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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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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).
Transcribed Image Text: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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