ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATY.CZNE XXX (1990)

**R****o sel i**** F****e r n a n d e z** (Sâo Paulo)

Characterization of the dual of an Orlicz space

**Our objective is to characterize the dual of an Orlicz space & A{ X , E, fi), with the only ****hypothesis that (X , I , ц) is a measure space with no atoms of infinite measure.**

**This work originates from the reading of [12] and [13] of M. M. Rao, where we found some **
**statements which were unclear to us. In particular the characterization in [13] seems to be **
**incomplete, possibly for some fault in the fundamental definition. Here we present another **
**characterization.**

**1. Preliminaries. If ф is a nondecreasing function from [0, co[ to [0, oo], ***such that 0 < il/(t0) < oo for some t0 Ф 0, the function A defined on [0, oo[ by *
*the equality A(u) = jo ÿ{t)dt is called a Young function.*

*(1.1) R em ark. Let A be a Young function. Then*
*(i) A is nondecreasing and convex;*

*(ii) the right derivative A' of A exists, is nondecreasing and is finite-valued *
on [0*, b[, where b is as in (iv) below;*

*(iii) the function A defined on [*0, **oo[ ** as

*is a Young function, called the complementary function of A;*

(iv) *uv ^ A(u) + A(v) for u, ve [0, ***oo[, **with equality holding if and only if
*at least one of the relations v = p(u) or и = q(v) is satisfied by u and v, where*

*In what follows А, Я, b and b will be as in (1.1), and we will write*
*a = sup (ne [0, ***oo[: ** *A(u) = 0}, * *â = sup {ue [0, ***oo[: ** *A(u) = 0}.*

*Moreover, (X, Z, p) will be a fixed but arbitrary measure space with no atoms *
*of infinite measure (i.e. if F e Z and p(E) = oo, then there exists F eZ, F a E, *
*such that 0 < p(F) < oo). Unless otherwise stated, our functions will be from *
*X to R, and we shall employ the conventions that 0/0 = l/oo = 0 - oo = 0 and *
inf 0 = oo.

*A(u) = sup {uv — A(v): ***p g****[**0**, oo[},**

b = inf (ug [0, **oo[: ** Л(ы) = **oo} ** and *В = inf {ue [0, ***oo[: ** *A(u) = ***oo}.**

**70** **R. Fe r nan de z**

The Orlicz space *A is the space of all measurable functions f such that *

*§x A(k[f\)dn < со, for some /се]0, oo[. This is a complete space with the *
seminorm Ц-Ц^ defined by

*\\f\\A = inf {/ce] 0, oo[: J A(\f\/k)dp < 1}. *

**x**

(1.2) **P****r o p o s i t i o n****, ** *(i) I f E e l and £,Ee££A, then \\ÇE\\A ^ bp(E);*

(ii) *for E e l the relation p(E) < oo implies that ÇEeJ?A, the converse *
*holding if a = 0; moreover, if p(E) = oo and a > 0, then \\ÇE\\A = 1/a;*

*(in) for f e ^ A one has §x A(\f\)dp ^ *1* if and оп^У if \\f\W ^*
*(iv) if <5e[0, oo[ and /ejz?^, then*

*A(S)-ix{{xeX: |/(x)| > <5 l l / U H 1 ;*
*(v) if f e <fA, then * 1!/||л *\ x Ai\f\)dn+ *1.

*P ro o f, (i) follows immediately from A(u)/u * for u e ]0 , oo[, and for
*this, observe that A(u)/u is nondecreasing.*

To prove (iv), use (1.1 .i) and (iii). For the remaining assertions, see [2]. ■
(1.3) **P****r o p o s i t i o n****. ***For f in 3?A there is a sequence (s„) of simple functions in *
*S£**a** such that*

*(i) 0 ^ s„ ^ sn + u for all n e N , if f ^ 0;*

*(ii) If-s„\ < | / | and |s„| ^ \f\ for all ne N;*

*(iii) (s„) converges to f ;*

*(iv) (s„) converges in p-measure to / ;*

*(v) if /се]0, oo[ is such that A ( \ f \ / k ) e t h e n*
*lim J A ( \ f - s n\/k)dp = 0.*

**oo x**

*P ro o f. Discarding the trivial case, let \\ДА Ф0. The measurability of *
/ guarantees the existence of a sequence (s„) of simple functions satisfying

(i)—*(iii). Moreover, if E e l is such that/ is bounded on E, then (s„ÇE) converges *
*uniformly to fÇE. From (ii) it follows that each sne ^ A.*

To establish (v) it suffices to use (ii) and (iii) and apply Lebesgue’s Convergence Theorem.

*It remains to prove (iv), and we first consider the case in which a > 0. Let *
*B = {xeZ : \f(x)\ >2a\\f\\A}. Clearly (snÇBC) converges uniformly to /£ BC. By' *
*(1.2.iv) we have p(B) < oo and hence, by Egorov’s Theorem, (s„£B) converges to *
*f £ B in /r-measure. So (iv) holds in this case.*

*If a = 0, let c e ]0 , *^{o o}^{[ }*be such that 0 < A(c) < oo. Replacing к by \\f\\A in*
(v) we have

*lim $ A ( \ f - s n\/\\f\\A)dp = 0, *

**n-+ oo x**

and so there is a subsequence (s„J such that

*f A ( \f -s J I \\ f\ \A)iii < 2 - t A(c/2k).*

*X*

*From this we see that p({xeX\ |/(x) —s„k(x)| > c\\f\\A/2k}) < \/2k for all k e N , *
and so (s„J converges to / in /^-measure. Replace (s„) by (snJ. ■

**2. Characterization of the dual space (J?A)*. Denote by JtA the closed ***subspace of f£A spanned by the simple functions.*

*In [13] the author gives a representation for x*e(JtA)* as an integral, in *
the sense of Dunford and Schwartz [3], relative to a measure G defined as
*G(E) = x*{ÇE). Clearly this defines G(X) only if ^x Ef£A. Since this does not *
*occur when a = 0 and p(X) = oo, the measure G is not defined on I in this *
case, and so we cannot apply the theory of [3]. Thus in [5] we introduce the
concept of integration relative to a measure defined on an ideal.

To facilitate the reading, we transcribe from [5] some definitions and one
*proposition. For this, let sé be an algebra of subsets of X, and Ж an ideal of *
*s4 (i.e. Ж is a ring, and for Е**е** Ж and F е Ж one has E n F e Ж). Moreover, let *
*G be an extended real-valued finitely additive measure defined on Ж , and let *
*v(G, •) be the total variation of G, which is obtained by replacing I by Ж in *
III. 1.4 of [3].

(2.1) **D****e f in it io n****. ***A function s is Ж -simple if there is a pairwise disjoint, *
*finite sequence (Elt E2, ..., En) in Ж and there exists (c*l5* c2, ..., cn)e R n such *
*that s = ^ " =iC ^ £.. If v(G, (x e X : s(x) = cj) < oo for ct ф 0, we shall say that *
*s is G-integrable and for E e s f we define*

*П*

*\sdG = £ CiG{E n Et).*

*E * *i = 1*

(2.2) **D****e f in it io n****. ***A function / is G-integrable if there is a sequence (sj of *
G-integrable simple functions such that

(i) (sn) converges in G-measure to / , i.e. for every <5e]0, oo[ one has
*lim (inf{w(G, F): Р е Ж and {xeX : |/„(x)-/(x)| > 0} c F}) = 0;*

*n~> <x>*

*(ii) lim J \sn — sm\dv(G, •) = *0.

*m, n~* *oo *x*

*If / is G-integrable and F e s / we define*
*j / dG = lim j sndG.*

*E * *n ~* со E*

(2.3) **P****r o p o s it io n** *(Vitali Convergence Theorem). Let f be a function and *
*(/„) a sequence of G-integrable functions such that*

*(i) {fn) converges in G-measure to f ;*

*(ii) \imviG'E)^ 0\E\fn\dv{G, •) = *0* uniformly in neiV;*

*(iii) for each e e ]0 , oo[ there is a set Fe in Ж with v(G, FJ < oo and such*
*that* *j If„\dv(G, •) < e, * *for all ne N.*

*FÎ*

*Then f is G-integrable and for all *^{Е}^{е}^{ Ж }*one has*
*S f d G = lim J f J G .*

*E * n_>0° £

*Next we take up our characterization of {JtA)*.*

**72** **R. Fe r na n de z**

*From here on we agree that sé = Z and Z 1 = { E e l : ц(Е) < ***oo}; **

*moreover, if a — 0 we set Ж — Z t , and if a > О, Ж = Z.*

(2.4)** D****e f in i t i o n****. ** *We shall say that G**e**&**a**(X, Z, ***p , ***Ж) if G is a real*

*valued, finitely additive measure defined on Ж such that*
*(i) G « in, i.e. if Е е Z and p(E) = 0, then G(E) = 0;*

*(ii) 0 ^ v(G, E) < ***oo ** *for all Е е Ж ;*

*(iii) <X**q** = inf{fce]0, ***oo[: ** *IA(G/k, X) ^ 1} < ***oo, ** where

*I-A(G/k, E) = sup *

### j

*£ A ( r ~ § f ) ME,):*( £ „ £ 2...E J e â f t ,

### £)j,

*for E e Z , and @(Zl} E) is the set of all pairwise disjoint finite sequences *
*(E1, E2, ..., En) in Z t such that (J*"= 1* E} c= E.*

Unless otherwise stated, G will denote a real-valued, finitely additive
*measure defined on Ж such that G « /л.*

(2.5) R em ark s, (i) For /се]0, **oo[, ***the function IA(G/k, •) is a real-valued *
finitely additive measure defined on I .

(ii) If 0 < *ocq* < **oo, ** then ^{I}^{a}^{( G /}^{o l g}*^{,}* X) ^ 1.

(iii) If a *g** = 0, then G(E) = 0 for all E e Z l .*

*(iv) The function a<?> is a seminorm on the vector space &A(X, Z, p, Ж), *
*and is a norm if Ж = Z l (i.e. a = 0).*

*(v) If G e ^ A(X, Z, p, Ж), then by (1.2.Ü) and (2.4.ii), every simple *
function in *A is Jf’-simple and G-integrable.*

*In (2.8) below, for a fixed / e E£A we shall find a sequence (s„) of simple *
*functions in S£**a* such that

*\ f d G = lim J sndG * *for every E e Z and all G**e**&**a**(X, Z, p, Ж).*

*E * *n~>co E*

First, some propositions.

(2.6) **P****r o p o s i t i o n****. ** *I f b = ***oo ** *and G is such that * *olq* < **oo***, then G is *
*p-continuous, i.e. И т*^(Л)_ 0* G (A) = 0.*

P ro o f. See Lemma 6 in [12]. ■

(2.7) **P****r o p o s i t i o n****. ** *Let G**e**&**a**( X , Z, p, Ж). I f seL£A is a simple non*

*negative function, then*

*J sdv{G, •) ^ c\\s\\A, *

**x**

*where с = 2а£ if a — 0 or p(X) < ***oo, ** *and c — 2aG + av(G, X) if a > 0 and *
*p(X) **= o o .*

P ro o f. Eliminating the trivial case, suppose j|sj|^ / 0.

*Let s = Yj = ^ ci^Ei be as in (2.1) with ct > 0.*

*First we observe that if (F1} F2, ..., Ft) is a pairwise disjoint finite *
*sequence in Z 1 and * # 0, then by (l.l.iv) we have

(1) X С|В Д | ^ а § И л
*j =i*

**M** **L**

*Let J = {ieN: 1 ^ i < n and Eie Z 1}. Using (1.2. iii), (2.5. i), (2.5. ii), (1) and *
observing (2.5. iii) it follows that

(**2**^{) } *Y ,c A G ,E ^ 2 a .l\\s \\A.*

*ie J*

By (1.2.Ü) we obtain

(3) *£,) = а ^ с М в Л л ^ С , E,) « a||sLi>(G, X).*

*i$J * *i$J*

From (2), (3) and (1.2. ii) we obtain the desired result. ■

(2.8) **T****h e o r e m****. ***For f in f£A there is a sequence (s„) of simple functions in f£A *
*satisfying (1.3.i), (1.3.ii), (1.3.iii) and such that for all G**e**^**a**{X, Z, p, Ж) one *
*has:*

*(i) (s„) converges in G-measure to /;*

*(ii) / is G-integrable and*

*I f dG = lim j s ndG, * *for Е е Z.*

*e* *" - >o° E*

*/Is a consequence, the following hold:*

*(iii) \\x fdG\ < с ||/ ||А, with c as in (2.7);*

*(iv) )xfd(yG1 + G2) = y$xfdG1 + Sx fdG2, for * *y e R * *and * *Gt , G2e*

** A X , z , ii, * ) •*

P roof. To establish (i) and (ii), it suffices to consider / ^ 0 .

*If b — oo, let (s„) be as in (1.3). From (2.6) we deduce that (s„) converges in *
G-measure to / .

*If b < oo, we have |/ | ^ 2b \\f\\A /i-a.e. by (1.2.iv) and, since/is measurable, *
we can take a sequence (s„) of simple functions satisfying (1.3.i), (1.3.iii) and
*converging uniformly to / except on a set F e F with p(E) — 0. Clearly sne л *
*for n e N, v(G, ' ) « p and (s„) converges in G-measure to /.*

Now we will prove that the sequence (sn) above also satisfies (2.3.ii) and (2.3. iii), and thus obtain (ii).

*For all n e N , let yn = §x sndv(G, •). Since (y„) is bounded by (2.7), and *
nondecreasing, given s e ]0*, oo[ there is an n0e N such that*

*$sndv(G, ■) = j sndv(G, • ) - J sndv(G, •) < e/2 + J snodv{G, •),*

*E * *X * *Ec * *E*

*for all n ^ n0 and E e Z .*

*If M — 1+max {|s„*0*(x)|: x e l } , ô = e/(2M) and Fe = ( x e l : s„*0*(x) Ф 0}, *
we have F£e J a n d (s„) satisfies (2.3.ii) and (2.3.iii).

**74** **R. F e r n a n d e z**

For (iii), use (ii), (2.7) and note that

|Js„dG| ^ J|sJd » (G .-), for n eN .

*X * *X*

To establish (iv) it suffices to use (ii). ■

*Our next objective is to define a norm on УА(Х, E, g, Ж) and to prove *
*that this space is isometrically isomorphic to { J if f.*

(2.9) **P****r o p o s i t i o n****. ** *The function defined by*

*|| G ||л = sup{|J/dG |: feS eA and ||/||л < 1} *

*x*

*is a norm on @A(X, E, g, Ж). Moreover, one has*

*IIGh = sup {|f fdG\: f e J t A and \\f\\A ^ 1}. *

*x*

P ro o f. The first assertion follows trivially from (2.8.iii) and (2.8.iv). For
*the second, observe that simple functions in <£A belong to JiA and use (*2.8. ii). ■
(2.10) **P****r o p o s i t i o n****. ***I f G**g**^**a**(X, E, g, Ж), then the function x* defined on *
*MA{_ie^\ by x*(f) = \ x fdG, belongs to { J i f f [(Jzf*)*] and ||x*|| = ||G||^.*

(2.11) **P****r o p o s i t i o n****. ** *Let x* G{Mff and let G be the function defined on *
*Ж by G{E) = x*{ÇE). Then Ge&A(X, E, g, Ж) and*

*(i) x*(f) = fx f d G for all f e M A, and x* = ||G||^.*

P ro o f. We may suppose ||x*|| # 0 . It is clear that G is a real-valued,
*finitely additive measure, and G « g.*

*To prove that v(G, E) < oo for all *^{E}^{g}*Ж , it is enough to observe that if *

**E****g***Ж and (Et , E2, .* *En) is a pairwise disjoint finite sequence in Ж with *
и ?=1*^г ^ E then taking J — (ieiV: *1* ^ i ^ n and x*(£Êi) ^ *0}, we have

**î ***|G(£,)I = x*(« v * *u e ) ^ 2 Иx*|| I ie jx.*

*i = 1 * *i eJ * *i $J*

Next we will prove that < ||x*|| < oo.

*Let c e ]0 , 1[ and let (£ ls E2, ..., En) be an arbitrary pairwise disjoint *
*finite sequence in Г 1. If щ — c|G(Ei)|/(||x*||jit(£j)) belongs to [0, * (1.2.i),

**c {**** = ***limA'(t) for ie { l, *2, ..., *^{n ]}* and

**/ = £**

**C i S g a ( G ( E ^ E t i**i= 1

then and by (l.l.iv) and (1.2.v) we have

> c|x*(/)|

i= z1

*A{c^ + A* *c\G(EÙ\ '*
*х *\\ дЩ*

* »* II/IL - 1 + Z Л| №

*i = 1* 1**11

From this relation it is easy to see that a*q* ^ ||x*||/c, and since c e ]0 , 1[ is
*arbitrary, we conclude that a§ ^ ||x*||. Thus G e ^ ( l , Z, p, Ж).*

To obtain (i) it is enough to observe that (2.10) is true and that
*x*(s) = Jx sdG, where s is a simple function in &Л- ■*

From (2.10), (2.11) and (2.8.iv) we obtain in (2.12) below a characterization
*of {Jl'A)*; it will follow that &A{X, Z, pi, Ж) is a Banach space.*

(2.12) **T****heorem****. ** *There exists an isometric isomorphism of { J i f f onto *
*УА(Х, Z, p, Ж), given by the mapping x*i—>G where G(E) = x*(££)for Е**е**Ж, *
*and the following holds:*

*(i) x*(f) = \ x fdG for all f e J i A and ||x*|| = ||G||*.*

(2.13) **C****omment****. **In [13], with a given *х*****е**{Ла)* the author associates *
*a measure G defined on Z, satisfying some conditions. To obtain (2.12) it was *
*necessary to replace the domain of G by Ж , and condition (i) of Definition 3 *
in [13] by (2.4. ii).

*3. Characterization of the dual space (ЖА)*. Let JTA = .J}A \JtA, write *
*/ ~ f \ M A for a n y / e i ^ , and denote by d(-) the quotient norm on JfA. One *

can easily show that

*d(f) = inf {||/+s||^: s is a simple function in * *A).*

(3.1) **D****efinition****. ***Let / , де<ЖА. We write / < g if there exist f e f and gxE§ *

*such that f x ^ gx.*

*In [13] it is proved that the above relation is a partial ordering on JfA and *
*that { J f f f is a vector lattice. Two results which are used in the proofs, and are *
not obvious to us, are established in (3.4) and (3.5). First, two propositions.

**(3.2) P****roposition****. ***Let a *^{>}** 0 ***or p(X) *^{<}** oo. ***I f f is a measurable function, *
*bounded p-a.e, t h e n f e J i A.*

P ro o f. We may suppose / to be bounded. Since there exist x e ]0 , **oo[ **

*such that A(x) < \/(p{X)+ 1), and a sequence (s„) of simple functions, conver*

*ging uniformly to /, it is easy to verify that lim ,,^ ||/ — s„||x = 0. From a > 0 or *
*p{X) < ***oo ** *it follows that / , sne f£A for all n e N . Thus f e J i A. ■*

**(3.3) P****roposition****. ***Let*

*f A ***= ***{ f e & A: { A(k\f\)dp < ***oo ***for all /се]***0, oo[}. **

x

*Then f A is a closed subspace of JiA and*
*(i) f A a M A cz <£a,*

*(ii) if a = 0 and b = ***oo, ** *or if p(X) < ***oo ***and b — ***oo, ***then f A ~ JiA\*

*(iii) if a > 0 and b < ***oo, ** *or if p(X) < ***oo ** *and b < ***oo, ***then f£A = JiA.*

P ro o f. From (1.3.v) it follows easily that The remaining assertions are proved in Lemma 2 in [13]. ■

**76** **R. F e r na n de z**

(3.4) **P****r o p o s i t i o n****. ** *I f g s J t A and f is a measurable function such that *
*l/l ^ \g\ g-a.e, then f t J ( A.*

P ro o f. We may suppose | / | ^ *\g\. Let * (s„) be a sequence of simple
*functions in S£A such that lim,,-^ \\g — s j ^ = *0.

First we observe that for /се]0, **oo[, ***if n e N is such that 2\\g — sn\\A ^ k, *
then by (l.l.i) and (1.2*.iii), for all H e l we have*

(1) *f A{\f\/k)dg ^ f A(\g\/k)dg ^ i + i j A(2\s„\/k)dg.*

*H * *H * *H*

*Suppose b = ***oo ** *and a > 0. From lim,,-,^ Il g — * = 0 there exists an
*n0e N such that \\g — * ^ 1.

*Let B = **{**x e* *X : * *\g(x) — sno(x)\ ^ 2a}. Then g(B) < ***oo ** (1.2.iv), and the
*inequalities |/ | ^ \g\ ^ \g — s„0| + |s„0|, together with (3.2), imply that fÇBceJtA. *

*Moreover, it follows from (1) that f£,Bz f A <= Л А. So f ^ M A in this case.*

*Let b < o о, a — 0 and En = {xeX : *s„(x) *ф 0} for neiV. Then *
*l/l ^ 2b\\f\\A g-a.e. (1.2.iv), g(En) < ***oo ***(1.2.Ü) and f ^ Ene J I A (3.2), for all ne N. *

*Using (1) we obtain lim,,-,.,» ||/—№**е**„\\**а** = 0, and since JtA is closed in ££A, we *
*conclude that / e MA also in this case.*

The remaining cases are easy (3.3). ■

(3.5) **P****r o p o s i t i o n****. ** *I f f , geJ?A and \f\ ^ \g\ g-a.e, then d{f) ^ d(g).*

*Pro.of. We may suppose | / | ^ \g\ and, by (3.4), that д ф ^ А.*

It suffices to prove that for all e e]0 , oo[ we have

(1) *d ( f ) ^ d ( g ) + s.*

For a fixed e e ]0 , c o [ , let *s e**J£**a * be a simple function such that

0* < d(g) ^ Hgf + sll^ < d(g) + s and let * = { x e l : s(x) # 0}.

*As дфЛ А we have b < ***oo ** *and a = 0, or b = ***oo ** (3.3.iii).

*If b < ***oo ** *and a = 0, then giHf) < ***oo ** *(1.2.Ü) and |/ | ^ 2b\\f\\A g-a.e. *

*(1.2.iv). Since f £ HlEJfA (3.2) and*

*Ш**н**\\\**а** < Ы* *щ**\\**а* < *\\(9** + **s**)^H**c**\\**a** < d(g) + £,*
we conclude (1).

*For the remaining cases, we first note that if H**e**Z and if we have *
*A(|s|/((a — l)\\g + s\\f))ÇHE & 1 for all <xe]l , oo[, then*

(2) *d ( ( f ( BT ) K d ( g ) + s .*

In fact, by (l.l.i) and (1.2.iii),
*I -I ( , 11 T ) d l , - s i л ( * lgl )

*h * *\a|l 9 + **s**\\**a**J * *h* *\<**x**\\9 + **s**\\aJ*

*, * 1 a —1 *c*

*dg ^ -H---j A* N _{dg,}

sa|l *0 + **s \ \ J * *h* *\ot\\g **+ **s\\AJ * a « я \(а-1)110 ^{+ }«1и,
*and hence A(\f\/(<x\\g + **s**\\**a**))Ç**h e**J ? l .* From (1.3) there exists a simple function

*s**0**e**^* *a* such that | | ( / -s0KhIIa ^ a Il0 + «L- Thus

*d((f£HT) < \ \ / - З**о**)£**н**\\**а** ^ <**x**\\9 + **s**\\**a** < a(d(g) + e)*
and, since a s ] l , **oo[ ** is arbitrary, (2) holds.

*If b = ***oo ***and a = 0, then * *< ***oo **(1.2.Ü) and we obtain (1) replacing
*H by X in (2).*

*If b — ***oo ** *and a > 0, let H2 = **{**x e* *X* *: \g{x) +s{x)\ ^ 2a\\g+ s\\A}. Then *
*(1.2.iv) tells us that g(H2) < ***oo ***and we may replace H by H2 in (2). Moreover, *
it follows from |/1* ^ \g\ ^ *|0* + s| + |s| that f£ Hc is bounded, and thus ft,Hc2E.ÆA*
(3.2). Therefore (1) holds. ■

*Let £ be a pseudonormed vector lattice, x e E and z* **e**E*. Then we will set *
*as usual x + = x v 0, x_ = ( — x) v 0 and |x| = x + + x _ . Also, we will write *
z* ^ 0* if z*(y) ^ *0* whenever y ^ *0.

(3.6) **P****r o p o s it io n****, ** *(i) J f A is a vector lattice, and if / , §**е**ЛГа, then *
*/ v g = (max {/, g}J, f л g = (min {/, g}J and ( | / / = *1/ 1;

*(ii) J fA and {jVf f are Banach lattices;*

*(iii) if х*е(ЖА)* and x* ^ *0*, then*

*||x*|| = sup(x*(/): / ^ 0 and j A(f)dpt < *00};

**x**

*(iv) if x *é ( J^ A)*, then ||x*|| = ||(x*)+|| + ||(x*)_||;*

*(v) if x*, y*e(jVA)* and x*, y* ^ *0*, then*

llx* + y*ll = llx*ll + lly*ll;

*(vi) if f, g e J f A and /, g ^ Ô, then d ( f v g) = max{d(f), d(g)}.*

P ro o f. Except for (iii), these assertions are proved in [13]. Assertion (iii)
follows from Lemma 6 in [13], upon observing that |/ | = .fl/l/for ■
*From (3.6. ii) and (3.6. v), and from (3.6. ii) and (3.6.vi) we have, respectively, *
*that (**j**Va)* is an L-space and that Jf A is an M-space.*

From here on v will denote a real-valued, finitely additive bounded
*measure defined on I , v(v, •) the total variation of v, vt = (v + u(v, *-))/2 and
v2* = (a(v, •) —v)/2. Also we will denote by 0 the set of all pairwise disjoint *
*finite sequences (El , E2, ..., En) in I such that (J"=1£ f = X.*

(3.7) **D****e f in i t i o n****. ** *L e t/e jS ^ . I f f is nonnegative, we define*
*j f d v j = inf{ £ d((ftcy)vj(Ei: (£ 1; E2, .... E„)e3>},*

*X * **i = 1**

f o r j6{ l ,2}.

In the general case we define

*I fdv = ( J /+ dvx - J /+ dv2) - (J /_ dv1 - j /_ dv2).*

X X X X X

**78** **R. F e r n a n d e z**

(3.8) R em ark s, (i) From Lemma 10 in [13] and (3.5), one has
*I (yf+g)dv = y J f d v + §gdv, for y e R and*

*x * *x* *x*

and

*If f d v I ^ f | / | dv, + f l/l dv2 « d(\f\')(v, + v2)(X) = d ( f Mv, X), * for /<= *.*

*X* *X* *X*

*(ii) If x*(f) — J f d v , for f e j V A, then x * e ( J rA)* and ||x*|| ^ u(v, X).*

*x*

*(iii) If v is nonnegative, E e l and vE is defined on I by vE(F) — v(E nF), *
*then j x f ^ E dv = \ x f d v E for all 0 ^ f e F F A.*

(iv) If v and v are real-valued, finitely additive bounded measures defined
*on E , f e F £**a** and ye R , then*

*if d ( yv + v) = y \ f d v + \ f d v .*

*X * *X* *X*

(V) If / e * JTA *a n d / ^ 0, there exists a nonnegative representative of /.

*Next we take up our characterization of {JFff.*

(3.9) **D****e f in i t i o n****. ** *We shall say that ve'Y'jffF, I , pi) if v « pi and there *
*exists f e J ^ A, f ^ 0 with d(f) ^ *1, for which the following holds:

*(i) if E e l and v(v, E) ф *0*, then d((fÇEf) = *1.

*It is easy to see, using (3.6.vi), that Уд(Х, I , pi) is a vector space.*

(3.10) **C****o m m e n t****. ***In [13] (page 573), in place of i^A(X, I , pi), the author *
*uses &A{fi), the space of all v « pi such that there exists / e J*2^ with
*l x A{\f\)dpi < a o ,f 4 J ? A and the following holds:*

(i) the support of v lies in the support o f/, where “support of v” is defined
*in [13] (page 571) as “the sets E for which v(v, E) > 0”.*

*The author claims that if v e & A{pi) is nonnegative and x*(/) = \ xfdv for *
*a ll/e Jf A, then ||x*|| = v(X). To prove this, he considers л: = (Ex, E2, ...,£ „ ) *
*e 0* with v(Et) > 0 and defines f = f x *+ / 2* + ••• +/„ where \ xA(\f\)dpi < oo, *
*d(fy = 1 and support o f/ is Et, for ie{ 1 , 2 , . . . , n}. Then he writes: “Thus *
*Y!i=i d((f%EiT)v(Ei) = v(X) and refining the partition n on the left yields *
*x * ( f ) = v(X)”. This last assertion is not clear to us, for / depends on the *
partition 7i. However, we observe that if there is an / satisfying (3.9. i), / ^ Ô,
*then it is easy to see that x*(f) = v(X). Thus we have replaced, in our *
characterization, (3.10. i) by (3.9. i).

(3.11) **N****o t a t i o n****. ** *If E e l and х*е(.Л/1)*, we will denote by x | the *
*function defined on J fA by x E(f) = x*((f£Ey).*

(3.12) **P****r o p o s i t i o n****. ** *Let 0 ^ х*е(ЖА)*. Then there exists geJ?A, g ^ 0, *
*with d(g) ^ *1*, such that*

*(i) ||xl|| = x*((g£EY) for all E e l ;*

*(ii) if E e l and ||x||| ^ 0, then d((gÇEy) = 1.*

*Proof. By (3.6.iii), for n e N there exists a nonnegative/„eJS*? 4 such that
*JxA(f„)dp < *00* and x*{fn) > ||x*|| — l/n. Moreover, from (1.3) we know that *
*there is a nonnegative hnef£A such that \AA(hn)d/x < 1/2" and /*j„ =/„.

*Let h = lim„_>*00*(max{/i1, . . . , hn}) and observe that if E = {**x e**X: *

*h(x) = *00*}, then p(E) = 0. In fact, if p(E) > 0, there exists F e l l , F <= E such *
*that jx{F)> 0. By (1.1. iv) and (1.2. iii) we have*

**oo-ju(F) = j ****ÇF h d p**** ^ ****U****f****\\****a**** lim**

*X * *k- + 00*

**H** ^{ï} **&)**

^{ï}

*dp + E J A(hn)djx <*

^{00},

*п = 1 X*

*and so p(F) = 0, which is a contradiction.*

Let *g = h£Ec * Then *j x A(g)dp ^ * sc 1, and since
*x*(g)^ x*(£n) = x*(fn) > \\x*\\ — l/n, for all neN, we conclude that *

*||x*|| = x*(g). So for E**e**E, using (3.6. v), we obtain*

*0 ^ llxlll ~ х * ( Ш ) = x*((gÇECT) - Hxtll ^ 0,*
and thus

*llxlll = х * ( Ш ) < WxiWdfaZJ) ^ llxlll. .*

(3.13) **T****h e o r e m****. ** *Let 0 ^ x* e (.Ж^)*. Then there exists a unique *

*v e**'VA(X, E, fx), nonnegative, defined by v(E) = ||x |||, such that*
*(i) x*{f) = $x fd v for all f e J f A, and ||x*|| = v(X).*

*P ro o f. Let v{E) = ||x||| for E**e**E. Then * *v e**'Ÿ'A(X, I , p) ((3.6.v) and*
*(3.12)). To show that (i) holds define z*(f) = $xfdv for f e J A A\ we shall prove *
that I)z* — x*I) = 0.

For Ô 4* and ж = (E1, E2, * *En)e0> we have*

Î d((/^,r)v(£,.) = £ d((/{E,r) llxl.ll » £ **((/<W) = * * (Л

*i = 1 * *i = 1 * *i = 1*

*and from this it is clear that z * ^ x * . Thus * ||z*|| = ||x*||, for

*||z*|| ^ v ( X ) = ||x*|| (3.8.ii), and also ||(z* — x*) + x*|| = ||z* — x*|| + ||x*||*

(3.6.v). Hence ||z* — x*|| = 0 .

To prove the uniqueness of v suppose that *v e**'V A{X, E, p) is nonnegative *
*and x*(f) = \ x f dv for all f c J T A. Then by (3.8.iii) and (3.8.i) we have*

*x u f ) = x*((fiEj) = и м и < d ( f ) m ,*
*x*

*for j > 0 and Ee E. So v —v is nonnegative, belongs to УА{Х ,Е ,р ) and *
*i x f d { v - v ) = 0 for all f E f £ A (3.8.iv).*

*Since v - v e f ^ ( I , E, p), there exists Ô ^ g e*/ 4* such that d((g^Ef) = 1 for *
*E**e**E * with (v — v)(£) # 0. Thus *\xQd{v — v) = (v — v)(X), * and hence
(v-v)(X) = 0. ■

(3.14) **T****h e o r e m****. ** *There exists an isometric isomorphism of (JfA)* onto *

*^д(Х, E, p), given by the mapping x*i—>v, such that*
*(i) x*(f) = Jx f d v for all f e J f A, and ||x*|| = v{v, X).*

**80** **R. Fer nan de z**

*P ro o f. If х * е ( Ж А)* we will write y* = (x*)+, z* = (x*)_ and define, for *
*all E e Z , v3(£) = ||y*|| and v4(£) = ||*ze||. Let v = v3 —v4.

To prove (i), use (3.13) and (3.8. iv) to write
*x*(f) = Sfdv * *for all f e JTA, *

*x*
and (3.6.iv) and (3.8.ii) to obtain

v3(X) + v4*(X )= №*11 + 11**11 = ||x*|| ^ v(v, X) ^ v3(X) + vA(X).*

*To show that the mapping х*ь->v is onto, consider x e ' V A{X, Z, g) and *
*define x*(f) = $ x f d v for /еЛ~А. Then, writing vl = (v + u(v,-))/2 and *
v2 = (u(v, •) — v)/2, we have

j *- J f dv2 = j f d v = x*(f) = \ f d v = \ f d v 3 - J f dvA,*

*X* *X* *X* *X* *X* *X*

*for all f e j T A, and thus*

*J /rf(Vi+v4) = J f d{y**2* T v2) for all / e ^ 4-

*X * *X*

*Since \ е У А{Х, Z, g), it is easy to verify that v*l5 v2 6*^ (X, Z, g), and *
thus vj + v4 = v3 + v2 (3.13). Hence v = v.

Finally, observing that x*i—>v has a linear inverse, we conclude that the mapping is linear. ■

*It is immediate, by (3.14), that ЖА(Х, Z, g) is a Banach space.*

*4. Characterization of the dual space (£ТЛ)* . As in Section 2, set Ж = Z if*
*a > 0, and Ж = Z x if a = 0.*

(4.1) **T****h e o r e m****. ***Let x* e ( ^ A)*. Then there exist a unique G e ^ A(X, Z, g, Ж), *
*defined by G(E) = x*(ÇE) for Е е Ж , and a unique z* e{Jtf)1 such that*

*(i) x*(/) = j x f dG + z*(f), for all f e S T A.*

P ro o f. The same as in Proposition 2 of [13], noting that (2.10), (2.11) and (2.1 2) hold. ■

(4.2) **P****r o p o s i t i o n****. ***Let x* e(JTA)*, G and z* as in (4.1), and y*(f) — \ x f d G*

*for all f e L£a. Then * *\*

*(i) if x* ^ *0*, then y* ^ *0* and z* ^ *0;
*(ü) Il \y*\ + \z*\ II = 11/41 + *11**11;

*(iii) |x*| = |y*| + |z*| and |y*| л |z*| = *0.

*P ro o f. If x* ^ 0, it is clear that y* ^ 0. Thus for (i) it suffices to prove *
*that x* ^ y*. For this let 0 ^ f e S T A and take (s„) as in (2.8). Since z*e{Æff- *
*and x* ^ *0 we have

*y*(f) = lim j sndG = lim x*(s„) ^ x*{f).*

*n~* 00 X * 00

*For (ii), let e e ]0 , oo[. Observing that if w*e(JTA)*, then ||w*|| = || |w*| || *

*(page 239 in [7]), thus there exist f g e £TA, nonnegative, with \\f\\A ^ 1, *

*\\g\\A ^ *1 and such that

Ilk*!! < l.y*l(/) + £/2 and *||z*|| < \z*\{g) + e/2.*

*From (1.3), for k e N fixed there exists se.J/A with 0 ^ s ^ g and such that *
*J A(g-s)dp < (1 + l/7c)-J A(f)dp.*

*X * *X*

*Let h = max {/, (g — s)]. Then hefTA and *

*f kh \ * *к * *к*

*\ A* 7TT ^ r r r f ^ 7 Т т [1 + f ^ L

**X *** \ k +* 1 /

**k + i x**

**k + l**

**x**

**x***Thus \\h\\A < (k+ l)/k.*

Hence we have

l l / I K 11**11* < \y*\(f) + \z*\(g) + e = \y*\(f) + \z*\(g-s) + e*

*< \y*\(h) + \z*\{h) + £ = (|y*l + |z*|)(/i) + e*

< Il *\y*\ + \z*\* II НЛИл + е < II ^{\У*\ }* ^{+ \г*\}* ll((/c + l)/fc) +

^{e.}

*Since s and к are arbitrary we conclude that (ii) holds.*

(iii) may be found in [11] (page 40). ■

(4.3) Pr o p o s it io n. *There is an isometric isomorphism of ( J i f f onto ( Л А)* *

*given by the mapping j : z * w h e r e x * is defined by x*(f) = z*(/).*

*In the next theorem we present our characterization of (J ’f * .*

(4.4) ^{T}h e o r e m. *There is an isometric isomorphism of (STA)* onto the Banach *
*space УА(Х, Z, p, Ж) x ЖА(Х, I , p) given by the mapping *x*h->(G, v*) such that*

*(i) x*{f) = $x f dG + ] x fd v for all f e <£A\*

*(ii) 11**11 = \\G\\**a** + **v**(**v**,X),*

*the first integral being as in (2.2) and the second as in (3.7).*

P ro o f. This is a consequence of (4.1), (4.3), (3.14) and (4.2), by observing
that ||x*|| = || |x*| || for *x*****e**(J>a)* [7]. ■*

*5. Concluding remarks. (I) The characterization of (f £ f f obtained by *
Andô in [1] goes through the spaces *and J AjjiA with the hypotheses *
*p(X) < oo and b = oo. From (3.3) we know that under these hypotheses the *
*spaces / A and JiA coincide.*

(II) Using (2.6), (2.8) and Radon-Nikodym’s Theorem, one can prove that
*if p(X) < oo and b = oo, given G**e c**$**a**{X, I , ***p, ***Г Д there is g e f f A such that*

*(i) §xfdG = §x fgdp, for all f e J A,*

*(ii) \\G\\A = swp{$x \gh\dp: hef£A and ||й||л <1}.*

*Hence, in this case, Theorem (4.4) above tells us that given x*e(J?A)*, there *
*exist gef£-A and ve r A( X , Z, p) such that*

*(iii) x*(f) = \xf9dp + ^xfdv, for all feS£A,*

*(iv) ||x*|| = sup{jx \gh\dp: h e f T A and \\h\\A ^ 1} + v(v, X).*

Thus the result obtained by Andô in [1] and Theorem (4.4) coincide when
*h{X) < oo and b = oo.*

*(III) In [4] we also prove that (iii) and (iv) above still hold when (X, Z, p)*

*~*** Commentationes Math. 30.1**

**82** **R. F e r na n de z**

*is a (j-finite measure space and # A — JtA, or when à = 0 and f A = JiA.*

(IV) *If f A Ф {0} and # A Ф MA, there exists Ge&A(X, I , g, Z) such that *
for all the following assertion is false:

*J f d G = J fgdu, * *for all f e J f A.*

*x * *x*

This will be proved below.

*From (IV) it follows that (iii) is not true if / А Ф {0} and J A Ф JiA. *

*P r o o f of (IV). By (3.3) we know that fi{X) = *go, *b = *oo *and a > 0. *

*Therefore E,x e J i A and %ХФ</А>* also я = 0.

By Hahn-Banach’s Theorem and (2.12) there exist *x*****e* *(J^a)** and
*Ge&A(X, Z, ji, Г) such that*

^*(^лг)#0 5 **(/) = 0* for all f e f A,*
*x*(f) = j fdG, * *for all f e J t A. *

*x*

*Suppose there is a ge££A such that*

*{ fd G = \f g d g , * *for all f e M A.*

*X * *X*

Let y e ]0 , oo[ *and F = { x eX : \g(x)\ > y} be such that g(F) > 0 (this *
*F exists because x*(^x) ф 0).*

*As à = 0, and*
**A**

*we conclude that g{F) < *oo, and so ^ s g n g e ^ .
Thus

*0 = x*(Çpsgng) = $(ÇFsgng)gdn = j ÇF\g\d[i ^ yg(F) > 0,*

*X * *X*

which is impossible. ■

This work is part of our Master’s Dissertation written under the guidance of Dr. Iracema Martin Bund.

**References**

**[1] T. A n d ô , Linear functionals on Orlicz spaces, Nieuw Arch. Wisk. 8 (3) (1960), 1-16.**

**[2] I. M. B u n d , Birnbaum-Orlicz spaces, IME-USP, Sâo Paulo, 1978 (Notas do Instituto de ****Matemâtica e Estatistica da Universidade de Sâo Paulo, Série Matemâtica, 4).**

**[3] N. S. D u n f o r d and J. T. S c h w a r tz , Linear Operators, part I : General theory, Interscience, ****New York 1967.**

**[4] R. F e r n a n d e z , Caracterizaçâo do dual de um espaço de Orlicz, Dissertaçâo (Mestrado), ****Instituto de Matemâtica e Estatistica da Universidade de Sâo Paulo, 1986.**

**[5] —, Integraçâo em relaçâo a medidas definidas em ideais, Trabalhos apresentados, 23° **

**Seminârio Brasileiro de Anâlise, Campinas, 1986, 199-214.**

**[6] H. H u d z ik , Orlicz spaces o f essentially bounded functions and Banach-Orlicz algebras, Arch. **

**Math. 44 (1985), 535-538.**

**[7] J. L. K e lle y and I. N a m io k a , Linear Topological Spaces, Graduate Texts in Mathematics, ****36, Springer, New York 1963.**

**[8] M. A. K r a s n o s e l’s k ii and Ya. B. R u tic k ii, Convex Functions and Orlicz Spaces, ****Noordhoff, Groningen 1961.**

**[9] J. M u s ie la k , Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, ****New York 1983.**

**[10] W. O r lic z , On integral representability o f linear functionals over the space of (p-integrable ****functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 567-569.**

**[11] A. L. P e r e s s in i, Ordered Topological Vector Spaces, Harper&Row, New York 1967.**

**[12] M. M. R ao, Linear functionals on Orlicz spaces, Nieuw Arch. Wisk. 12(3) (1964), 77-98.**

**[13] —, Linear functionals on Orlicz spaces: general theory, Pacific J. Math. 25(3) (1968), 553-584.**

INSTITUTO DE MATEMÂTICA E ESTATISTICA UNIVERSIDADE DE SAO PAULO

SÀO PAULO, BRASIL