View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 23, 119-134 (1976) Representation of Linear Functionals on a Class of Normed K&he Spaces EP DE JONGE Mathcmatisch Instituut, Katholieke Universiteit, Toernooiveld, Nijmegen, The Netherlands Communicated by the Editors Received October 1, 1974; revised July 9, 1975 Under the condition that LF,, (th e set of singular functionals on a normed Kijthe space LP) is an abstract L-space, it is proved in this paper that there exists a set of purely finitely additive measures -(v, such that Lz,, 2 AP holds. It follows that Lg, II is an abstract L-space if and only if Lg,+ is Riesz isomorphic and isometric with a band in L*,,, . 1, INTRODUCTION Throughout this whole paper (a, L’, p) will be a u-finite measure space such that the p-null sets are measurable. By a we shall denote the Riesz space consisting of all real-valued finitely additive measures v on (d, r) which satisfy 1 Y [ (d) < co and which are absolutely continuous with respect to p. Here 1v 1 = V+ + v-, where Y+ and Y- are the components of v in the Jordan decomposition v = v+ - v- of v into two positive measures. If we define 11v 11= 1v 1 (d) for all v E 2, then g becomes a normed Riesz space and even an abstract L-space (so 0 < pi , vs E g implies II v1 + y2 II = II y1 II + II v2 II). In the rest of the paper the reader is assumed to be familiar with the theory of finitely additive measures as developed in [2, Chap. III] and in [9]. Since all spaces we deal with are actually normed Riesz spaces, we note that if in the sequel R, and R, are two normed Riesz spaces, then R, z R, always means that there exists a norm and order preserving linear operator I from R, onto R, such that I-l does exist and such that I-l is a norm and order preserving linear operator from R, onto R, . It is well known (cf. [9]), that the dual space L,* of L,(d, r, p) satisfies L,* E a. If we denote by CA(@) the collection of all countably additive measures in ~?8and by PFA(a) the collection of 119 Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. 120 EP DE JONGE all purely finitely additive measures in a, then g = CA(a) @ PFA(&?), and in Riesz space terminology it is even so that CA(a) and PFA($?) are norm closed bands in g being each others disjoint complements. It is clear from the Radon-Nikodym theorem that L,(d, I’, CL)E CA(g). Now note that if we consider L, as a normed K&he space (cf. [lo, Chap. 15]), then L,* = Lz,C @ Lz,S, where L$,+ denotes the set of integrals on L, and L z,S denotes the set of singular functionals onL,. Moreover, since L, is the first associate space of L, it follows that Lz,+ g L, (cf. [lo, Theorem 69.31). Thus Lz,s g PFA(g). It will be the purpose of this paper to generalize this last result. In fact, our first main result will be: THEOREM 1.1. Let L,,(d) r, p) b e a normed K6the space. Then there exists a band A0 of PFA(@ such that A%$g LzS if and only ifL$ is an abstract L-space. We shall explain the terminology of this theorem. From now on M will be the set of all real-valued p-measurable functions on 4, where the values + CO and -co are allowed for functions in M. It is to be understood that p-almost everywhere equal functions in M are identified. Next, let p be a nontrivial function norm on M, so for all f,gEM and for all BEIR we have (i) 0 <p(f) < co, p(f) = 0 if and only if f E 0, and there exists an element f. E M such that 0 < dfd < a. (3 &f) = I a I df 1. 64 P(f + g) G f(f) + dg). f$x:flLf I < I g 1, then 4 f ) ,< p(g) (in particular d f 1 = p(If 1)). , -W, r, P) = {f~ M : p(f) < ~0)= Lo , then L, becomes a normed Riesz space with Riesz norm p. The space L, is called a normed Kothe space. Let L,,” be the Banach dual of L, , provided with the Riesz norm p*, so L,* is again a normed Riesz space. A functional F EL,* is called an integral whenever ui > uz 3 e-e 3 0 and inf, u, = 0 (notation U, .J,0) inL, impliesF(u,) --+O. The collection of all integrals on L, is denoted by LzC . If follows that L:C is a norm closed band in L,*. Next, we denote by LzS the disjoint complement of LzC in L,*. Elements of L2s are called singular func- tionals. It follows that L$ is a band in L,* and that L,* = Lz,@ LX8 . From now on we shall also assume that the function norm p is saturated, i.e., any set E E r satisfying p(E) > 0 contains a subset FE r such that p(F) > 0 and such that p(xF) < co (xF denotes the characteristic function of F). As follows from ([lo, Theorem 67.21) this is no restriction at all. If L,’ is the first associate space of L, (so L,’ is a SINGULAR FUNCTIONALS ON KiiTHE SPACES 121 normed Kothe space by itself), then L,’ s L$ . Hence, it follows from Theorem 1.1, that we have a complete representation of L,* for a large class of normed K&he spaces. In the remaining part of this paper the reader is assumed to be familiar with the theory of normed Kothe spaces as developed in [IO, Chap. 151. All facts on K&he spaces can be found there. Returning to Theorem 1.1, it is obvious that this theorem generalizes Lz,s z PFA(99). Also it is obvious that ~.44g L$ implies L$ is an abstract L-space (since ,N, is a band of the abstract L-space PFA(9Y)). The question arises under which conditions L$ is an abstract L-space. In [6] this problem is solved completely. It is proved there that L$ is an abstract L-space if and only if L, is a semi-M-space (i.e., fi , fi ELo+, P(fJ = P(f2) = 1 and 4 $4 implies lim p((sup(f, ,fJ)xd,) d 1 as n ---f co), cf. [6, Theorem 5.11). It is also shown in [6] that the well- known Orlicz spaces L, as well as the less well-known Lorentz spaces L, are semi-M-spaces, so there are many examples. Finally we note that similar work on this subject is done in [l, 3, 71. However, since our definition of singular functionals differs from the definition used in [3, 71 it follows that our results are different. In fact, the results of [3, 71 are also generalized, since L$ is in general larger than the class of “singular” functionals N,* of [3], and since [3] generalizes [7]. For example, considering the space L, , it follows that in [3] the class N,* = (01, but L& # (01. The fact that N,,* C L$ in general follows from Loa = J-{L$} and from L,” C MO (using the terminology of [3]). W e 1eave verification to the reader. Also we note that our assumption L& is an abstract L-space is much weaker than property (I) required in [3]. Furthermore we do not require p to have the Fatou property. It must be mentioned that our methods to derive Theorem 1 .l are very similar to the methods used in [1, 3, 71. 2. THE LINEAR OPERATORS I, From now on p will be a fixed saturated function norm on M. Furthermore, L, and $zYwill be as in Section 1. By B we shall denote the collection of all disjoint finite partitions of d into measurable sets, and P, will be the trivial partition {A}. If PI , Pz E 8, then PI < Pz will mean that PI is a refinement of Pz . It follows that P < P,, for all P E 9, and that < defines a partial ordening in 8. If E E r and f E L, , then fE = fxE . Furthermore, if FE L,* and E E r, then FE(f) = F(fE) for all f E L, . It is obvious that for all E E r and for all FE Lo*, FE < F and FE E Lo* holds. 122 EP DE JONCE Let now (T be a function seminorm on L, such that u(f) < p(f) for allf E L, (so u = p is allowed), and define i=l for all f E Lo+, for all v E S?+ (the positive measures in B’) and for all P = {Ei : i = 1, 2 ,..., n} E 9. Note that QO(*, v, -) is linear on SY+ and that Pl , Pz E 9, P, < P, implies QJf, V, Pl) < Q&f, v, P2) for all f E L,+ and for all v E k@+. Next, we define for all v E a+ and for all f EL,+ F”(f) = inf-Q(f, v, P): P E 9'). If f E L, and v E S? are not necessarily positive, then F,(f) = F”+(f+> - F”+(f-> - F”-(f+) + F”-(f->.