Journal of Convex Analysis Volume 3 (1996), No.1, 25–44

Two Generalizations of Komlós’ Theorem with Lower Closure-Type Applications

Erik J. Balder

Mathematical Institute, University of Utrecht, 3508 TA Utrecht, the Netherlands. e-mail: balder@math.ruu.nl

Christian Hess

Ceremade, URA CNRS No. 749, Université Paris Dauphine 75775 Paris Cedex 16, France. e-mail: hess@paris9.dauphine.fr

Received March 30, 1994 Revised manuscript received January 19, 1996

Dedicated to R. T. Rockafellar on his 60th Birthday

Two generalizations of Komlós’ theorem for functions and multifunctions with values in a Banach space are presented. The first generalization is partly new and the second one is a Komlós-type result of a completely new nature. It requires the Radon-Nikodym property for the Banach space and its dual. In both cases our approach relies on using Komlós’ theorem by means of a diagonal extraction argument, as introduced in [5], [6], [7]. Two quite general lower closure-type results, which follow immediately from our main results, are shown to generalize or substantially extend a number of results in the literature.

1. Introduction

It is well-known that weak lower semicontinuity and lower closure results for integral functionals over a finite measure space (Ω, A, μ) can be obtained from relative compactness results for K-convergence [5], [6], [7]. In themselves, such relative compactness results can be used to develop new relative compactness results, e.g. for narrow convergence of Young measures [9] or to refine classical results for relative weak compactness in L_{1}-spaces. As a fine example of the latter we mention the recent work of Saadoune [44], where the well-known characterization of relative weak L_{1}-compactness of Diestel, Ruess and Schachermayer [28] is generalized. Recall that a sequence of functions or multifunctions (F_{n}), F_{n} : Ω → Y, is said to K-converge to another function or multifunction F_{0} : Ω → Y

(written as F_{n} →^{K} F_{0}) if for every subsequence (F_{n_{j}}) of (F_{n}) there exists a null set N such that for every ω ∈ Ω\N

lim_{m→∞} \frac{1}{m} ∑_{j=1}^{m} F_{n_{j}}(ω) = F_{0}(ω).  (1.1)

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

26 E.J. Balder, C. Hess / Two Generalizations of Komlós’ Theorem

In other words, K-convergence (which is itself a nontopological notion) constitutes almost everywhere pointwise convergence of arithmetic averages (i.e., Cesaro-convergence) over any subsequence. Here the image space Y of the functions or multifunctions is a convex cone in the sense of e.g. [30] (i.e., (Y, +) is an abelian semigroup satisfying the usual laws of multiplication with nonnegative scalars), equipped with a topology. Concretely, this paper uses for Y either a separable Banach space X or a subset of its hyperspace 2^{X}. As initiated in [5], [6], [7], relative compactness results for such K-convergence can be obtained by exploiting a deep and fundamental theorem of J. Komlós [40] for realvalued functions. In turn, this produces a powerful abstract version of Komlós’ theorem, which is characterized by a convex version of tightness and the presence of a class of affine continuous integrands with a point-separating property (see [6], [7] and Appendix A).

This paper presents essentially two generalizations of Komlós’ theorem for functions and multifunctions taking values in a separable Banach space X. The first generalization, Theorem 2.1, subsumes the two separate main results Theorems A, B of [5] (see Corollary 2.2). A result similar to Theorem 2.1 (a) can already be found in Theorem 2.2 of [6], and, actually, Theorem 2.1 (a) follows from the abstract version of Komlós’ theorem in [6], [7]. This is demonstrated in Appendix A to this paper by proving what is actually an extension of Theorem 2.1 (a). Our Theorem 2.1 (b) is new, and we suspect that its proof contains elements that may be useful in other, related arguments as well.

Theorem 2.5, the second principal generalization of Komlós’ theorem given in this paper, is completely new. It depends on the Banach space X and its dual X^{∗} having the Radon-Nikodym property, and our proof depends on the Radon-Nikodym theorem for multimeasures. On the other hand, it is interesting to note that essential parts of the reasoning followed in proving Theorem 2.1 are also used (e.g., part 1 in section 3 serves in the proof of both theorems).

As mentioned in the first paragraph, there are natural by-products of these principal results in the form of lower closure-type results. In section 5 these results, Theorems 5.1 and 5.2 have been derived. In section 6 it is shown that together (or in conjunction with Appendix A) they generalize a number of results in the literature, and sometimes quite substantially. These include the Fatou-type lemmas in [18], which formed for us the starting point of the present paper.

The remainder of this introduction is used to establish some notation: Let ∥ · ∥ stand for the norm on the separable Banach space X; the associated dual space is denoted by X^{∗} and the usual duality between X and X^{∗} by < ·, · >. Respectively by s and w we shall indicate the strong and the weak topology on X. A subset of X is said to be w-ball-compact if it has a w-compact intersection with every closed ball. Recall that the sequential weak Kuratowski limes superior w-Ls C_{n} of a sequence (C_{n}) of subsets of X is defined as the set of all w-limits of subsequences (x_{n_{j}}), with x_{n_{j}} ∈ C_{n_{j}} for all indices n_{j}. Further, let P_{wkc}(X) be the collection of all nonempty w-compact convex subsets of X. For any C in P_{wkc}(X) we set

s(x^{∗} | C) := sup_{x∈C} < x, x^{∗} >, ∥C∥ := sup_{x∈C} ∥x∥,

to define for the set C respectively its support function s(· | C) and radius ∥C∥. On P_{wkc}(X) we use the following topology: a sequence or generalized sequence (C_{n}) converges scalarly (alias weakly [26]) to C_{0} in P_{wkc}(X) if

lim_{n→∞} s(x^{∗} | C_{n}) = s(x^{∗} | C_{0}) for every x^{∗} ∈ X^{∗}.  (1.2)

A function F : Ω → P_{wkc}(X), also called a multifunction, is said to be Effros measurable (or measurable for short) if the set {ω ∈ Ω : F (ω) ∩ U ≠ ∅} belongs to A for any s-open subset U of X. As we shall see later, the Effros measurability of a multifunction taking values in P_{wkc}(X) is equivalent to its being scalarly measurable. By L_{P_{wkc}(X)}^{1} we denote

the set of all measurable multifunctions F : Ω → P_{wkc}(X) such that ∫_{Ω} ∥F (ω)∥μ(dω) < +∞ (the integrand ∥F (·)∥ is measurable by III.9 of [19]). Obviously, the singleton-valued multifunctions in L_{P_{wkc}(X)}^{1} can be identified with elements of the space L_{X}^{1} of all Bochnerintegrable functions from Ω into X. Finally, recall that for a set A ∈ A the integral of a multifunction F : Ω → 2^{X} over A is the set given by

∫_{A} F dμ := {∫_{A} f dμ : f ∈ L_{X}^{1}, f(ω) ∈ F (ω) a.e.}.

As such, this general definition still allows the set in question to be empty; however, it is well-known (Theorem 3.6 (ii) of [39]) that for F ∈ L_{P_{wkc}(X)}^{1} the set ∫_{A} F dμ is nonempty and belongs to P_{wkc}(X) for every A ∈ A.

2. Main results

In this section we state our main results; sections 3 and 4 are devoted to the proofs. Our first principal result is an extension of Komlós’ theorem in the spirit of [5], [6], [7], but with pointwise ball-compactness conditions in what can be seen as a multivalued elaboration of Remark 2.3 in [8]. A generalization of part (a) of this theorem will be derived from [6],

[7] in Appendix A.

Theorem 2.1. Let (F_{n}) be a sequence in L_{P_{wkc}(X)}^{1} satisfying the following hypotheses:

(i) sup_{n} ∫_{Ω} ∥F_{n}(ω)∥μ(dω) < +∞, (ii) clco ∪_{n}F_{n}(ω) is w-ball-compact a.e.

Then there exist a subsequence (F_{n′}) of (F_{n}) and F_{∗} ∈ L_{P_{wkc}(X)}^{1} satisfying

(a) F_{n′} →^{K} F_{∗},

(b) F_{∗}(ω) ⊂ clco w − Ls F_{n′}(ω) a.e.

In the single-valued case Theorem 2.1 takes the following form, in which it subsumes the two main results of [5]. ^{1}

Corollary 2.2. Let (f_{n}) be a sequence of Bochner-integrable functions in L_{X}^{1} satisfying the following hypotheses:

(i) sup_{n} ∫_{Ω} ∥f_{n}(ω)∥μ(dω) < +∞, (ii) clco{f_{n}(ω) : n ∈ N} is w-ball-compact a.e. Then there exist a subsequence (f_{n′}) of (f_{n}) and f_{∗} ∈ L_{X}^{1} satisfying

1 It is enough to observe that w-compactness always implies w-ball-compactness, and that w-closedness

implies w-ball-compactness in case X is a reflexive Banach space.

(a) f_{n′} →^{K} f_{∗},

(b) f_{∗}(ω) ⊂ clcow−Ls f_{n′}(ω) a.e.

Remark 2.3. If one systematically replaces the w-topology by the s-topology, that is, one replaces P_{wkc}(X) by P_{skc}(X) (the set of all nonempty s-compact convex subsets of X) and one replaces w-ball-compactness in (ii) by s-ball-compactness, then the conclusions (a) and (b) can be improved as follows:

(α) F_{n′} →^{K_{h}} F_{∗},

(β) F_{∗}(ω) ⊂ clcos−Ls F_{n′}(ω) a.e.,

where F_{n′} →^{K_{h}} F_{∗} means now K-convergence as in (1.1) when Y := P_{skc}(X) is equipped with the Hausdorff metric h; i.e., for every subsequence (F_{n′_{j}}) of (F_{n′}) there exists a null

set N such that for every ω ∈ Ω\N

lim_{m→∞} h(\frac{1}{m} ∑_{j=1}^{m} F_{n′_{j}}(ω), F_{∗}(ω)) = 0.

This observation follows by adapting the proof of Theorem 2.1 given below, in particular by replacing part (a) of Lemma 3.2 by its part (b).

Remark 2.4. Both in Theorem 2.1 and in Remark 2.3 the convexity condition for the values of the multifunctions can be lifted.^{2} In Theorem 2.1 this is evident by elementary properties of support functions and Krein’s theorem. As for Remark 2.3, the removal of convexity is made possible by applying Lemma p. 307 of [2] or Lemma 2 of [37]. We now state our second principal result, a Komlós-type theorem of a completely new kind, characterized by the fact that X and its dual X^{∗} now have the Radon-Nikodym property (RNP). Recall that X is said to have the RNP (with respect to (Ω, A, μ)) if each μ-absolutely continuous vector measure ν : A → X with bounded variation has a density f ∈ L_{X}^{1} with respect to μ, i.e.,

ν(A) = ∫_{A} f dμ for all A ∈ A.

Actually, following [17], [18] it can be observed that in this paper the full force of the RNP for X is not really needed: it is enough to require the above density property only for those ν which satisfy

ν(A) ⊂ clco ∪_{n} ∫_{A} F_{n} dμ for all A ∈ A.

Theorem 2.5. Suppose that both the Banach space X and its dual X^{∗} have the RNP. Let (F_{n}) be a sequence in L_{P_{wkc}(X)}^{1} satisfying the following hypotheses:

2 We thank G. Krupa (Utrecht) for calling this point to our attention.

(i) sup_{n} ∫_{Ω} ∥F_{n}(ω)∥μ(dω) < +∞, (ii’) clco ∪_{n} ∫_{A} F_{n} dμ is w-compact for every A ∈ A, Then there exist a subsequence (F_{n′}) of (F_{n}) and F_{∗} ∈ L_{P_{wkc}(X)}^{1} satisfying

(a) F_{n′} →^{K} F_{∗},

(b’) F_{∗}(ω) ⊂ ∩_{m} clco ∪_{n′≥m}F_{n′}(ω) a.e.

Theorem 2.5 replaces the pointwise ball-compactness conditions (ii) by the w-compactness conditions (ii’) for integrals, which is of an essentially different, more macroscopic nature. But it imposes additional RNP-conditions on X and its dual. For singleton-valued multifunctions we immediately have the following corollary, which is also new:

Corollary 2.6. Suppose that both the Banach space X and its dual X^{∗} have the RNP. Let (f_{n}) be a sequence in L_{X}^{1} satisfying the following hypotheses:

i) sup_{n} ∫_{Ω} ∥f_{n}(ω)∥μ(dω) < +∞, ii’) clco{∫_{A} f_{n} dμ : n ∈ N} is w-compact for every A ∈ A. Then there exist a subsequence (f_{n′}) of (f_{n}) and f_{∗} ∈ L_{X}^{1} satisfying

a) f_{n′} →^{K} f_{∗},

b’) f_{∗}(ω) ∈ ∩_{m} clco{f_{n′}(ω) : n′ ≥ m} a.e.

Remark 2.7. Evidently, (b’) is a little weaker than (b); see p. 2.17 of [14] for examples, due to M. Valadier, which illustrate this, both in finite and infinite dimensions (see also Lemma 3.3 for a special situation where (b) and (b’) are equivalent).

Remark 2.8. For a separable Banach space X it is well-known, by a theorem of Stegall (see p. 195 of [29]), that X^{∗} has the RNP if and only if X^{∗} is separable for the dual norm.

Example 2.9. In the presence of (i) and the following condition: for every x^{∗} ∈ X^{∗}

(s(x^{∗} | F_{n}(·))) is uniformly integrable over Ω,

hypothesis (ii) implies the validity of (ii’). This can be easily proven by means of Young measures, for instance, since (i)–(ii) constitute a tightness condition for the corresponding relaxations of any sequence of selectors. Conversely, hypothesis (ii’) does not necessarily imply (ii), as the following examples show. For Ω := [0, 1] with Lebesgue measure and X := ℓ^{1} we define f_{n}(ω) := 1_{A_{n}}(ω)e_{n}. Here (A_{n}) is a sequence of independent subsets of [0, 1] with Lebesgue measure μ(A_{n}) → 0 and ∑_{n=1}^{∞} μ(A_{n}) = +∞ (e.g., see Example 4.14 of [11] for a concrete instance). By the second Borel-Cantelli theorem we have that for almost every ω the sequence (f_{n}(ω)) contains an infinite number of e′_{n}s, which implies that it cannot be compact for σ(ℓ^{1}, ℓ^{∞}). Therefore, (ii) does not hold. On the other hand, (ii’) certainly holds, because ∥ ∫_{A} f_{n} dμ∥ = μ(A∩A_{n}) → 0 for every A ∈ A. The previous example is not entirely convincing, because its dual space X^{∗} = ℓ^{∞} does not have the RNP. However, the following modification addresses this shortcoming. Let X := J, the space defined in Example 1.4.2 of [41], with canonical basis (e_{n}). Let (A_{n}) be as above, and define f_{n}(ω) := 1_{A_{n}}(ω) ∑_{m=1}^{n} e_{m}. Neither (∑_{m=1}^{n} e_{m}) nor any of its subsequences converges weakly [41], so the above argument can be repeated. This time, both X and X^{∗} have the RNP. ^{3} Remark 2.10. Notice that in the presence of (i), hypothesis (ii’) is equivalent to (ii′′) clco ∪_{n} ∫_{A} F_{n} dμ is w-ball-compact for every A ∈ A, which resembles hypothesis (ii) a little more.

3 We thank G. Godefroy, H. Klei and G. Pisier (Paris) for providing us with this information.

Remark 2.11. Hypothesis (ii′) is equivalent to the following two conditions: (ii′′′) the set clco ∪_{n} ∫_{Ω} F_{n} dμ is w-compact, (ii′′′′) there exists a sequence (f_{n}) in L_{X}^{1} such that f_{n} is a selector of F_{n} for each n and

such that C(A) := clco{∫_{A} f_{n} dμ : n ∈ N} is w-compact for every A ∈ A. It is evident that (ii′) implies (ii′′′)– (ii′′′′). For the converse, note that for any A in A the set ∫_{A} F_{n} dμ + ∫_{Ω\A} f_{n} dμ is contained in ∫_{Ω} F_{n} dμ. Hence, clco ∫_{A} F_{n} dμ is contained

in clco ∫_{Ω} F_{n} dμ − C(Ω\A); the latter set is w-compact by (ii′′′) –(ii′′′′), so (ii′) follows.

Remark 2.12. Unlike Theorem 2.1, Theorem 2.5 does not have a variant when the s-topology systematically replaces the w-topology (cf. Remark 2.3). This is due to the fact that in this case the Radon-Nikodym theorem for multimeasures (which will play a crucial role in the proof of Theorem 2.5) only provides the existence of a derivative multifunction with w-compact values; see Example 1 in [22].

3. Proofs of the main results

We shall denote by τ the Mackey topology on the dual space X^{∗}. Since X is supposed separable, the unit ball of X^{∗} has a countable, τ-dense subset D with D = −D; also, the set H spanned by all rational linear combinations of vectors in D is τ-dense in X^{∗} (III.32 of [19]). For any C ∈ P_{wkc}(X) one has

C = ∩_{x^{∗}∈H}{x ∈ X : ⟨x, x^{∗}⟩ ≤ s(x^{∗} | C)},  (3.1)

by an application of III.34 of [19]. On the space P_{wkc}(X) we also consider the H-scalar convergence topology. This is the topology of pointwise convergence of support functions on H, obtained by requiring (1.2) to hold only for all x^{∗} ∈ H; evidently the H-scalar convergence topology is metrizable. For scalar convergence of sequences, the following consequences of (1.2) are immediate but useful:

∥C_{0}∥ ≤ lim inf_{n} ∥C_{n}∥, clco Ls C_{n} ⊂ C_{0} ⊂ ∩_{p} clco ∪_{n≥p}C_{n},  (3.2)

where it can be remarked that the first inclusion becomes an identity in case ∪_{n}C_{n} is relatively w-compact (by an obvious modification of Proposition 3.3 of [26]). For future use it is also important to note that equality (3.1) continues to hold for any closed convex and merely bounded subset C, provided that X^{∗} is separable for the dual norm topology; in this case the separable dense subset H is the obvious one. The set of all Bochner-integrable X-valued functions on (Ω, A, μ) is denoted by L_{X}^{1}. For any multifunction F ∈ L_{P_{wkc}(X)}^{1} we denote

∥F ∥_{1} := ∫_{Ω} ∥F ∥dμ.

It is well-known (Theorem 2.2 of [38]) that for Effros measurable F : Ω → 2^{X}, having closed values, the following identity holds:

s(x^{∗} | ∫_{A} F dμ) = ∫_{A} s(x^{∗} | F)dμ, x^{∗} ∈ X^{∗}, A ∈ A,  (3.3)

provided that ∫_{Ω} F dμ is nonempty (as is always the case when F ∈ L_{P_{wkc}(X)}^{1}). The following celebrated theorem, due to Komlós [40], will play a key role in the proofs of Theorems 2.1 and 2.5. A fairly short proof can be found in [20].

Theorem 3.1. Suppose that (ϕ_{n}) is a sequence in L1_{R} such that

sup_{n} ∫_{Ω} |ϕ_{n}| dμ < +∞.

Then there exist ϕ_{∗} ∈ L1_{R} and a subsequence (ϕ_{n′}) of (ϕ_{n}) such that for every further subsequence (ϕ_{n′_{j}}) of (ϕ_{n′}) there is a null set N with

lim_{m→∞} \frac{1}{m} ∑_{j=1}^{m} ϕ_{n′_{j}}(ω) = ϕ_{∗}(ω) for all ω ∈ Ω\N.

The next three lemmas will be used to prove Theorem 2.1. The first one is a Blaschke type compactness criterion for scalar convergence; it is Lemma 5.1 of [35] and can be seen as a consequence of e.g. Theorem 3.1 of [21]. The second one is recalled from Proposition 3.10 of [33]. For the sake of completeness, the proofs of both lemmas are given in section 4.

Lemma 3.2.

(a) For every K in P_{wkc}(X), the subset K := {C ∈ P_{wkc}(X) : C ⊂ K} of P_{wkc}(X) is metrizable and compact for the scalar convergence topology.

(b) For every K in P_{skc}(X), the subset K := {C ∈ P_{skc}(X) : C ⊂ K} of P_{skc}(X) is compact for the Hausdorff metric h.

Lemma 3.3. Suppose (C_{n}) is a sequence of subsets of X such that there exists K ∈ P_{wkc}(X) for which

∪_{n}C_{n} ⊂ K.  (3.4)

Then

clco(w − Ls C_{n}) = ∩_{k} clco(∪_{n≥k}C_{n})

and

s(x^{∗} | clco(w − Ls C_{n})) = lim sup_{n→∞} s(x^{∗} | C_{n}) for every x^{∗} ∈ X^{∗}.

The third lemma is the well-known biting lemma, which goes back to V.F. Gaposhkin [31], and has been rediscovered by several authors (e.g., [13]).

Lemma 3.4. [biting lemma] Suppose (ϕ_{n}) is a sequence in L1_{R} such that

sup_{n} ∫_{Ω} |ϕ_{n}|dμ < +∞.

Then there exists a nonincreasing sequence (B_{p}) in A such that μ(∩_{p=1}^{∞}B_{p}) = 0 and for every p

(ϕ_{n}) is uniformly integrable over Ω\B_{p}.

Proof of Theorem 2.1. Part 1. ^{4} Using Komlós’ theorem (Theorem 3.1) and the diagonal method, we obtain the existence of a subsequence (F_{n′}) of (F_{n}) and of functions ϕ_{x^{∗}}, x^{∗} ∈ H, and ψ in L1_{R} such that

ϕ_{x^{∗}}(ω) = lim_{m→∞} \frac{1}{m} ∑_{n′=1}^{m} s(x^{∗} | F_{n′}(ω))  (3.5)

and

ψ(ω) = lim_{m→∞} \frac{1}{m} ∑_{n′=1}^{m} ∥F_{n′}(ω)∥  (3.6)

for every x^{∗} ∈ H and every ω outside some null set N_{1}. By the same result, this also holds for every further subsequence of (F_{n′}). (Note already that then the limit functions ϕ_{x^{∗}}, x^{∗} ∈ H, and ψ do not change, but that in general the null set N_{1} will have to be replaced by another null set.) We define the partial sums G_{m} : Ω → P_{wkc}(X) by

G_{m}(ω) := \frac{1}{m} ∑_{n′=1}^{m} F_{n′}(ω)  (3.7)

for all m ∈ N. Relationship (3.6) implies that for every ω ∉ N_{1}

∥G_{m}(ω)∥ ≤ ψ(ω) + 1 for m large enough (say m ≥ m_{ω}).  (3.8)

Part 2. We apply the previous part 1. Denote G(ω) := clco ∪_{n}F_{n}(ω); by hypothesis (ii) this set is w-ball-compact whenever ω is outside some null set N_{0}. Hence, for any ω ∉ N := N_{0} ∪ N_{1} and for every m ≥ m_{ω} the set G_{m}(ω) is contained in the w-compact set K(ω) := G(ω) ∩ {x ∈ X : ∥x∥ ≤ ψ(ω) + 1}. Thus, Lemma 3.2 gives the existence of a subsequence (G_{m_{i}}(ω)) of (G_{m}(ω)) (possibly depending upon ω), which converges scalarly to some limit set F_{∗}(ω) in P_{wkc}(X). But because of (3.1) and (3.5) it is easily seen that F_{∗}(ω) acts as the unique cluster point of (G_{m}(ω)), which therefore converges to it. Thus, we obtain that for all ω ∉ N

lim_{m} s(x^{∗} | G_{m}(ω)) = s(x^{∗} | F_{∗}(ω)) for all x^{∗} ∈ X^{∗}.  (3.9)

On N we set F_{∗}(ω) identically equal to some fixed singleton (say). Evidently, this implies the measurability of s(x^{∗} | F_{∗}(·)) for every x^{∗} ∈ X^{∗}. Effros measurability of F_{∗} then follows easily from

d(x, F_{∗}(ω)) = sup_{x^{∗}∈X^{∗},∥x^{∗}∥^{∗}≤1}[< x, x^{∗} > −s(x^{∗} | F_{∗}(ω))] = sup_{x^{∗}∈D}[< x, x^{∗} > −s(x^{∗} | F_{∗}(ω))],

4 This part of the proof will also serve in the proof of Theorem 2.5.

where the first identity is well-known and the second one follows by τ-denseness of D in the unit ball of X^{∗} and τ-continuity of the support function (by F_{∗}(ω) ∈ P_{wkc}(X)). Also, it follows from (3.9), by (3.2) and the classical Fatou lemma, that

∥F_{∗}∥_{1} ≤ ∫_{Ω} lim inf_{m} ∥G_{m}∥dμ ≤ lim inf_{m} ∥G_{m}∥_{1}.

Here the limes inferior on the right is finite by hypothesis (i); hence, we conclude that F_{∗} belongs to L_{P_{wkc}(X)}^{1}. By part 1 it follows that the above argument can be repeated

for the partial sums H_{m} := \frac{1}{m} ∑_{j=1}^{m} F_{n′_{j}} corresponding to any subsequence (F_{n′_{j}}) of (F_{n′}).

This gives the existence of a null set N′ of Ω and a multifunction F_{∗∗} ∈ L_{P_{wkc}(X)}^{1} such

that for all ω ∉ N′

lim_{m} s(x^{∗} | H_{m}(ω)) = s(x^{∗} | F_{∗∗}(ω)) for all x^{∗} ∈ X^{∗}.  (3.10)

From (3.5), (3.9) and (3.10) we conclude that for all ω ∉ N ∪ N′

s(x^{∗} | F_{∗}(ω)) = s(x^{∗} | F_{∗∗}(ω)) = ϕ_{x^{∗}}(ω) for all x^{∗} ∈ H.

Therefore, F_{∗∗} = F_{∗} a.e., so (a) has been proven. We comment that, apart from the use of Komlós’ theorem, the above technique is quite standard (e.g., see Lemma 5.1 of [35]). Part 3. Next, we prove (b). We begin by observing that the multifunction w-Ls F_{n′} is measurable by Theorem 4.4 of [33]. Suppose one did not have (b). Then there would exist x^{∗} ∈ D and a nonnull set A such that

s(x^{∗}|F_{∗}(ω)) > s(x^{∗}|w-Ls F_{n′}(ω))

for all ω ∈ A. For each n′ there exists a measurable (and integrable) selector of F_{n′} such that s(x^{∗}|F_{n′}(ω)) =< f_{n′}(ω), x^{∗} > a.e. Without loss of generality we may suppose that the nonnull set A is such that on A the three sequences (s(x^{∗}|F_{n′}(·))), (∥f_{n′}∥) and (< f_{n′}, x^{∗} >) are uniformly integrable (apply the biting lemma three times: after three sufficiently small bites there is still a nonnull set left). Then (f_{n′}), restricted to A, is a uniformly integrable subset of L_{X}^{1}(A) which is R_{w}-tight in the sense of [1]. Therefore, Theorem 6 of [1] implies that a subsequence (f_{n′′}) of (f_{n′}) converges weakly to some f_{∗} ∈ L_{X}^{1}(A). Further, by [1, Theorem 8] f_{∗}(ω) ∈ clco w − Ls f_{n′′}(ω) a.e. in A (see Appendix B). On the one hand, elementary properties of K-convergence [7] imply

lim_{n′′} ∫_{A} s(x^{∗}|F_{n′′})dμ = ∫_{A} s(x^{∗}|F_{∗})dμ.

On the other hand, the stated properties of f_{∗} imply

lim_{n′′} ∫_{A} s(x^{∗}|F_{n′}) = lim_{n′′} ∫_{A}⟨f_{n′′}, x^{∗}⟩ = ∫_{A}⟨f_{∗}, x^{∗}⟩ ≤ ∫_{A} s(x^{∗}|w-Ls F_{n′}) < ∫_{A} s(x^{∗}|F_{∗}).

This gives a contradiction.

We remark that the proof of part (b) can also be given by well-known results on Young measures. These results actually generalize the cited theorems from [1]; e.g., see Theorem 6.2 and Remark 6.5 in [9]. Here it should be observed (Remark 6.6 (a) of [9]) that L^{1}-boundedness and R_{w}-tightness as in [1] imply tightness in the sense of [3], [6]. We also remark that for the partial sums (G_{m}) appearing in the proof of Theorem 2.1 the identity F_{∗}(ω) = clco G_{m}(ω) holds a.e.; this follows from the remark after (3.2).

To prove Theorem 2.5, we need a version of the Radon-Nikodym theorem for multivalued measures taking values in P_{wkc}(X). Recall that a map M from A into P_{wkc}(X) is called additive if

M(A ∪ B) = M(A) + M(B),

whenever A, B ∈ A are disjoint sets. If in addition the function s(x^{∗} | M(·)) is a finite measure for every x^{∗} ∈ X^{∗}, then M is said to be a weak multimeasure (or multimeasure for short); this notion was introduced by Costé and Pallu de La Barrière in [24], [25]. Interestingly, by a result of Costé (p. III.4 of [23]) this definition of a multivalued measure with values in P_{wkc}(X) is equivalent to the following one: for every sequence (A_{n}) of pairwise disjoint sets in A one has

lim_{m→∞} h(M(∪_{n}A_{n}), ∑_{n=1}^{m} M(A_{n})) = 0,

where h stands for the Hausdorff distance on P_{wkc}(X). The multimeasure M is said to be μ-absolutely continuous if μ(A) = 0 implies M(A) = {0} for every A ∈ A. The total variation V_{M} of M is given by

V_{M}(Ω) := sup ∑_{i=1}^{p} ∥M(A_{i})∥,

the supremum being taken over all finite measurable partitions {A_{1}, A_{2}, …, A_{p}} of Ω. If V_{M}(Ω) is finite, the multimeasure M is said to be of bounded variation. The following result, due to Costé Théorème 3 of [22] or Théorème 8, p. III.31 of [23], is a multivalued analogue of the classical Radon-Nikodym theorem:

Theorem 3.5. Suppose that X and X^{∗} both have the RNP. Let M be a weak multimeasure of bounded variation with values in P_{wkc}(X). If M is μ-absolutely continuous, there exists a multifunction Γ ∈ L_{P_{wkc}(X)}^{1}, (a version of) the Radon-Nikodym derivative of M with respect to μ, such that

M(A) = ∫_{A} Γ dμ for every A ∈ A.

Actually, in [22], [23] the measure space (Ω, A, μ) is assumed to be complete. However, by reasoning as in Remark, p. 163 of [38] this condition is seen to be removable. Klei has given a very interesting generalization of the above theorem (Theorem 5.3 of [39]); his result also demonstrates the basic necessity of the RNP for both X and X^{∗}. In the absence of (ii), Lemma 3.3 can no longer be used, and an additional tool is required to obtain the pointwise inclusion in (b’):

Lemma 3.6. Let (C_{n}) be a sequence in P_{wkc}(X). Then

∩_{p} clco (∪_{m≥p} \frac{1}{m} ∑_{n=1}^{m} C_{n}) ⊂ ∩_{p} clco ∪_{n≥p}C_{n}.

Proof of Theorem 2.5. By Remark 2.8 we can take H to be a countable, strongly dense subset of X^{∗}, closed for taking rational linear combinations. Let (F_{n′}) be as obtained in part 1 of the proof of Theorem 2.1. By (3.6) the sets G_{m}(ω), m ∈ N, of (3.7) are of course (pointwise) uniformly bounded; hence (s(· | G_{m}(ω))) is equicontinuous on X^{∗} (recall that the strong topology is used on X^{∗} in the current proof). Hence, it follows from (3.5) that

ϕ_{x^{∗}}(ω) := lim_{m→∞} \frac{1}{m} ∑_{n′=1}^{m} s(x^{∗} | F_{n′}(ω)) exists ,  (3.11)

not only for x^{∗} ∈ H, but for all x^{∗} ∈ X^{∗}. As is clear from (3.8), ϕ_{·}(ω) is Lipschitzcontinuous on X^{∗} with Lipschitz-constant ψ(ω) + 1 for every ω ∉ N_{1}. By (3.11) the function ψ_{A}, defined on X^{∗} by ψ_{A}(x^{∗}) := ∫_{A} ϕ_{x^{∗}} dμ, is subadditive on X^{∗} and it satisfies

ψ_{A}(x^{∗}) ≤ s(x^{∗} | K(A)) for all x^{∗} ∈ X^{∗},  (3.12)

where K(A) := clco ∪_{n} ∫_{A} F_{n} dμ is w-compact by hypothesis (ii’). Hence, by classical properties of sublinear functions, ψ_{A} is τ-continuous on X^{∗}. Therefore, it is also w^{∗}-lower semicontinuous, which by Theorem II.16 of [19] implies the existence of a nonempty closed convex subset M(A) of X such that

ψ_{A}(·) = s(· | M(A)).  (3.13)

By (3.12) it follows that M(A) is contained in the w-compact set K(A), so M(A) itself is also w-compact. We now show that M : A → P_{wkc}(X) has the following properties: (P_{1}) M is additive, (P_{2}) M is absolutely continuous with respect to μ, (P_{3}) M has bounded variation. (P_{4}) s(x^{∗} | M(·)) is σ-additive for every x^{∗} ∈ X^{∗}, As for (P_{1}), note that this property follows from

s(x^{∗} | M(A ∪ B)) = ∫_{A} ϕ_{x^{∗}} dμ + ∫_{B} ϕ_{x^{∗}} dμ = s(x^{∗} | M(A) + M(B))

for every x^{∗} ∈ H by the definition of ϕ_{x^{∗}} and (3.13). So by (3.1) it follows that M(A∪B) = M(A) + M(B). Also, (P_{2}) follows easily, for if A ∈ A satisfies μ(A) = 0, then ψ_{A}(x^{∗}) = 0 for all x^{∗} ∈ H. To prove (P_{3}) we observe that for all x^{∗} ∈ D

s(x^{∗} | M(A)) = ∫_{A} ϕ_{x^{∗}} dμ ≤ ∫_{A} ψ dμ  (3.14)

by (3.5)–(3.6). Therefore, as in the case of scalar-valued measures (p. 101 of [29]),

∥M(A)∥ = sup_{x^{∗}∈D} s(x^{∗} | M(A)) ≤ ∫_{A} ψ dμ.

Of course, this implies V_{M}(Ω) ≤ ∫_{Ω} ψ dμ < +∞, which proves (P_{3}). By sublinearity of support functions, (3.14) gives

|s(x^{∗} | M(A))| ≤ ∫_{A} ψ dμ

for all x^{∗} ∈ D (by D = −D). Hence, the total variation V_{x^{∗}} of the additive set function s(x^{∗} | M(·)) satisfies V_{x^{∗}}(A) ≤ ∫_{A} ψ dμ for all A ∈ A. Thus, (P_{4}) follows. In view of (P_{1})–(P_{4}), we may now invoke Theorem 3.5, the multivalued Radon-Nikodym theorem. This yields the existence of a multifunction F_{∗} ∈ L_{P_{wkc}(X)}^{1} which satisfies

M(A) = ∫_{A} F_{∗} dμ for all A ∈ A.  (3.15)

Moreover, F_{∗} is unique up to null sets. The proof of (a) is easily finished by substituting (3.3) in (3.14) and using the Lipschitz-continuity of ϕ_{·}(ω) on X^{∗} for ω ∉ N_{1}. To prove (b’), note first that by Lemma 3.6 it is enough to prove that for every p

F_{∗}(ω) ⊂ L_{p}(ω) := clco ∪_{m≥p}G_{m}(ω) a.e.  (3.16)

We observe that for every p ∈ N and for a.e. ω ∈ Ω the set L_{p}(ω) is bounded. Indeed, one has ∥G_{m}(·)∥ ≤ \frac{1}{m} ∑_{n′=1}^{m} ∥F_{n′}(·)∥, which, by (3.6), implies that there is a null set N_{1} such that for every ω ∉ N_{1} and every p ∈ N ∥L_{p}(ω)∥ ≤ sup_{m} \frac{1}{m} ∑_{n′=1}^{m} ∥F_{n′}(ω)∥ < +∞. On the other hand, (3.15) gives ∫_{A} ϕ_{x^{∗}} dμ = ∫_{A} s(x^{∗} | F_{∗})dμ for every A ∈ A and x^{∗} ∈ H. Hence, for every ω outside a suitable null set N_{2}, ϕ_{x^{∗}}(ω) = s(x^{∗} | F_{∗}(ω)) for all x^{∗} ∈ H. Fix ω ∉ N_{1} ∪ N_{2}; then (3.5) gives s(x^{∗} | F_{∗}(ω)) = lim_{m} s(x^{∗} | G_{m}(ω)), which entails s(x^{∗} | F_{∗}(ω)) ≤ s(x^{∗} | L_{p}(ω)) for every p and x^{∗} ∈ H. By boundedness of the L_{p}(ω) this gives F_{∗}(ω) ⊂ L_{p}(ω) for all p, in view of (3.1) and the separability of X^{∗}. This proves (3.16), so the proof of Theorem 2.5 is finished.

4. Proofs of auxiliary results

Proof of Lemma 3.2 (a). (See also the proof of Lemma 5.1 of [35].) It is obvious that K is closed for scalar convergence; so let us show that it is also relatively compact. Just as was done in the proof of Theorem 2.5, one shows that the collection {s(· | C) : C ∈ K} is τ-equicontinuous (see also (4.1) below). Consequently, on K, the scalar convergence topology coincides with the metrizable H-scalar convergence topology. Thus, it remains to show that an arbitrary sequence (C_{n}) in K has a convergent subsequence. By definition of K one has for every n ∈ N and every x^{∗} ∈ X^{∗} that

−s(−x^{∗} | K) ≤ s(x^{∗} | C_{n}) ≤ s(x^{∗} | K).  (4.1)

By the diagonal method this gives the existence of a subsequence (C_{n′}) of (C_{n}) such that the limit π(x^{∗}) := lim_{n′→∞} s(x^{∗} | C_{n′}) exists for every x^{∗} ∈ H. From (4.1) we deduce that the sublinear function π satisfies

|π(x^{∗})| ≤ max[s(x^{∗} | K), s(−x^{∗} | K)]  (4.2)

for every x^{∗} ∈ H. Hence, π is τ-continuous on H; therefore, it admits a τ-continuous extension to X^{∗}. This extension, denoted by π̃, still satisfies (4.2), but now for all x^{∗} ∈ X^{∗}. As in the proof of Theorem 2.5, by invoking II.16 of [19] we can see that this yields the existence of C_{∗} ∈ P_{wkc}(X) such that π̃(x^{∗}) = s(x^{∗} | C_{∗}) for every x^{∗} ∈ X^{∗}. In view of (4.1), C_{∗} is a subset of K (observe that π̃(x^{∗}) ≤ s(x^{∗} | K) for all x^{∗} ∈ X^{∗}). It is now elementary to conclude that the subsequence (C_{n′}) converges scalarly to C_{∗}. b. Repeating similar arguments as in the proof of part a, but replacing the topology τ on X^{∗} by the topology τ_{c} of uniform convergence on s-compact subsets of X, we can prove the existence of a subsequence (C_{n′}) such that π(x^{∗}) := lim_{n′→∞} s(x^{∗} | C_{n′}) for every x^{∗} ∈ H. Likewise, we obtain existence and τ_{c}-continuity of π̃, the extension of π to X^{∗}, and this implies the existence of an s-compact subset C_{∗} of X such that π̃ = s(· | C_{∗}). By Ascoli’s theorem, the τ_{c}-equicontinuity of (s(· | C_{n′})) causes the D-scalar convergence of (C_{n′}) to C_{∗} to coincide with uniform convergence of (s(· | C_{n′})) to s(· | C_{∗}) on any τ_{c}-compact subset of X^{∗}. By the Banach-Dieudonné theorem (Théorème 1, p. IV.24 of [12]) the τ_{c}- and w^{∗}-topologies coincide on the unit ball of X^{∗}. So it follows that (s(· | C_{n′})) converges uniformly to s(· | C_{∗}) on the dual unit ball; thus, (C_{n′}) converges in the Hausdorff metric to C_{∗}. An alternative proof can be given, based upon the remark following Theorem II.4 in [19].

Proof of Lemma 3.3. Let C be the right-hand side of the first identity that we have to prove. From the definition of w-Ls C_{n} it is clear that C contains the left-hand side of that same identity. It remains to prove the opposite inclusion. By definition of C it easily follows that for any x^{∗} ∈ X^{∗}

s(x^{∗} | C) ≤ lim sup_{n→∞} s(x^{∗} | C_{n}).  (4.3)

Further, it is possible to find a subsequence (C_{n_{j}}) of (C_{n}) such that lim_{j} s(x^{∗} | C_{n_{j}}) = lim sup_{n} s(x^{∗} | C_{n}). Now, for any j, there exists x_{j} ∈ C_{n_{j}} with < x_{j}, x^{∗} >≥ s(x^{∗} | C_{n_{j}}) − 1/j; then clearly lim_{j} < x_{j}, x^{∗} >= lim sup_{n} s(x^{∗} | C_{n}). By condition (3.4), the sequence (x_{j}), contained in K, has a subsequence which w-converges to some x̄ ∈ K. Clearly, such x̄ belongs to w-Ls C_{n}, so

lim sup_{n→∞} s(x^{∗} | C_{n}) =< x̄, x^{∗} >≤ s(x^{∗} | clco(w − Ls C_{n})).

Hence, the desired inclusion is obtained, by the arbitrariness of x^{∗}. Finally, the second identity to be proven is an obvious consequence of (4.3) and our last inequality.

Lemma 4.1. Let (C_{n}) consist of w-compact and (D_{n}) of w-closed subsets of X. Assume that (C_{n}) and (D_{n}) are both nonincreasing. Then ∩_{n}(C_{n} + D_{n}) = ∩_{n}C_{n} + ∩_{n}D_{n}.

Proof. Evidently, the right-hand side is contained in the left side. The proof of the converse inclusion is straightforward: let x be an element of the intersection on the left, i.e., for every n there exist c_{n} ∈ C_{n} and d_{n} ∈ D_{n} with x = c_{n}+d_{n}. Now by w-compactness of C_{1} and the Eberlein-S̆mulian theorem, a subsequence (c_{n_{j}}) of (c_{n}) will converge to some c̄ ∈ X. Of course, then (d_{n_{j}}) converges to x−c̄. It follows immediately from the monotone inclusions that c̄ ∈ ∩_{n}C_{n} and x − c̄ ∈ ∩_{n}D_{n}.

Proof of Lemma 3.6. For every p ∈ N

clco ∪_{m≥p}(\frac{1}{m}C_{1} + \frac{1}{m} ∑_{n=2}^{m} C_{n}) ⊂ clco ∪_{m≥p} \frac{1}{m}C_{1} + clco ∪_{m≥p} \frac{1}{m} ∑_{n=2}^{m} C_{n}.

The right-hand side is closed, because clco ∪_{m≥p} \frac{1}{m}C_{1} is w-compact. Upon taking the intersection over all p one thus obtains

∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=1}^{m} C_{n} ⊂ ∩_{p}(clco ∪_{m≥p} \frac{1}{m}C_{1} + clco ∪_{m≥p} \frac{1}{m} ∑_{n=2}^{m} C_{n}).

By Lemma 4.1 this gives

∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=1}^{m} C_{n} ⊂ ∩_{p} clco ∪_{m≥p} \frac{1}{m}C_{1} + ∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=2}^{m} C_{n}.  (4.4)

It is easy to check that clco ∪_{m≥p} \frac{1}{m}C_{1} is actually identical to co (\{0\} ∪ \frac{1}{p}C_{1}) (note that

\{0\} ∪ \frac{1}{p}C_{1}, being w-compact, has a convex hull that is also w-compact). Therefore, it

follows immediately that ∩_{p} clco ∪_{m≥p} \frac{1}{m}C_{1} is equal to \{0\}, in view of the boundedness of C_{1}. Substitution in (4.4) gives

∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=1}^{m} C_{n} ⊂ ∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=2}^{m} C_{n} ⊂ ∩_{p} clco(∪_{m≥p} clco ∪_{n=2}^{m}C_{n}),

which leads to

∩_{p} clco ∪_{m≥p} \frac{1}{m} ∑_{n=1}^{m} C_{n} ⊂ ∩_{p} clco ∪_{n≥2}C_{n} ⊂ cl co ∪_{n≥2} C_{n}.

The proof is now easily completed by induction.

5. Applications: lower closure results

This section gives immediate lower closure-type applications of Theorems 2.1 and 2.5.

Theorem 5.1. Let (F_{n}) be a sequence in L_{P_{wkc}(X)}^{1} satisfying the following hypotheses:

(i) sup_{n} ∫_{Ω} ∥F_{n}(ω)∥μ(dω) < +∞, (ii) clco ∪_{n}F_{n}(ω) is w-ball-compact a.e. Also, let (v_{n}) be a sequence of scalarly measurable functions v_{n} : Ω → X^{∗} such that

(iii) v_{n}(ω) →^{τ} v_{0}(ω) a.e.,

where τ again denotes the Mackey topology on X^{∗}. Then the subsequence (F_{n′}) of (F_{n}) and F_{∗} ∈ L_{P_{wkc}(X)}^{1} of Theorem 2.1 satisfy, next to (a)–(b),

c) lim inf_{n′} ∫_{Ω} s(v_{n′}(·) | F_{n′}(·))dμ ≥ ∫_{Ω} s(v_{0}(·) | F_{∗}(·))dμ,

provided that min(0, s(v_{n′}(·) | F_{n′}(·))) is uniformly integrable.

Theorem 5.2. Suppose that both the Banach space X and its dual X^{∗} (equipped with the dual norm) have the RNP. Let (F_{n}) be a sequence in L_{P_{wkc}(X)}^{1} satisfying the following hypotheses:

(i) sup_{n} ∫_{Ω} ∥F_{n}(ω)∥μ(dω) < +∞, (ii’) clco ∪_{n} ∫_{A} F_{n} dμ is w-compact for every A ∈ A. Also, let (v_{n}) be a sequence of strongly measurable functions v_{n} : Ω → X^{∗} such that (iii’)∥v_{n}(ω) − v_{0}(ω)∥^{∗} → 0 a.e.

Then the subsequence (F_{n′}) of (F_{n}) and F_{∗} ∈ L_{P_{wkc}(X)}^{1} of Theorem 2.5 satisfy, next to

(a)-(b′),

(c) lim inf_{n′} ∫_{Ω} s(v_{n′}(·) | F_{n′}(·))dμ ≥ ∫_{Ω} s(v_{0}(·) | F_{∗}(·))dμ,

provided that min(0, s(v_{n′}(·) | F_{n′}(·))) is uniformly integrable.

Proof of Theorem 5.1. Observe that in the proof of Theorem 2.1 the sequence (s(· | G_{m}(ω)) was a.e. equicontinuous for the τ-topology thanks to the intervention of the compact sets K(ω). Now there exists a subsequence (F_{n′_{j}}) of (F_{n′}) such that the left hand

side of (c) is equal to lim_{n′_{j}} ∫_{Ω} s(v_{n′_{j}}(·) | F_{n′_{j}}(·))dμ, and without loss of generality we may

suppose that this limit is finite. By the equicontinuity observed above and the definition of K-convergence as in (a), we have

lim_{m} s(v_{m}(ω) | G_{m}(ω)) = s(v_{0}(ω) | F_{∗}(ω)) a.e.

So by the Fatou-Vitali lemma and the uniform integrability hypothesis for the negative parts (which obviously transfers to the negative parts of the s(v_{m} | G_{m})), we get

lim_{n′_{j}} ∫_{Ω} s(v_{n′_{j}} | F_{n′_{j}})dμ = lim_{m} ∫_{Ω} s(v_{m} | G_{m})dμ ≥ ∫_{Ω} s(v_{0} | F_{∗}),

which amounts to (c).

The proof of Theorem 5.2 is completely similar and will be omitted (by pointwise uniform boundedness of the G_{m}(ω) and the strength of the dual norm topology we again have equicontinuity).

6. Comparisons with other work

In this section we compare the results of this paper with those in the literature. Most of the literature can only be compared with our Theorems 5.1 and 5.2, and throughout such comparisons it is instructive to keep in mind that those lower closure type theorems are of a less general nature than our principal Komlós-type results of section 2.

1. Theorem 2.1 subsumes the two Komlós-type Theorems A and B of [5]. Theorem 2.5 generalizes Theorem A of [5] (which has X separable and reflexive). It also implies the Komlós-type Théorème 3.2 of [15] (its condition (2), which amounts to our (ii), is not needed – see also the remarks on p. 3.9). Two similar extensions of Komlós’ theorem were also given in section 3 of [7]. However, Theorem 3.2 of [7] is only correct under an additional separability condition for the dual Banach space.^{5}

2. Théorème 2.8 of [16] follows from Theorem 5.2, if we combine it with the biting Lemma 3.4. The remark following Théorème 2.7 of [16] (but not that theorem itself) follows from Theorem 5.2. Théorème 2.9 in [16] follows from the singleton-valued version of Theorem 7.1 by setting h(ω, x) := ϕ(ω, x) + ∥x∥, with ϕ as in [16], and by invoking Lemma 3.4 again.

3. Theorem 5.1 generalizes Théorème 4.2 of [18]. A similar comment applies to Théorème 3.4 of [16]. Here it is important to note that the RNP-conditions in [18] cause the sequential and topological Kuratowski limes superior to coincide Corollary 3.9(i) of [33]. Incidentally, by the same comment we also conclude that Theorem 2.1 immediately implies an improvement of Théorème 3.5 of [15] Theorem 5.2 applies as well to the context of Théorème 4.2 of [18], but it only generalizes partially: the first part of Théorème 4.2 follows directly, but this time the second part seems a little stronger than our (b′).

4. Theorem 5.2 improves upon Théorème 3.4 of [16], which takes, as in the case of Théorème 4.2 of [18] discussed above, the sets in (ii) to be locally compact, not containing any lines and which supposes the v_{n} to take their values in the unit ball of X^{∗}. Here again the comment involving Corollary 3.9 of [33] should be kept in mind.

5. Théorème 3.5 of [16] presents us with an interesting open problem. In Remark 6.6 (a) of [9] it has been shown that the R-tightness notion of [1], also used in [16] and related papers, is for all practical purposes – it is invariably accompanied by uniform L^{1}-boundedness – a tightness condition in the sense of [3]. Therefore, Théorème 3.5 of [16] follows from well-known lower semicontinuity results for Young measures. However, we cannot use the function h, constructed in Remark 6.6 (a) of [9], in Theorem 7.1 (applied to singletonvalued multifunctions), because it lacks convexity. We suspect that the solution to this problem lies in adapting ideas from [44].

6. Of course, several other, more traditional Fatou-type results can be found in the literature, for ordinary functions as well as multifunctions. Recently, the present authors gave comprehensive results of this type, including cases where the F_{n} have unbounded values [10]. In such Fatou type lemmas the prime interest lies in obtaining inclusions for the integrals of the multifunctions F_{n}. Roughly, such inclusions are of the type

5 The first author is indebted to S.I. Suslov (Novosibirsk), M. Talagrand (Paris), and S. Diaz (Sevilla),

who independently brought this omission to his attention.

w − Ls _{n} ∫_{Ω} F_{n} dμ ⊂ cl ∫_{Ω} F_{0} dμ,

where F_{0}(ω) := w-Ls F_{n}(ω). While this gives a greater precision per point (because pointwise F_{0} is a subset of the “upper bound” for the multifunction F_{∗} – see Theorems 2.1 (b) and 2.5 (b’)), precision about ∫_{Ω} F_{0} dμ is reduced by the presence of the additional closure operator. Objects like our F_{∗} sometimes also figure in the statements of such approximate Fatou lemmas (e.g., see Theorems 2.1 and 2.2 of [4]), and as a rule they can at least be found in the proofs of such results, although they do not involve K-convergence explicitly.

7. The above comments apply to approximate Fatou lemmas corresponding to situations similar to the one addressed in Theorem 2.1. Whether a true approximate Fatou lemma is also associated to Theorem 2.5 remains to be studied. As yet, we know of no such result, but Theorem 2.5 could certainly be considered as a first step in this direction. The novel method of proof employed here, which combines K-convergence methods with the Radon-Nikodym theorem for multivalued measures, could well turn out to be significant in this regard.

7. An Extension of Theorem 2.1 (a)

As we mentioned before, Theorem 2.1 (a) (and of course Corollary 2.2 (a)) can also be derived from the abstract Komlós-type theorem in [7]. We now show this by actually deriving the following extension of Theorem 2.1 from [7]. It can be seen as a multivalued version of Theorem 2.1, Remark 2.3 of [8], which itself implies Corollary 2.2. Its formulation uses outer integration over Ω (indicated by ∫_{Ω}^{∗}), so as to avoid unnecessary measurability considerations.

Theorem 7.1. Let (F_{n}) be a sequence in L_{P_{wkc}(X)}^{1} satisfying the following hypothesis: there exists a function h : Ω × P_{wkc}(X) → [0, +∞] such that

(i’) sup_{n} ∫_{Ω}^{∗} h(ω, F_{n}(ω))μ(dω) < +∞, (i”) h(ω, ·) is convex and inf-compact ^{6} on P_{wkc}(X) for almost every ω, (i”’) sup_{n} ∫_{Ω} |s(x^{∗} | F_{n}(ω))|μ(dω) < +∞ for every x^{∗} ∈ X^{∗}.

Then there exist a subsequence (F_{n′}) of (F_{n}) and F_{∗} ∈ L_{P_{wkc}(X)}^{1} satisfying

(a) F_{n′} →^{K} F_{∗},

(b) F_{∗}(ω) ⊂ clco w − Ls F_{n′}(ω) a.e.

This result implies Theorem 2.1, because under the hypotheses (i) and (ii) the function h, given by

h(ω, C) := \{ \begin{matrix}∥C∥ & if C ⊂ clco ∪_{n}F_{n}(ω), \\ +∞ & otherwise, \end{matrix}

meets the conditions of Theorem 7.1 by the definition of ball-compactness and by Lemma 3.2.

6 I.e., for every β ∈ R the set {C ∈ Pwkc(X) : h(ω, C) ≤ β} is compact.

Proof of Theorem 7.1. Part (a) of this result follows directly from Theorem 2.1 of [7] by setting its collection (a_{j}) equal to the set of all C ↦ s(x^{∗} | C), x^{∗} ∈ D. Observe that this family satisfies the conditions of that result, since the s(x^{∗} | ·) are affine and continuous on P_{wkc}(X) and also separate its points. Part (b) now follows as in the proof of Theorem 2.1 (b).

8. Addendum on sequential limits

Let X be a Hausdorff topological space (e.g., a Banach space with the weak topology). The sequential closure of a subset Y of X is defined by

seq cl Y := {x ∈ X : x = lim_{n} y_{n}, y_{n} ∈ Y }.

Let (x_{n}) be a given sequence in X. In part 3 of the proof of Theorem 2.1 we need the following reformulation of the sequential closure in terms of a concept used in [1]:

Ls (x_{n}) = ∩_{k=1}^{∞}seq cl Y_{k},  (8.1)

where Y_{k} := {x_{n} : n ≥ k}. We prove this identity as follows: For any k ∈ N, we let (x_{n}^{k}) be the sequence given by x_{n}^{k} := x_{n+k}, n ∈ N. Then it is not difficult to check that

seq cl Y_{k} = Y_{k} ∪ Ls _{n}(x_{n}^{k}) = Y_{k} ∪ Ls (x_{n})

for every k ∈ N. Consequently, we have

∩_{k}seq cl Y_{k} = (∩_{k}Y_{k}) ∪ Ls (x_{n}).

The proof of (8.1) is finished by noting that ∩_{k}Y_{k} is actually the repetition set of (x_{n}), defined to consist of all x ∈ X for which x = x_{n} for infinitely many indices n. This repetition set already belongs to the left side of (8.1).

Acknowledgment. We are grateful to CEREMADE and the Department of Mathematics, University of Utrecht, for providing the necessary financial support for our cooperative efforts. We thank the two referees for twice calling our attention to a serious gap in the proof of Theorem 2.1(b). We also thank one of the referees for several pertinent remarks, which led us to add sections 5 and 6, and the other referee for removing, with painstaking precision, some remaining flaws from the final revision.

References

[1] A. Amrani, C. Castaing and M. Valadier: Méthodes de troncature appliquées à des problèmes de convergence faible ou forte dans L^{1}. Arch. Rational Mech. Anal. 117 (1992) 167–191.

[2] Z. Artstein, J.C. Hansen: Convexification in limit laws of random sets in Banach spaces. Ann. Prob. 13 (1985) 307–309.

[3] E.J. Balder: A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570–598.

[4] E.J. Balder: Fatou’s lemma in infinite dimensions. J. Math. Anal. Appl. 136 (1988) 450–465.

[5] E.J. Balder: Infinite-dimensional extension of a theorem of Komlos. Probab. Th. Rel. Fields 81 (1989) 185–188.

[6] E.J. Balder: Unusual applications of a.e. convergence. in: Almost Everywhere Convergence (G.A. Edgar, L. Sucheston, eds.) Academic Press, New York (1989) 31–53.

[7] E.J. Balder: New sequential compactness results for spaces of scalarly integrable functions. J. Math. Anal. Appl. 151 (1990) 1–16.

[8] E.J. Balder: On uniformly bounded sequences in Orlicz spaces. Bull. Austral. Math. Soc. 41 (1990) 495–502.

[9] E.J. Balder: Lectures on Young Measures. Cahiers du Centre de Recherche de Mathématiques de la Décision No. 9517, (1995) CEREMADE, Université Paris IX-Dauphine.

[10] E.J. Balder, C. Hess: Fatou’s lemma for multifunctions with unbounded values. Math. Oper. Res. 20 (1995) 175–188.

[11] P. Billingsley: Probability and Measure. Wiley, New York, 1986.

[12] N. Bourbaki: Espaces Vectoriels Topologiques, Chapitres 1 à 5. Masson, Paris, 1981.

[13] J. Brooks, R.V. Chacon: Continuity and compactness of measures. Adv. in Math. 37 (1980) 16–26.

[14] C. Castaing: Quelques résultats de convergence des suites adaptées. Travaux Sém. Anal. Convexe 17 (1987) 2.1–2.24.

[15] C. Castaing: Sur la décomposition de Slaby, applications aux problèmes de convergence en probabilités, economie mathématiques. Théorie du contrôle. Minimisation. Travaux Sém. Anal. Convexe 19 (1989) 3.1–3.35

[16] C. Castaing: Méthode de compacité et de décomposition, applications: minimisation, convergence des martingales, lemme de Fatou multivoque. Ann. Mat. Pura Appl. 164 (1993) 51–75.

[17] C. Castaing, P. Clauzure: Compacité faible dans L_{1}(E) et dans l’espace des multifonctions intégrablement bornées, et minimisation, Ann. Mat. Pura Appl. IV, 140 (1985) 345–364.

[18] C. Castaing, P. Clauzure: Lemme de Fatou multivoque. Atti Sem. Mat. Fis. Univ. Modena 39 (1991) 303–320.

[19] C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.

[20] S.D. Chatterji: Les martingales et leurs applications analytiques. in: Ecole d’Eté de Probabilités: Processus Stochastiques. (J.L. Bretagnolle et al., eds.) Lecture Notes in Math. 307 (1973) Springer-Verlag, Berlin, 27–164.

[21] J.P.R. Christensen: Topology and Borel Structure. North-Holland, Amsterdam, 1974.

[22] A. Costé: La propriété de Radon-Nikodym en intégration multivoque. C.R. Acad. Sci. Paris 280 (1975) 1515–1518.

[23] A. Costé: Contribution à la théorie de l’intégration multivoque. Thèse d’Etat, Université Pierre et Marie Curie, Paris, 1977.

[24] A. Costé, R. Pallu de La Barrière: Un théorème de Radon-Nikodym pour les multimesures à valeurs fermées localement compactes sans droite. C.R. Acad. Sci. Paris 280 (1975) 255–258.

[25] A. Costé, R. Pallu de La Barrière: Radon-Nikodym theorems for set-valued measures whose values are convex and closed. Roczniki PTM Prace Mat. 20 (1978) 283–309.

[26] F. De Blasi, J. Myjak: Weak convergence of convex sets in Banach spaces. Arch. Math. 47 (1986) 448–456.

[27] C. Dellacherie, P.-A. Meyer: Probabilities and Potential, North-Holland, Amsterdam, 1978.

[28] J. Diestel, W.M. Ruess and W. Schachermayer: Weak compactness in L^{1}(μ, X). Proc. Amer. Math. Soc. 118 (1993) 447–453.

[29] J. Diestel, J.J. Uhl: Vector Measures. AMS Mathematical Surveys 15, Providence, 1977.

[30] B. Fuchssteiner, W. Lusky: Convex Cones. Math. Studies 56, North-Holland, Amsterdam, 1981.

[31] V.F. Gaposhkin: Convergence and limit theorems for sequences of random variables. Theory Probab. Appl. 17 No. 3 (1972) 379–400.

[32] C. Hess: Lemme de Fatou et théorème de la convergence dominée pour des ensembles aléatoires non bornés et des intégrandes. Travaux Sém. Anal. Convexe 16 (1986) 8.1–8.56.

[33] C. Hess: Measurability and integrability of the weak upper limit of a sequence of multifunctions. Travaux Sém. Anal. Convexe 10 10.1–10.38 and J. Math. Anal. Appl. 153 (1989) 226–249.

[34] C. Hess: Convergence of conditional expectations for unbounded random sets, integrands, and integral functionals. Math. Oper. Res. 16 (1991) 627–649.

[35] C. Hess: On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence. J. Multivariate Anal. 39 (1991) 175–201.

[36] C. Hess: Multivalued strong laws of large numbers in the slice topology. Application to integrands. Set Valued Anal. 2 (1994) 183–205.

[37] F. Hiai: Strong laws of large numbers for multivalued random variables, in: Mulitfunctions and Integrands (G. Salinetti, ed.) Lecture Notes in Math. 1091, Springer-Verlag, Berlin, 1984.

[38] F. Hiai, H. Umegaki: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7 (1977) 149–182.

[39] H.-A. Klei: A compactness criterion in L^{1}(E) and Radon-Nikodym theorems for multifunctions. Bull. Sc. Math. 112 (1988) 305–324.

[40] J. Komlós: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967) 217–229.

[41] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I], Springer-Verlag, Berlin, 1977.

[42] J.J. Moreau: Fonctionnelles Convexes. Sé]minaire sur les équations aux dérivées partielles, Collè]ge de France, Paris, 1966.

[43] R.T. Rockafellar: Level sets and continuity of conjugate functions. Trans. Amer. Math. Soc.] 123 (1966) 46–63.

[44] M. Saadoune: Une nouvelle extension en dimension infinie du Théorème de Komlós. Applications: compacité faible dans L_{X}^{1}, convergence en mesure. Preprint, Université Ibnou Zohr, Agadir, Marocco, 1995.