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Question 3 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 gn = |fn| + |ƒ| − [ƒn − ƒI).

Question 3 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 gn = |fn| + |ƒ| − [ƒn − ƒI).

Algebra & Trigonometry with Analytic Geometry
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 3 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| + |f|-|fn-f|).
Transcribed Image Text:Question 3 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| + |f|-|fn-f|).
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