Assume that a sequence of integrable functions (fn)n converges pointwise to an integrable function f, a.e. and (fn)n converges to f absolutely in mean. Prove that (fn)n converges strongly to f. (Hint: Apply Fatou's Lemma to the sequence 9n = |fn| + |ƒ|- |fn-f).

Assume that a sequence of integrable functions (fn)n converges pointwise to an integrable function f, a.e. and (fn)n converges to f absolutely in mean. Prove that (fn)n converges strongly to f. (Hint: Apply Fatou's Lemma to the sequence 9n = |fn| + |ƒ|- |fn-f).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 73E

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Question

Assume that a sequence of integrable functions (fn)n converges pointwise to
an integrable function f, a.e. and (fn)n converges to f absolutely in mean.
Prove that (fn)n converges strongly to f.
(Hint: Apply Fatou's Lemma to the sequence 9n = |fn| + |ƒ|- |fn - f).

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