By S. Kusuoka, A. Yamazaki
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Additional resources for Advances in Mathematical Economics Volume 11 (Advances in Mathematical Economics)
3, 1–44 (2001) 6. : Uniform scalar integrability and strong law of large numbers for Pettis integrable functions with values in a separable locally convex space. J. Theoret. Probab. 13(1), 93–134 (2000) 7. : Dunford–Pettis–types theorem and convergences in set-valued integration. J. Nonlinear Convex Anal. 1, 37–71 (1999) 8. : Convex analysis and measurable multifunctions. Lectures Notes in Math 580 (1977) 9. : Fatou’s Lemma for unbounded Gelfand integrable mappings. CERNSEM, Universit Paris 1 (2002) 10.
9 Since the consumption set R L is not bounded, the Hausdorff distance may be infinite. + Core convergence in economies with bads min a∈An a ∈ An | d n a, n a ≤ tn 53 →∞ as n → ∞. In the analysis of the core convergence property for monotone preference relations, the No Peculiar Individuals Condition (and its variants) is often defined using the metric of the closed convergence topology in place of the Hausdorff distance. The Hausdorff distance measures the difference between two preference relations that is applicable uniformly, regardless of the choice of consumption vectors at which the difference is measured, while (the metric of) the closed convergence topology allows the difference between the two to depend on the norm (length) of such consumption vectors.
Hypothesis (i) shows that h n (ω) ∈ (ω) for all n ≥ 1 and µ-almost all ω ∈ . On the other hand, we claim that lim sup |h n | is µ-integrable. Indeed, one has ⎛ ⎞ hn = βin ⎝ βin gi = i≥1 i≥n whence α ij f j ⎠ j≥i ⎞ ⎛ α ij | f j |⎠ . βin ⎝ |h n | ≤ i≥n j≥i This yields ⎛ ⎞ lim sup |h n | ≤ lim sup ⎝ n→+∞ n→+∞ βin ri ⎠ ≤ lim sup rn n→+∞ i≥n which, by hypothesis (iii) shows the desired integrability property. 5 of , which shows that Y admits at least one measurable and µ-integrable selection f ∞ .