By Jerrold E. Marsden

Easy complicated research skillfully combines a transparent exposition of center idea with a wealthy number of applications. Designed for undergraduates in arithmetic, the actual sciences, and engineering who've accomplished years of calculus and are taking advanced research for the 1st time..

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L(A) > 0 such that (x~, f(w)) > sup (x~, y), wE E A. L. Thus the first statement is proved. A with Chapter 2 46 The Aumann Integral and the Conditional Expectation Now assume that X* is separable, and let FE L 1 (n; Kc(X)]. Let {x~} be a countable dense subset of X*. 3) holds. The rest of the proof is same as above. 2(1) the following result is obtained immediately. 11 Let F1,F2 E L 1(0;Kc(X)]. L. = 0. L. 8 we have s(x*, F) is measurable for any E X*. L ln. L fESp ln. ;~£. 13 partition of n. L) .

3. 12 and assume that X reflexive or the dual space X* is separable. If {Fn,n 2:: 1} is a sequence of convex set-valued random variables, w-lim supn-+oo Fn is a set-valued random variable. 17 (3) we have w-limsupFn(w) = n-+oo U nw-cl( U (Fn(w) npU)). p~l m~l n~m Let Gmp = w-cl(Un~m(Fn(w) npU)). 10, G;~(V) E A for any weakly open set V. Thus G(Gmp) E Ax 13x. 3, we have that Gmp is a set-valued random variable. 11, we have w-limsupn-+oo Fn is a set-valued random variable. D 3. L) be a a-finite measure space and (X, ll·llx) be a real separable Banach space.

We thus obtain llx- fm,n(w)llx :S llx- Xmllx + llxm- fm,n(w)llx < 2-n+l. (ii) =? (i) Assume that there exists a countable family Un : n E N} of measurable selections ofF such that F(w) = cl{fn(w)} for all wEn. Since for any given x EX, llx- fn(w)llx is measurable for each n EN, d(x,F(w)) = inf{llx- fn(w)llx: n EN} 24 Chapter 1 The Space of Set-Valued Random Variables is measurable. 3, we have F is a set-valued random variable. 1) {in} is also called a representation of F. About above results, please refer to the work of Castaing [36], Himmelberg and Van-Vleck [97], and also the book of Klein and Thompson [115].