By Takuji Arai (auth.), S. Kusuoka, A. Yamazaki (eds.)

A lot of monetary difficulties can formulated as limited optimizations and equilibration in their options. a variety of mathematical theories were delivering economists with essential machineries for those difficulties coming up in monetary concept. Conversely, mathematicians were influenced by way of quite a few mathematical problems raised via monetary theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new not easy stimuli from financial theories with these economists who're looking for powerful mathematical instruments for his or her researchers. individuals of the editorial board of this sequence contains following well-known economists and mathematicians: coping with Editors: S. Kusuoka (Univ. Tokyo), A. Yamazaki (Hitotsubashi Univ.) - Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier II), F. H. Clarke (Univ. Lyon I), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Fukuoka Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Hitotsubashi Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), okay. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), T. Maruyama (Keio Univ.), H. Matano (Univ. Tokyo), okay. Nishimura (Kyoto Univ.), M. ok. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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

**Example text**

Proof. 4 and we only explicit the arguments to be modified. First, we change the definition of the set Anq by setting now Anq = {ω ∈ : X n (ω) ⊆ q (ω)}. 5) we deduce for all ω ∈ Dq d(0, Z q (ω)) ≤ lim inf 1 Anq (ω) d(0, X n (ω)∩ n→+∞ q (ω)∩q B)+1(Anq )c (ω) | f q (ω)| ≤ lim inf d(0, X n (ω) ∩ q B) + 1(Anq )c (ω) | f q (ω)| . o. 4 of [11] (in fact, a slight extension of it) it follows d(0, Z q (ω)) ≤ lim inf d(0, X n (ω)) + 1(Anq )c (ω) | f q (ω)| . 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).

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. (i) There exists a sequence (rn ) in L 0 (µ) with rn ∈ co{ f i : i ≥ n} such that lim sup rn ∈ L 1 (µ). (ii) for each x in E the sequence (< x, f n >)n≥1 is uniformly integrable in L 1 (µ) (iii) there exists b ∈ E ∗ such that b = w ∗ − lim f n dµ. n→+∞ Under the above hypotheses, there exists f ∞ ∈ L 1E ∗ [E] such that (j) b = f ∞ dµ and (jj) for µ-almost all ω ∈ one has w ∗ − cl co{ f n (ω) : n ≥ m}.

Although it is most common to take P = p ∈ R L | p = 1 , where p = L=1 | p |, we only require inf p∈P p > 0. In fact, since there are only two types of commodities, of which the first one is a good and the second one a bad, in our examples, we will take P = { p ∈ R L | p1 = 1}. For each z ∈ R, denote max {z, 0} by z + . Define ψ : P × R L × X × P → R+ by ψ(Q, w, x, p) = | p·(x−w)|+(sup { p · (x − y) | y ∈ X, y Qx, but not x Qy})+. (1) Thus ψ(Q, w, x, p) measures, in monetary terms, the gap between the given consumption vector x ∈ X and the demand of the consumer with the preference relation Q and the initial endowment vector w under the price vector p ∈ P, where the first term penalizes the violation of the budget-balancing condition and the second term penalizes the violation of the utility maximization condition.