For each n E N, let fn → R be a (F,B(R))-measurable function.(i) Then, each of the functions supnen fn, infnen fn, lim supn→∞o fn, andlim infn oo fn is (F, B(R))-measurable.(ii) The set A = {w : limn→∞o fn(w) exists and is finite} lies in F andthe function h = (limn→∞ fn) · IA is (F, B(R))-measurable.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

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e functions supnen fn, inf s (F, B(R))-measurable. limn→∞o fn (w) exists a = (limn→∞ fn) IA is (F,