By S. Kusuoka, A. Yamazaki
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Extra resources for Advances in Mathematical Economics Volume 11 (Advances in Mathematical Economics)
3 is measurable. It is useful to note that statement (b) implies statement (a) and, as shown by the previous result, that the converse implication holds when X has w∗ -closed ∗ values. Further, for any subfamily C of 2 E , we denote by M(C) the set of all C-valued measurable multifunctions. The following two theorems provide measurability properties for the w∗ -sequential upper limit of a sequence of multifunctions. In the first one, the multifunctions are assumed to be contained in a fixed w ∗ -compact valued multifunction.
10) Tightness conditions and integrability 31 Proof. Consider the sequence (rn ) of condition (iii). For each n ≥ 1, there exists a sequence (αin )i≥n of reals, such that rn = αin | f i | αin = 1 i≥n αin ≥ 0 i≥n where αin > 0 only holds for a finite number of indices i. Now, consider the sequence (gn )n≥1 defined by gn = αin f i i≥n Further, let D ∗ be a countable w ∗ -dense subset of E ∗ . From hypothesis (ii), we know that for each y ∈ D ∗ the sequence (< y, gn >)n≥1 is uniformly integrable, because uniform integrability is preserved under the convex hull operation.
N→+∞ n→+∞ Thus, d(0, Y ) is µ-integrable. Any µ-integrable selection of Y is also a selection of X , which yields the desired result. 3. 7. Tightness conditions and integrability 41 As in Section 4, we provide an application to the Fatou Lemma in infinite dimension, this time for functions taking on values in a dual space. As in the primal case, the L 1 -boundedness hypothesis is not needed. In the next theorem, it is replaced by a (weaker) Mazur type condition. 9. Let ( f n )n≥1 be a sequence in L 1E ∗ [E], which satisfies the following conditions.
Advances in Mathematical Economics Volume 11 (Advances in Mathematical Economics) by S. Kusuoka, A. Yamazaki