2.1.1. 2.1. 2. On spaces L*v based on the notion of a finitely additive integral

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X I I (1968)

ANNALES SOCIETATIS MATHEMATICAL POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I (1968)

W. ^O

(Poznań)

On spaces L*v based on the notion of a finitely additive integral

1. Let T denote an abstract set on which real functions s, x, y, z, ...

are defined. Let ё denote an algebra of subsets of T and p(:) an additive measure on ё which assume non-negative values only and such that 0 < fjb{T) < oo. By ё а we shall denote the algebra of all subsets of T;

we define on ё а [/(a ) = m ip(e) where the infimum is taken for all e in ё which cover a. Clearly, p*(e) is a non-negative, subadditive set func­

tion (the ^-measure) and ju*(a) = ju{a) if a belongs to ё . ё° stands for the ring of subsets of T which are of the ^-measure zero. We denote by the characteristic function of a set e e T. A function s on T is called у -simple if it is of the form

(*) s (b = ^'lZe1(b + Д 2Хе2( Ь+ - « * + ^и%еп(Ь>

where ё and щ are real scalars. The canonical form of a simple function s is the form (*) provided that e* are non-empty, pairwise disjoint and

U ei = T.

2. On the algebra of sets ё а the Badon-Nikodym pseudometric can be defined by setting for the distance-function g

Q(ei, — P ег) ?

where ex~ e2 denotes the symmetric difference of sets and e2 (cf. [2]).

2.1. (a) If д(вп, ef)

^O as n -> oo, g{e\, e2) -> 0 as n -> oo, then g(eln ^ el,, ег w e2) -> 0 as n oo, g(eh el, ег ^ e2) -> 0 as n -> oo.

(b) If д(еп, e) -> 0 as n -> oo , then g(e£, ec) -> 0 as n -> oo, where e£, ec denote the complements of set ecn or ec with respect to T, respectively.

This lemma is known.

2.1.1. The function p*{e), as a function of point e on ё а, is uniformly

continuous on ё а.

In fact, the following inequalities hold

—/**(еа)1 < \f**M — f**fa ^ e2')| + \p*fa) — /**fa ^ e2)\

^ f* C^i ^ 2 )4 " ( ^ 2 ^ 1 )

2.2. Let us denote by S0 the class of sets in T which are of the form 00

U Ci where / / fa) = 0 for i = 1 , 2 , . . . Clearly, S0 is a cr-ring.

1

2.3. (a) For a sequence e1, e 2, . . . of sets in Sa let e = limsupen, п— уоо e =lim infen. If en c= e or en => e respectively for n = 1 , 2 , ... and

n—>00

g f a , e) 0 as n -> 0 0 , then e — e e S 0 or e —e e S 0 respectively.

Assume en c e for w = 1 , 2 , ... Then e — (en^> en+1 w ...) c= e — ek for Tc > n and consequently / / (e — (en w en+1 ^ . . . ) ) = 0 for n = 1 , 2 , . . . ,

O O

e —e — U (e—(en w en+1 w ...)) and we obtain e — e e S 0. If en => e for

1 O O

n = 1 , 2 , . . . , then / / f a rv en+1 ... — e) = 0 and e — e = U f a rs en+1 гл i

r\ ... —e). Hence e — e e S 0.

2.4. A set e is called p-measurable if there exists a sequence of sets en belonging to S such that g f a , e) -> 0 as n -> 0 0 . The collection of subsets of T which are /л-measurable, constitute an algebra of sets (because of 2.1) and it will be denoted by Sm. Clearly, every set in Sa with the //-measure zero belongs to Sm and S cz Sm. Moreover, Sm is closed in Sa with respect to the Eadon-Mkodym pseudometric.

2.4.1. The algebra Sm is called complete with respect to the metric q

if g f a , em) -> 0 as n , m -> 00 implies the existence of a set ее Sm such that g f a , e) -> 0 as n -» 0 0 , and is called C-complete if it is complete and besides the following condition is fulfilled:

(o) If /л* fa) = 0 for i = 1 , 2 , . . . , then p*fa fa e2 w ...) = 0.

If the condition (o) is fulfilled, then every set in S0 belongs to Sm and has the //-measure zero.

2.4.2. A necessary and sufficient condition for the algebra Sm to be C-complete is the fulfilment of the inequality

00 у*

(*) Y y f a )

i - 1 1

for any sets e* in Sm.

П Necessity. Suppose e4e Sm, p*fa)-\-p* f a ) < 0 0 . Hence ^ ((J e.*,

m 1

U et) -* 0 as n , m ->

and, in view of the (7-completeness of Sm, there

1

On spaces L *(p 101

n 00

exists a set ее £ m for which q ( { J ei, e) Q as n-> o o . For e = we

i i

П

have the inclusion e — e a e — [ J eiy thus /Z(e — e) = 0. Moreover, en—e к i

c U e% — e for h > n, what implies /Z(en—e) = 0 for n = 1 , 2 , . . . and,

i

by 2.4.1 (о), /л* (e — e) — 0. Hence g (e,e ) = 0 , ^q ( [ J ei,e) = /л*( e%) ->0

1 i i = n + 1

as w -> oo. From the inequality

n 00

/ / ( e ) < yV*(e<) + /**( U e<),

г=1 г=и+1

passing with тг to infinity, we get (*).

S u fficiency. Let en belong to £ m and g(en

em)

0 as n , m -> oo.

Choose nx < n 2 < ... in such a manner that — c„2) + д(еП2 — enf) +

+ ^{. . .} < ^{o o} and set ak — еПк—еПк+1 for h — 1 , 2 , . . . Let ^ё = limsupc^,

n ^ o o

e = liminf enjc) the inclusions e—enjc c: ak w afc+1 w . .. , e — еПк <= w u*+1

nk—>oo

w . . . hold and by (*) we have

OO oo

[л*(ё— еП]с) < ^ / / ( a n), / / ( е — еП]с) < £ / * * ( a n).

n = k n = k

From the inequality g(e, си) < j>(e„, Cnfc) + £>(ё, we obtain д(ё, en) -> О as n -» ^{o o ,} Ъе Sm. Analogically we have g (e ,e n)- > § as w -> ^{o o ,} ее £ m.

Obviously (*) implies condition 2.4.1 (o).

2.4.3. There exist algebras of sets Sm which are complete but not C-complete. For example: let us assume the interval <a, by as T and the algebra of all finite subsets of T and their complements as £. We define /i(e) on £ setting /л(е) = 0 if e is finite and /i(e) = 1 in the opposite case.

Clearly //(e ) equals either 0 or 1 and //( e ) — 0 if and only if e is a finite set. We infer from this immediately that £ m = £. Let @(cw,em) - > 0 as n , m -> ^{o o ,} where ene £ m. If en is finite for infinitely many indices, then g(en, e0) -> 0 where e0 = 0 . If en are infinite for sufficiently large indices, then T —en is finite and setting eQ = T we obtain д(еп, e0) -> 0 as n -> o o . Thus £ m is complete but the condition 2 . 4 . 2 ( * ) is not fulfilled.

2.4.4. The set function / / ( :) is additive on the algebra £ m and if £ m is C-com/plete, then £ m is a a-algebra and / / ( :) is в-additive on £ m.

Let e1} e2e £ m, ex rs e2 = 0. Choose sequences en, e2n of sets belonging to £ such that g(e„, ex) -> 0 as n -> oo, g(e2n, e2) -> 0 as n -> oo. We have

en г\ en ex e2 c (en ef) w (en e2),

en kj e^—ex^s e2 cz (en— ex) w (en e2),

thus e2n, ex rs e2) -> 0 as n -> oo and д{е1, w e|, ex e2) -> 0 as n -» oo. By 2.1.1 we have p{el r\ e\) -> /л* [ex ^ e2) as n ->■ oo, /i(eln w e2n) ->

-> p*(ex w e2) as w ->■ oo, //(e^) -> /z*^) as n -> oo, р{е2п) -> ju*{e2) as n-> oo;

moreover, the equation

^ ^n ) == A*(e») ^ &n ) -f- [ л { е п )

holds. Passing with % to infinity we obtain f / { e x w e2) = [л*(ех)-\-/л*(е2).

Suppose now that <fm is (7-complete. Let the sets en belong to and be pairwise disjoint. By the additivity of //*(:) on we have //*{ex)Ą-

OO

+ //*(e2) + ^{. . .} </z(T) < o o , and by 2.4.2 ( * ) , //* ( U ą) ^-> 0 as тг -> o o .

Thus

71 OO OO

e (U

U

0 as n

г=1 г=1 г=п+1

0 0

and consequently U belongs to «fTO. Because of

1

0 0 n OO У 00

iW*(U ei) = /‘* ( U eł) + /ł*( U ei) = У р*(е€) + р*{ U ef)

г= 1 г=1 г=п +1 i - n+ 1

00 00

the equation //*((J ег) = V //*(%) holds.

1 i

3. The sequence {#»(<)} is said to be convergent in p-measure to x if for every A > 0, we have //* (ете(А)) -> 0 as n ^{o o ,} where ere(A)

= {L \xn{t) — x(t)\ > A, teT}. We shall use the notation 4- x as n -> o o

to denote that converges in //-measure to x. A function x is called /и-measurable if there exists a sequence of //-simple functions sn such that sn 4- x as n -> ^{o o .} It can be easily verified that the characteristic function xe °f a set e is //-measurable if and only if e a $ m.

3.1. I f x is p-measurable, 0 < x(t) for teT, then there exists a sequence of simple functions sn such that sn(t) < s « +1(/), 0 < sn(t) < x(t) if teT, n = 1 , 2 , . . . , sn -> x as n -> o o .

Indeed, x being //-measurable, we can find //-simple functions sn for which /л*{ёп) -> 0 as n -> ^{o o ,} where en = {t: \sn(t) — x(t)\ ^ I In}. Let ene Ć, ёп CZ en, ^{p ( e n)} < //*(ёта)+ 1> ._ Setting *„($) = (en(i)— 11п)%т-еп^ we get //-simple functions such that sn(/) < a? (4) in T for n — 1 , 2 , . . . ; s»(ż)

= sup(^(i), 0) are //-simple functions for which 0 < sn(t) < x(t), sn(ź)4-

Д . sup(x(t), 0) = x(t) as n -> ^{o o .} Let us define the //-simple function

On spaces L *<p 103

sn(t) = sup(sx(ź), s2(t), ..., sn(t)). Then, the inequality 0 < x {t,)- s n{t)

< x(t) — sn(t) for teT implies sn Д- x as w->oo, moreover, the sequence sn(t) is non-decreasing for teT, and 0 < sn(t) <ж(£).

3.2. In connection with the last lemma we will make the following remark which explains the difference between the /«-measurability of functions under assumptions that £ is an algebra and /«(:) additive on

£, and under assumptions that £ is a c-algebra and /«(:) cr-additive on it. If xn are /«-measurable, ocn(t) non-decreasing for teT and xn(t) converges pointwise in T to a function x(t), then, assuming £ to be a «т-algebra and /«(:) to be <r-additive on £, x converges in /«-measure to x and the limit x{t) is /«-measurable. This statement does not be true in general if £ is only an algebra. Assume for T the closed interval (a, by for £ the algebra J , elements of which are the finite unions of the intervals (c, d), a <

< d < b, <d , by, a < d < b, and the empty set. Let /«(:) be the Lebesgue measure on J . Let { ą , w2, ...} be an enumeration of the set of all rational numbers in <«, by. Define on {a, by a function xn setting x (t) — i if t — u\, i — 1 , 2 , . .. , n, xn(t) — 0 elsewhere. Evidently, xn{t)

< xn+1(t) for te (a , by, xn{t) converges pointwise in T, xn converges in /«-measure to у = 0, but not to x(t) = lima?n(I). Moreover, as easily

П seen, x is not /«-measurable.

3.3. The essential supremum of a function x on T, in symbols sup0x(t), will be defined as the lower upper bound of the numbers supa?(ź), where a

teT—a

runs over all set for which /л* (a) = 0. From this definition it follows that sup°|;r(I)| = 0 if and only if for any set ex — {t: \x{t)\ > Я > 0} we have /л*(ел) = 0. This yields that e0 = {t: x(t) Ф 0} can be expressed as the union of finite or countably many sets of /«*-measure zero. However, it may happen that [л*{е0) > 0, for /«*(:) is in general not «y-subadditive.

The function x is said pt-equal to y, written x — у [/«], whenever sup°|a?(2) —

— y{t) I = 0 . It can be easily verified that the relation . = . [ / « ] is an equivalence relation. We will say that x is less or equal to у in the sense of the measure /« (briefly, in the sense /«) whenever sup (x, у) =y[ /«]

and we shall write x < у [/«]. If x is /«-measurable and /«-equal to y, then у is /«-measurable, too. Kow, let us denote by Ж the collection of classes of /«-equivalent functions which contain /«-measurable functions. It can be easily shown that, defining the addition of classes of /«-equal functions and their multiplication by real scalars in a natural way, Ж becomes a real linear space. Moreover, provided with the order relation “ . ^ the space Ж becomes a linear lattice. The supremum of two classes repre­

sented by the functions x or у respectively, is the class represented by

sup(x, y), and analogously their infimum is the class which is represented

by the function inf (x, у). In accordance with the usual habit of saying

“function” instead of “equivalence class of functions”, when speaking on the members of SC, we shall speak rather on functions as on the corre­

sponding classes of functions. Therefore in the sequel we shall use the letters x , y , z, ... either as symbols of individual functions or as symbols of classes of /^-equal elements. But to avoid some misinterpretations we will preserve the notation x = у [у,], x < у [у] to denote that the class x equals the class y, respectively is less or equal to y.

4. How, for the purposes we have in mind, a notion of an integral of a scalar function x with respect to a measure y, defined on an algebra ё , must be fixed. We shall use that one of an additive (but not u-additive in general) integral. First of all, for a /^-simple function s of the form 1 (*) we define the integral in usual way, that is to say, we set

П

j s(t)dp = Cl'i pi for e € ё I

e %= 1

this integral does not depend on the particular representation of the function s. Following the definition given in [1], [3], [5], a function x will be called y-integrable over T with respect to у (briefly /г-integrable or integrable) if there exists a sequence sn of /^-simple functions which satisfies the following two conditions:

(1) sn 4- x as n -> ^{o o} ,

(2) the set functions

(+) Xn(e) — J* sn(t)dy

e

converge uniformly on «f to a set function A(e).

The set function A(:) defined on ё will be called the indefinite integral of x with respect to у on ё and the customary notations

X{e) = J x(t)d y , X{\) = j x(t)dy

e

will be used for it. Clearly X(e) = f x{t)%e{t)dt. It can be shown that con­

ditions (1), (2) are equivalent to another pair of conditions, namely to i’

the condition (1) and simultaneously to one of the conditions:

(2') the set functions Xn(e) are equiuniformly absolutely continuous with respect to y, i.e. for any e > 0 there is a <5 > 0 such that

\K{e)\ < e for n = 1 , 2 , ... if y(e) < <5,

f \sn(t) — sm(t)\dy -> 0 as n, m -> ^{o o .} T

( 2 " )

On spaces L *,p 105

Assuming conditions (1), (2") as the starting-point we can prove imme­

diately that if x is //-integrable, then the function \x\ is //-integrable too.

Let us remark, moreover, that the definition of an integral based on (1), (2) or (1), (2') respectively, when applied to a vector-valued func­

tion x instead to a scalar-valued function (then, of course, the sign of the absolute value in (1), (2), (2") etc. means the norm) leads to the definition of the integral in sense of Dunford, and that based on (1), (2") to the definition of the “strong” integral (of the Bochner-Dunford type).

It is well known, that for vector-valued functions x from (1), (2"), (1), (2) follows (or equivalently (1), (2')) but not conversely.

4.1. The definition given in 4 have been formulated for individual functions x. However, the property of being //-integrable is a class-prop­

erty, i.e. if x — у [//] and x is an //-integrable function, then у is also //-integrable, moreover, jx(t)dy = jy(t)dy for every ее S’. Let us denote

e e

by SCX(T) (or briefly by SCX) the collection of all classes of equivalence in SC which contain integrable functions. It is known that SCX{T) is a real linear space and it can be shown that SSX(T ) is a linear lattice (more precisely: a linear sublattice of SC) with respect to the order relation in­

troduced in 3.3 (cf. 5.1). For a fixed set e in S the integral is a linear bounded functional over SCX{T) provided with the norm |[ж|| = j\x(t)\dy,

T

and for a fixed member xeSCx(T) the integral is an additive set function on S, which is absolutely continuous with respect to //(:).

4.2. I f x , у e£Cx{T) and x У then

J x (t )d y < Jy (t)d y for e e S .

e e

if x means an individualy given /^-integrable function and x (t) > 0 for te e , ее S’, then jx(t)dy > 0.

e

Indeed, this is true obviously if ж is a simple function. By Lemma 3.1 we can find simple functions sn, 0 < sn(t) < x(t) for teT, вп Л- x as n -> oo.

We can assume that x(t) — sn(t) < 1/w on a set en, for which y ( T —en) -> 0 as n -> oo. For the simple functions sn = sn%6n we obtain

j jx (t)d y — j s n(t)d(A ] < ljny {en)-\-1 J x(t)dy |,

e e (T ~ en)^e

by the application of the standart properties of /г-integral. Since / oc(t)dy-*0&$n-+oo, j s n(t)dy ^ 0, the inequality jx (t)d y ^ 0 follows.

(5Р-еи)ле T e

How, suppose that x , yeSCx{T) and x < у [у]. For a given Я > 0 let us set ex — {t: y{t) — x{t) < — A} = [t: sup(«(/), y(t)) — y(t) > A}. Then y*{ex)

— 0 and there is a set ax in S for which eA c= ax, y{ax) < S, where <3 is

arbitrarily prescribed. On е — ал the function (у — x) -f- A is non-negative and consequently / (y(t) — x(t))dp > —/ Ad/г, with arbitrary small A. The

е -« д e~ aX

integral / (y(t) — x(t))dp can be made arbitrarily small in virtue of the

«л

absolute ^-continuity of the integral. Hence the integral J [y(t) — x(t))dp

e

is non-negative.

4.3. I f хпе ^ х{Т) for n — 1, 2 , . . . , xn < Яп+Лр] for n — 1 , 2 , . . . (xn > sCn+iilA f or n — 1? 2, ...) хп Л- x as n oo and the sequence

J x n(t)dp is bounded then х е ^ г{Т) and

T

( + ) lim f xn(t)dy = \ x(t)dy,

n->oo i i uniformly with respect to e.

Assume, for example, that xn < xn+l [/г] for n — 1 , 2 , . . . The sequence of the integrals / xn (t) dp is convergent, for it is non-decreasing and bounded.

T

Thus, for p = 1 , 2 , . . . and any e e i , we get

0 < J xn+p(t)dp— J xn(t)dp < f%n+p(t)dp— j Xn(t)dp < £,

e e T T

for sufficiently large n. Thus the indefinite integrals j x n(t)dp converge uniformly on €. Choose a sequence of simple functions sn in such a manner that p*(en) < 1 jn, where en = {t: \sn(t) — xn(t)\ > 1 jn}, and

(i) J ' (*n(t) xn(t)jdp 1

< -

n for C e , n — 1 , 2 , . . .

Consequently sn - ^ x as n -> oo and the sequence js n(t)dp converges

e

uniformly on «f. Therefore x is /г-integrable and by (i) it follows the rela­

tion (+).

4.4. I f xeJ.^ (T ), у is p-measurable, and 0 <#[/*], then уе£Рх{Т).

Since у > 0[/г] the function u{t) = sup(y(/), 0) is /г-equivalent to y\

since гг<ж[/г] the function v{t) = sup (a? (4), u(t)) is /г-equivalent to x.

For any teT the inequalities v(t) > u(t) > 0 hold, moreover, the function и is /г-measurable. By 3.1 there is a sequence sn of simple functions, such that sn(t) < sn+1(t), 0 < u(t) ^ v(t) for t e T , n = l , 2 , . . . , s n - ^ u as n -> oo. But by 4.2 the inequality j s n(t)dp < jv(t)dp — Jx(t)dp holds

T T T

and thus by 4.3 the function u, and у as well, is /г-integrable.

On spaces L *,p 107

4.5. I f у is ^-measurable, \y\ < \x\ [//], and xeSCx(T), then yeS£x[T).

Since \y\ is //-measurable, \x\eSCx(T) we get \y\e<Fx(T), by 4.4. From 4.4 also follows that the functions u(t) = swp(y(t), 0), v(t) = — inf (//(£), 0) are //-integrable and consequently у — u + v is //-integrable.

4.6. If i m denotes the algebra of //-measurable sets, then by 2.4, 2.4.4 v(e) — y*{e) is an additive set function on it. Denote by the set of classes of r-equivalent functions which contain r-measurable function, and by SCX{T) the corresponding space of r-integrable functions. We claim that SC — , JFX{T) = JFX(T) and the integral of x over T with respect to //, S' is identical with the integral of x over T with respect to v, € m. Moreover, jx(t)dv = jx (t)x a{t)dy, for any ae <Sm. As easily

a T

seen x = y[y] implies x = y[v] and conversely. If sn ^>x as n ^{o o ,} where sn are //-simple, then sn Л- x as n -> o o , for v*(e) — y*(e) for any ее i a. Hence SC ^c= . The opposite inclusion is a consequence of the iden­

tity of convergence in //-measure with the convergence in r-measure and of the following lemma:

П

For given positive e, А, d and a v-simple function *= 'E K X a v, Mere i

exists a p-simple function s such that

(i) < <5 where ex = {t: \s(t) — s(t)\

(ii) IJ s(t)dr— js(t)d p ] < e for e e i .

e e

Indeed, since av

i m we can find the sets ev

& in such a manner

n n

that ^ju*{av — ev) < 6/2v+1, //*(«,— ev) < e, thus, setting s(t) =

i i

we obtain (i) and

П

j J s(t)dv — J s(t)dp I < ^ |A„| ju*{av — ev) < e for ее S’.

e e v=l

Assume that x is integrable with respect to v , i m that sn are r-simple and that conditions 4 (1), (2) (with v instead of //, <fm instead of ё) are satisfied. By the foregoing lemma there is a sequence of //-simple func­

tions sn such that sn Д- x as n -> ^oo and Jsn(t)dv —js n(t)dp 0 as n -> ^oo,

e e

uniformly on ё . Therefore the integral jx (t)d p exists and jx(t)dv

e

= jx(t)dp for any e lying in ё . How, let xeSC. Let sn be //-simple for

n = 1 , 2 , ... and let conditions 4 (1), (2 ) be satisfied. Clearly sn 4 - x as n -> oo, and

(iii) J sn(t)dv— J sn(t)dy

a e

< sup \sn(t)\v{a — e)

T

for any set ae«fTO, ее S’. For any a e S m and n = 1 , 2 , ... we can find en{a )e S such that sup |sw(ź)|r (« — en(a)j -> 0, v(a — en{a)) -> 0 as n ^ ^oo, uniformly on <fm. B y 4. (2 ) and (iii) we have

I J sn(t)dv — j Sm(t)dv I < £

a a

for n > n0. Thus x is integrable with respect to v, €m. For a given set

a in sn Xen(a) converges in //-measure to x x a - B y (iii) (iv) j sn{t)dv— j sn(t)dfi 0 as n -> o o ,

ar\e en(a)r\e

uniformly with respect to ее S’. Therefore j sn(t)d/u, = fs n(t)xen(a)(t)dp

en(a)r-,e e

converges uniformly on $ to jx {t)x a{t)d[i, and, by (iv) this integral is

e

equal to j x(t)dv.

(Xr\C

5. B y a ^-function we mean a continuous, non-decreasing function 9o(u), defined for и > 0, vanishing only at и — 0, and tending to 00 with и oo. 99-functions will be denoted by <p, у, ... and their inverse functions by 9

If 99(au + @v) < 0*9?(u) + /5*99(v), where 0 < s < 1 for a, (3 > 0, aSjr ( f = 1 and arbitrary u, v > 0, then a 99-function 9?

will be called s-convex. Any 99-function of the form y^(us), 0 < s < 1 , where \p is a convex 99-function, is s-convex. Any s-convex 99-function is strictly increasing in <0, 00).

5.1. Let 99 be a 99-function; we denote by ^ ( T ) (briefly by i? 95) the set of classes of equivalence xeS£ such that the integral j<p[\x{t)\)d[j, exists for a function x(t) which represents the corresponding class of equivalence. Let us set =2?*<P(T) = {x: Xxe£Pę (T) for some Я > 0}. Func­

tions (corresponding classes of equivalence) belonging to are called 99-integrable. Clearly, if <p(u) = \u\, then ^ * ę(T) = £ >4>{T) — ^ ( T ) . These definitions and the value of the integral f<p(\x(t)\)d/u, ее S’, do

e

not depend on a. representative of a class of equivalence, for if x = У Ш , where x is //-measurable, then

(i) < p №\) = <p(\y\)W-

Indeed, if the simple functions sn converge in //-measure to x, then, by

our assumption, they converge also in //-measure to y. Therefore, for

On spaces L * v 109

arbitrarily prescribed <5 > 0 there exists a simple function s{t) such that

\x{t) — 8(t)\ < 1 \y(t) — s(t)\ < 1 for te a e S, where ц {Т —а ) < <5. On a we have \oo(t)\ ^ \y(t)\ sjC h, with Jc — 1 — (- snp |s (1i) I. But, given a positive

T

number e, we have \(p(u) — <p(v)\ < e for 0 < u, v < k, \u — v\ <Я(е), and consequently the set as = {t : HK*)|)-<p(|y(J)|)| > e}, is contained in eXK j(T — a), where ex = {t: \x{t) — y(t)\ >Я(е)}, р*(ея) — 0. Since p* (ex w w (T — a)) < d we obtain p*(ae) — 0. Let us remark that (i) also shows that the indefinite integral fcp[\x(t)\)dp does not depend on a represen­

tative of the class of equivalence.

5.2. The collection ST*ę{T) is a linear sublattice of S£, the extended- valued functional q , defined by q { x ) — f(p[\x{t)\)dp if x e ^ ę(T), q ( x ) = oo

T

in the opposite case, is a modular on it, i.e. q has the following properties:

A. q { x ) = 0 if and only if x = 0[p]',

в. eW < ^q ( x 2 ) if \®i\ < \x% \ W i

C. д{хг + х2) = if Ы л \x2\ = 0 [>];

D. q {X x ) ->

if Я ->

.

First of all, ST*41 is a linear space. In fact if x±, x2€^>*4>, then Я ^ е ^ , Я2а?2е=279’ for some positive Я’а8. Let а,Ъ Ф 0 are given. Set Я = т ш (Я 1/2 |«|, Я2/2 |&|). Since cp(X\auĄ-bv\) < ^ (Я 1 |#|) + 9р(Я2|г?|) for any reals и, v, we have

<P (Я Iax(t) + by(t)\) < p (Ях |a?(#)|) + <p (Я2 \y (t)\),

and since (р(Цах-\- by\) is a ^-measurable function and 9?(ЯХ |г»1), <р{Л2\у\) are /л-integrable, by 4.5, it follows a x b y eJ?**. If xeJ?*(p, then \x\ eJT*9 and consequently x v у , x л у belong to <£?*’’, which means that ST** is a linear sublattice of ЗГ.

Ad A. It is a simple matter to prove that f\x{t)\dp = 0 if and only

T

if x = 0 [^]; the property A follows immediately from this remark.

Ad B. It is enough to apply Lemmata 4 .2 , 4.5.

Ad C. In order to prove the property C let us choose simple func­

tions such that 0 ^ s^ ^ s2 ^ ... ^ (^) 1 for 16 j 0, s^ —^ N , 0 < sx(t) < ś 2(4) < ... < \oo2(t)\ for teT, sn f> у as n -> 00. From the equation |aq| л \x2\ = 0[//] it follows sn Asn ~ 0 [p\ for n = 1 , 2 , ..., what, as easily seen, is equivalent to the equation sn-sn — 0[/t] for n = 1 , 2 , ... This implies

f <p(8n (t)+ 8n(t))dp = f (p(sn(t))dp+ J(p(s(t))dp,

T T T

and letting n -> 00 we get by 4.3 Q(x1Jr x2) = gix^ -j- q ( x 2), provided

that q (X j ) < 00, q ( x 2) < 00. If e.g. q ( x ±) — 00

then, by B, and |#i + #2|

= \noi\ + \oo2\, £>(aq + #2) — 00 and the equation + = g (aq) + f> (a?2) is satisfied as well.

Ad D. This follows immediately from 4.3 and from the remark that x being /«-measurable, Xx -> 0 as X -> 0 (the last remark fails for unbounded and non /«-measurable functions).

5.3. Suppose the 99-function 99 be «-convex. It follows from the general theory of modular spaces [6], that in JF*,p{T) an «-homogeneous, norm can be defined by the formula

\\x\\l = inf {e > 0: Q(xjells) < 1 } .

As easily seen the norm ||*||^, is monotonie, i.e. || ćp ||^, ^ НЛ if N < M l>].

Moreover, ||a?„||® -> 0 as n -> 0 if and only if g(Xxn) 0 as n 00 for every X > 0. For a /«-simple function x equation ||ж||® — 1 implies the equation jcp[\x(t)\)dp —

(

) = 1 and conversely.

T

5.4. To define the set ^ ( T ) , the space J?*rp(T) and the norm ||*||*

on it we can always use the integral fq)(\x(t)\)dv instead of the integral J<p(\x(t)\)dfX, for by 4.6, Ж = <&, j<p(\0D{t)\)xa{t)d[A = $q>(\x{t)\)dv for

T a

a e ^ j . It follows from the last equation that \\x\\l = inf {e > 0 : l j . Suppose that $ is О-complete. B y 2 .4.4 Sm is

T

a a-algebra and v(:) «т-additive on $m. Applying the last remarks one can prove by familiar arguments the following theorem:

5.4. I f S is C-complete, then the space Ж*<г'{T) is complete with respect to the norm ||-||®.

6 . In this section we are concerned with the following questions.

Suppose two «-convex 99-functions 99, ip are given. O11 what conditions does the inclusion

(o) c

hold % Suppose that (o) is fulfilled, on what conditions does the inequality

(*) ⁱⁿⁱ ; < ⁱⁿⁱ ;

for hold ? Let us introduce the following notation: od ( u ) = <p(ip-i{u))..

6.1. If and only if the inclusion (0) and the inequality (*) for хеЖ**

hold when the following Condition (K) is verified:

For any partition ег, e21 ..., en, e^ «f, of T and arbitrary non-negative u1} u2, . .., un the equation

a ) ( u 1) p ( e 1) + c o ( u 2) p ( e 2) - \ - . . . - \ - w ( u n ) j u ( e n ) = 1

(a)

On spaces L *41 111 implies the inequality

(b) uxp {e1)-\-u2p{ef)-\-.. .Ą- unp{en) ^ 1 .

It is easily seen that (*) holds for any //-simple function s if and only if

(i) /<?( 1*001)^ = 1?

T

implies the inequality

(ii) j y(\8{t)\)dii < 1 .

T

Therefore, assuming (*) to be satisfied, the equation

9?(la il)iM(ei) + 9’ (la al)jM(e2) + - •• + (p{\an\) У'Ы ~ i ,

where are arbitrary, e* are in S disjoint and whose union is T 7 implies

y { \ a 1\)fj,{e1) + y { \ a 2\)[j,(e2) - \ - . . . + 'ip{\an\)p,(en) < 1.

Hence, setting ^(jflq|) = щ, we obtain Condition (K).

Now, assume Condition (K) to be satisfied. This is equivalent to the fact that (i) => (ii) for any //-simple function, which in turn is equi­

valent to the inequality (*) for any //-simple function. Let x eJS*9*, and let ||<r||® = 1 , what implies

< 1 . B y ^{3 .1} there exists a sequence of //-simple functions sn, 0 < sn(t) < sn+1 (t), ..., 0 K s n{t) < \x(t)\, for n = 1 , 2, ..., sn ff. \x\ as n -> oo. But jV(|Sn(t)|)d// < /ę>(|a?(ż)|)d// < 1, thus, by

T T

(ii), we have jy){sn(t))dp ^ 1 . Since y(sn(t)) are //-simple functions,.

T

w(sn(t)) < y(*n+i(<)) f ° r n = 1 , 2 , ..., teT , yj(sn) 4 - w(\x \) as n — > oo, we infer, owing to 4 .3 , that x is ^-integrable and fy(\x(t)\)dp < 1 . The last

T

inequality implies ||a?||® < 1 = ||a?||4 • We have proved xa£P*y! and the in­

equality (*) under the assumption ||a?||® — 1. Owing to the «-homogeneity of the norm ||-||® the general case can be reduced to this special case.

6.2. I f Condition (K) is verified, then

( + ) /o(MT)-1)

Supjjose w(//(T)_1) < p{T)~l, then //(T)_1 < co_1[p(T)~1) = u0, and (о(щ) — p (T )-1. Hence co(u0)p (T ) = 1 and by Condition (К) щ/л(Т) < 1 , a contradiction.

6.3. The measure p is said to fulfil the property В (with respect to the algebra S) if the set of values of p(e) is dense in the interval <0 , //(T)>

(cf. [8]).

6.4. I f the measure p fulfils the property S with respect to £ and Con­

dition (K) is satisfied, then

(++) u ^ .co(u) for и > p(T)~l .

In fact, if the measure p fulfils the property S , then the equation (o(u)p(e) = 1 is satisfied for u 1 s in a set U a ^oo_x (М 2 Г 1), ooj which is dense in <^co_x(p(T)~1), ocj, for со is continuous and strictly increasing.

Applying Condition (K) to u x = u, ex — e, u 2 = 0, e2 = T —e, we see that (++) is verified for ueU, and by the continuity of со for all и ^ co_x(p{T)~1) as well. But by 6.2 for и > р(Т)~г the inequality и > co_x(p{T)~1) holds.

6.5. Let со denote the greatest non-negative convex function for which co(u) < a>(u) for и ^ 0. I f CO (fi(T )-1) ^ p{T) then Condition (K) is satisfied.

Since co(0) = 0 the quotient a>(u)ju is non-decreasing and consequently со is strictly increasing in any interval <u0, oo) such that co(u0) > 0, especially we can set u0 = p{T)~1. If ex, e2, ..., en, e^ S, is a partition of T and ao(ux)p (ex) + co(u2)p (e2) + . . . + co(un)p(cn) = 1, u — uxp {ex)-\r

f - u 2p {e2) + . .. + unp{en), then

oo[up{T)~l) < (а>{их)р {ех) + а)(и2)р {е2) + ... + oo{un) p{en)) p{T)~x

< [ao{ux) p{ex) oo{u2) p {e2) ... -f a>(un)p(en))p(T)~x = p{T)~x.

We claim that и < 1, i.e. that (b) of Condition (K) holds. For, if it is not so, we have

со ^ p { T ) ~ <C со [tip(T)~~xj ^ p(T)~x, a contradiction.

6 . 6 . As a corollary to 6.2, 6.4 we obtain the following theorem.

T

. I f co(/c(T)-1) = co[p(T)~1), in particular if со is convex, the necessary and sufficient condition for the inclusion 6 (o), and for the inequality 6 (*) to be valid, is that co^(T)-1) ^ p{T)~\

6.7. Assume that the algebra £ is C-complete. Then, the following statements are equivalent:

(1) se*v c Jg?**;

(2) ||a?||® < fc||a?||® for some h > 0 and for any p-simple function;

(3) the Condition (K), with the function oo(u) = <p(kip_x(u)) is fulfilled.

If <p{u) = cpfku), then and ||a?||~ = fc||a?||®. Suppose £ to

be О-complete. B y 5.4 the space provided with the norm ||-||® and

the space £?*'р provided with the norm |)-||® is complete. The implication

On spaces L *cp 113

(1) => (2) follows by the classical arguments (application of the closed- graph theorem, and of the fact that the norm convergence in

implies the convergence in /^-measure). As we have observed in course of the proof of 6.1 the Condition (K) implies the inequality 6 (*) for any /^-simple function and conversely. Thus, setting in (*) tp for <p, d>(u) —

= (p[kip_i{u)) in (K), we obtain (2 ) о (3 ) and applying 6.1 we obtain (3) => (1 ).

6 . 8 . I f the algebra € is C-complete and the measure p fulfils the prop­

erty 2 with respect to «f, then the inclusion 6.7 (1 ) and the inequality ( + ) xp{u) K<p{ku) for и ^ p{T)~l

are equivalent.

6 .9 . Assume <p{u) = ua, a > 0 , weS(>*4’(T) = J?a(T). If 0 < a < 1, then applying the definition of the norm with s = a, we obtain ||a?||®

= J\x(t)\adp. If a ^ 1, then setting s = 1 we obtain the familiar norm

T

||a?||® = ( J \x(t)\adpYla. Let y>(u) — uai, <p(u) = u°2, 0 < ax < a2. Then

T

cd ( u ) = ua2,ai is convex and by the application of the theorem from 6.6 with an appropriated index of homogeneity s we obtain the following classical result:

The inequality

( J \x(t)\aidp)1Ia1 < (j\x(t)a2\dp)1,a2

T T

holds for any x e J? a(T) if and only if p(T) < 1 (cf. [8]).

R eferences

[1] R. G. B a r t l e , A general bilinear vector integral, Studia Math. 15 (1956), pp. 337-352.

[2] N. D in c u le a n u , Vector measures, Berlin 1966.

[3] N. D u n fo rd and J . T. S c h w a rtz , Linear operators, Part I, New York 1958.

[4] J . M u sielak and W. O rlicz , On modular spaces, Studia Math. 18 (1959), pp. 49-65.

[5] — Notes on the theory of integral, I , Bull. Acad. Polon. Sci. 15 (1967), pp. 329-337.

[6] W. O rlicz , A note on modular spaces, I, Bull. Acad. Polon. Sci., Ser. sci.

math., astr. et phys. 9 (1961), pp. 157-162.

[7] - ibidem, VII, 12 (1964), pp. 305-309.

[8] — On some classes of modular spaces, Studia Math. 26 (1966), pp. 165-192.

Roczniki PTM — P race M atematyczne X II 8