By S. Kusuoka, A. Yamazaki

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**Extra info for Advances in Mathematical Economics Volume 11 (Advances in Mathematical Economics)**

**Example text**

4, but we consider the sequence (u n ) defined this time by u n (ω) = d(0, X n (ω)) + 1(Anq )c (ω) | f q (ω)| and we appeal to condition (iii)’ instead of condition (iii). ω∈ 30 C. Castaing et al. 8. Let C = K(E w ) and (X n ) be a S+ (C)ε -tight sequence in M(2 E ) such that sup d(0, X n ) dµ < +∞. n≥1 Then w − ls X n admits at least one integrable selection. Proof. 7 is satisfied, whereas (iii)’ follows from the L 1 (µ)-boundedness hypothesis. The existence results of the beginning of this section allow for deriving new versions of the Fatou Lemma in infinite dimension, alias Fatou’s Lemma for Mathematical Economics.

The values of multifunction q ∩ q B are weakly compact and the following inclusions hold on for all n ≥ 1 Xn ∩ q ∩ qB ⊆ q ∩ qB n, q ≥ 1. 3 of [11], which entails the measurability of multifunction Z q and, in turn Dq ∈ F. 1) also imply Dq = lim sup dom(X n ∩ q n→+∞ ∩ q B). s. s. q≥1 In particular, this shows that the Dq s are nonempty for q large enough. Without loss of generality, we can assume that this holds for all q ≥ 1. Step 2. 2) 26 C. Castaing et al. 4) Indeed, suppose ω ∈ Fq and x ∈ w−ls X nq .

Define e : A → R++ by e(a) = w for every a ∈ A. Core convergence in economies with bads 55 L defines an atomless economy Letting χ = ×e : A → (Pco ∩ Plns ) × R++ ((A, A , ν), χ ). This economy, therefore, consists of a single type. For each positive integer n, let An = {0, 1, . . , n}, A n be the power set of An , and ν n be the uniform probability measure on An .