Weak Compactness in L 1 (μ, X)

On the convexity and compactness of the integral of a Banach space valued correspondence

“…Let φ : T → 2 X be an integrably bounded correspondence with weakly compact values, and let 5 S 1 φ = {f ∈ L 1 (μ, X) : f is a selection of φ}. Since φ has weakly compact values and is integrably bounded, it follows from (Diestel et al (1993), Corollary 2.6) that S 1 φ is weakly relatively compact in L 1 (μ, X). Hence T φ dμ is weakly relatively compact in X (since the operator f → T f dμ from L 1 (μ, X) to X is continuous for the weak topologies of these spaces).…”

Young Measures on Topological Spaces

“…as p → ∞. By theÜlger-Diestel-Ruess-Schachermayer theorem [Ülg91,DRS93], we conclude that ( 1l { hn <n} h n ) is σ(L 1 , L ∞ ) relatively compact.…”

“…Comments Propositions 6.4.5-6.4.6 are the analogs for Young measures of thë Ulger-Diestel-Ruess-Schachermayer characterization of weak compactness in L 1 X where X is a Banach space [Ülg91,DRS93]. Namely these authors proved that a bounded uniformly integrable subset H of L 1 X is relatively compact iff ( * ) given any sequence (u n ) n in H, there are v n ∈ co{u m ; m ≥ n} such that the sequence (v n (ω)) n is weakly convergent in X for almost all ω ∈ Ω.…”

“…Thirdly, some sequences are weakly Cauchy and the foregoing techniques give a candidate to be the limit. A condition in the line ofÜlger [Ülg91,DRS93] is often used -as in Propositions 6.4.4-6.4.5 where it takes the form "there exists a subsequence (ν n ) n satisfying ν n ∈ co{θ m ; m ≥ n} for every n, which is stably convergent".…”

An interim core for normal form games and exchange economies with incomplete information

“…The existence of the sup-bound of g j is ensured because sup a |u i (a, ·)| is integrably bounded. Following Diestel et al [1993], there is a sequence h n ∈ co{y k : k ≥ n} such that h n converges a.e. to y, and hence L(h n ) also converges a.e.…”