Advances in Mathematical Economics Volume 11 (Advances in - download pdf or read online

By S. Kusuoka, A. Yamazaki

ISBN-10: 4431777830

ISBN-13: 9784431777830

ISBN-10: 4431777849

ISBN-13: 9784431777847

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

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2E ∗ (i) If X admits a µ-integrable selection, then d(0, X ) is µ-integrable. (ii) Conversely, if d(0, X ) is µ-integrable, then X admits at least one µ-integrable (and F-measurable) selection. Proof. The proof of (i) is easy and analogous to that given in [11]. As to the proof of (ii), it is enough to explain why the selection can be chosen to be ∗ )-measurable). It suffices to conE-scalarly measurable (or equivalently B(E w ∗ sider a measurable µ-integrable function r such that d(0, X (ω)) < r (ω) for all ω ∈ and the multifunction Y defined by Y (ω) = X (ω) ∩ r (ω)B ∗ .

7) This holds for µ-almost all ω ∈ . 7) implies Tightness conditions and integrability w ∗ − cl {h n (ω) : n ≥ m} ⊆ f ∞ (ω) ∈ m≥1 43 w ∗ − cl co{ f n (ω) : n ≥ m}. m≥1 As to (j), we note that, due to hypothesis (ii), the sequence (< gn , x >)n≥1 is uniformly integrable for each x ∈ E, which entails < f ∞ , x > dµ = because the sequence ( D this yields lim n→+∞ < h n , x > dµ = < f n dµ, x > = < b, x > gn dµ)n≥1 also w ∗ -converges to b. By the density of b= f ∞ dµ. and finishes the proof. 4. 5 in the present section, we know that the multifunction w ∗ − ls f n is measurable.

These examples share some common ingredients, which we present in the first subsection. We then turn to the specifics of each of the three. 1. Common ingredients Let L = 2. We define preference relations for which the first commodity is a good and the second is a bad, and which is quasi-linear with respect to the good. The disutility from consuming the bad is defined by the following function. Let q and q be such that 0 ≤ q < q < ∞. Define r : (0, 1] → R++ by r (b) = q − q 4b. Then, for each b ∈ (0, 1], define qb : R+ → R+ by 54 C.

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Advances in Mathematical Economics Volume 11 (Advances in Mathematical Economics) by S. Kusuoka, A. Yamazaki


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