Economics

Advances in Mathematical Economics by Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

By Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

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We have that 11 n = in + 9n with In E w. 1 and 9n E w. 2 · Since w. e to a function I E L~(p,) (see, [9, 10]). i with Ai ~ 0, :E~~nAi = 1. e to 9 = 7L - i E L~(p,). Since w. 1 and w. 2 are convex and closed in measure, 1 E w. 1 and 9 E w. 2 . Thus u=l+gEW.. 0 The following is a minimisation result. 3. 11 w. 1 and w. ) which are closed in measure and il w. 1 is bounded, then there exist 1 E w. 1 and 9 E w. 2 such that Praof. Let us set m := inf{II11-vl Ir : 11 E w. 1 , v E W. 2 }. There exist a sequence (fn) in w.

24 C. Castaing, M. 2. 11 w. 1 and w. 2 are convex sets in L~ (p,) which are closed in measur'e and il w. 1 is bounded and has the WTP, then the set is closed in measure. Praof. We need to prove that if (u n ) is a sequence in W. e to a function U E L~(p,), then U E w.. , we have that 11 n = in + 9n with In E w. 1 and 9n E w. 2 · Since w. e to a function I E L~(p,) (see, [9, 10]). i with Ai ~ 0, :E~~nAi = 1. e to 9 = 7L - i E L~(p,). Since w. 1 and w. 2 are convex and closed in measure, 1 E w. 1 and 9 E w.

Now we are able to state the equivalence WTP{:::::::} WKP. 5. surable multifunction. s the WKP. Proof. Ey hypothesis Sf is obviously Rw(X)-tight. Ey [9,10] we know that Rw(X)-tight==}WTP. ) be a bounded sequence in Sf. Let us consider a dense sequence (e;)pEN for the Mackey topology T(X*, X). 1) . 2) for each subsequence (wt) = (vm[). 1) we see that (~~j=lWj('))n is pointwise bounded and by hypothesis ~~j=lWj(W) E r(w) for almost all wEn and for all n E N*. )II. ~j=lWj(W)) n = (e*,u(w)) for each subsequence thus showing that (u n ) has the WKP.

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