Please follow the format a-d provided 1. Prove that, for a countable family of measurable sets {Ax}ien contained in (a,

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Please Follow The Format A D Provided 1 Prove That For A Countable Family Of Measurable Sets Ax Ien Contained In A 1 (59.57 KiB) Viewed 25 times

Please follow the format a-d provided 1. Prove that, for a countable family of measurable sets {Ax}ien contained in (a, b), lim (UA (4-0) = UA U=1 A Roadmap to Glory (a) But first, let play with some toy sets. Let A1 = [1/4, 3/4, A2 = (1/2,7/8) and A3 = (3/4, 15/16). What is B) = A2 A1? • What is B; = A3\(A, U A2)? . If B = A1, does A, UA, UA3 = BUB, U B3 and does (A, U A, U A3) = f(B1) + (B2+H(B)? (b) Let us now return to the generic sets described in the prompt. Construct a family of disjoint sets { Bx}ue that have the property that for every neN, ÜA; - ÜB. Note that this implies that UA; = UB; . (c) Since {B;}jen are disjoint sets, use the additivity property of the measure to express J=1 HUB as a sum. (d) How are the quantities lim =1 and (04) -(ÜB) lim related? How can you use this relation to prove our goal?