There exists a sequence o f ^-simple functions (tk) such that lim j f - l Wkdfx = lim $tk'dg.

к E k E

From the Nikodym Theorem ([7 ], IV.10.6) we deduce that both sequences ( J sk dg) and ( j tk dg) are uniformly countably additive on 01. Hence by

(•) (■)

Theorem 2.4 and Definition 2.6 we obtain that lim jskdg — lim §tkdg = j / dg.

к E к E E

Thus, lim§ f - l Zkdg = limj f - l Wkdg = J / d /л, and the theorem is proved

к E к E E

completely.

Theorems 2.5 and 2.7 are crucial for our theory since they give many criterions for integrability of measurable functions. Furthermore they establish an equivalence o f two different ways o f defining the integral with respect to operator measures. The first way is to define the integral as the limit o f the sequence o f integrals o f some simple functions, while other one is to define it as the limit o f the sequence o f integrals o f the restrictions o f the previous function to some sets on which it belongs to S0.

oo

2.8. R em ark . In the case where X = (J X n, X ne0> from the remark n= 1

after Theorem 2.2 it follows that

.Мъ = !/: X measurable such that there exists a non-decreasing sequence of sets Z k from 0*, X \ Z k \ N 0, where N 0 is /л-null,

f ' l z ke<£>o and lim j f - l Zkdn exists for every £ e f ] ,

к E

N ow we extend the definition o f integral from $ to the <r-algebra 0t*.

2.9. De f i n i t i o n. For every the integral of / over E e M * is defined by the formula

dp = j / dfi,

E E'

where E = E ' u N ' , Е' еШ and N ' cz Nef fi such that N is /л-null. •

The integral is uniquely defined on 0t* since if E = E' u N' = E" u N ” and N' c N lt N " cz ]V2; let N = и N 2 so that E' и N = E" и N and thus

J fd\i = f fd\i.

E' E”

3. Principal properties.

3.1. Th e o r e m (on orthogonal additivity of integral on Jt{ii)). I f functions /, g belong -#(/л) and supp/ nsupp# = 0 , then f + g e J / { g ) and \{f + g)dg

к

= f f d g + \gdg.

E E

326 T o m a s z S z c z y p i n s k i

P r o o f. Let us put D l = supp/, D 2 = suppg, D = Dj и D2 = supp(/ -fg).

Since f , g , f + g are measurable functions, it follows that Dl , D 2, Deÿ? * and there exists a sets D\, D'2, D' eM such that D1 = D\ и Ni, Z) 2 = D2 u N2 and D = D' u N ', where N i, N 2, N ' are //-negligible.

Since f e , #(//), it follows that there exists a sequence o f sets Z ke&>

00

(k = 1 ,2 ,...) such that (J Z k = X \ N U N x = N 0u Z 0, N 0 is //-null and k= 1

Z 0 cz \x: f {x) = 0], f ■ l ZkE/:0 and lim \ f ‘ l Zkdg exists for every Ee$ . к £

Let us put Z k — Z k r\D'x and N'0 = N 0 nD\. Hence, because D\r \Z0

oo

= 0 , (J Z k = D [ \ No, where N'0 is //-null, f ' l ^ E tо and lim f/ - / 2- ////

fc= 1 к £

exists for every Ee^.

Let (Wk) denote the analogous sequence chosen for g. Obviously, 00

( f + g ) - l z kuwke tо and JJ (Z k u WJ') - D '\N, where N is //-null.

For every /с, supp/• n supper = 0 . Thus, by Corollary 1.3, for every к we have

$(f + g)-lz'kvwkdn = $ f - l Z'kdti+ $g-lW'kdg.

E E E

The limits on the right-hand side exists for every Eeffl, thus there exists the limit on the left-hand side as well. Therefore, ( /+ g) - l D>e Coming with к to infinity, we get

J(/ + g ) - l D’ dg = i f - l D\dfi+ \ g - l D'2dg.

E E E

Hence f + g e JP(g) and

$ ( f + 9 ) d n = j ( f + g ) - l D'dg = l f - l D'1dfi+ j g - 1D>2 dpi

E E E E

= j f d j n + \gdn-

E E

3.2. Th e o r e m (on absolute continuity o f integral on M (//)). I f f e Jt (g) and A„ —>0, A ne M * for n = 1, 2, ..., then j f dpi->0 as n-> + оо.

P r o o f. For every n, A n = A ^ v N'n, where A'„e3$, A'n (-^> 0 and N„ is //- negligible.

Let / e then there exists a sequence of sets Z ke ^ , к = 1 ,2 ,. . ., such that (J Z k = X \ N , N = N0u Z 0, N0 is //-null, Z0 <= Jx: f (x) = 0),

fc= l

f ‘ l Zke S 0 and \\m\ f • l Zkdpi — \ f dg. This limit is uniform with respect

£ £

to Re .# . Therefore we can choose kt such that for к > kt

|| \ f - h kdn~ J f d f i|| ^ je for every A'„.

A n A n

By Theorem 1.6 on absolute continuity o f integral on S0 for every fixed Z k

||J f - l Zkdy|| ^ je for n sufficiently large.

A n

Thus for к > ki we have

IlJ fM\ = III f M \ < \\i f df*- J

A n A ’n A ’n A'n A n

^ j e + j e = e.

33. R em a rk . It is easy to check that

(a) i i f e J t i n ) and sup||/(x)|| < a , then ||\ f dy\\ ^ sv(/ra, £ );

x e E E

(b) if /, geJ/( y), f and y are bounded on £ e « ^ and sup.||/(x) — g(x)||

x e E

^ <5, then || I’f d / i - \gdy\\ ^ s v ^ , E), where

E E

a = max {sup||/(x)||, sup||^(x)|||.

x e E x e E

It is obvious that every function in S0 is ./i-integrable. However, there are examples o f bounded measurable functions, which do not belong to Jï(fi) even if y is a linear operator valued measure, as it was shown by Dobrakov in [5], p. 524. Nevertheless, the following theorem is true.

3.4. Theorem. I f the semivariation s\(ya,j is continuous on M for every a > 0 and f is a bounded measurable function, then f e

P r o o f. Let

a

<a.

x e X

function, there exists a sequence o f sets Z ke ^ , к = 1, 2, . . . , such that 00

(J Z k = X \ N , where N = iV0u Z 0, N 0 is /2-null, Z0 c {x: f ( x ) = 0} and

k = 1

/ • 1 zk e <^o for к = 1, 2, ... For m > n and Eg 01 we have:

E E E

= || J s v ( ^ , ( £ \ N )n ( Z „ \ Z „ ) ) E'N

^ SV(/(a, { X \ N ) n ( Z m\Z„))

< s v ( « , , ( X \ N ) n ( A - \ Z J ) « e

if n is sufficiently large, because sv(/2a, •) is continuous and ( X \ N ) n ( X \ Z „ ) \ Q .

9 — Com m entationes M ath. 27.2