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Let (fn) be a sequence of functions in L[a, b]. Suppose f E L[a, b] and lim |fn – f| = 0. a If the sequence (fn) converges pointwise almost everywhere on [a, b] to the function g, show that f = g a.e. on [a, b].

Let (fn) be a sequence of functions in L[a, b]. Suppose f E L[a, b] and lim |fn – f| = 0. a If the sequence (fn) converges pointwise almost everywhere on [a, b] to the function g, show that f = g a.e. on [a, b].

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 72E
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Let (fn) be a sequence of functions in L[a, b]. Suppose f ∈ L[a, b] and limn→∞ b a |fn − f| = 0 . If the sequence (fn) converges pointwise almost everywhere on [a, b] to the function g, show that f = g a.e. on [a, b]. Suggestion: Consider the sequence (|f−fn|) and Fatou’s Lemma.

Let (fn) be a sequence of functions in L[a,b]. Suppose f E L[a, b] and lim | \fn – f| = 0. If the sequence (fn) converges pointwise almost everywhere on [a, b] to the function g, show that f Suggestion: Consider the sequence (|f– fn|) and Fatou's g a.e. on [a,b]. Lemma.
Transcribed Image Text:Let (fn) be a sequence of functions in L[a,b]. Suppose f E L[a, b] and lim | \fn – f| = 0. If the sequence (fn) converges pointwise almost everywhere on [a, b] to the function g, show that f Suggestion: Consider the sequence (|f– fn|) and Fatou's g a.e. on [a,b]. Lemma.
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