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].
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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F01416e22-b897-4a9e-9b93-2d1738197d91%2F19a98aec-dc24-4e03-9d82-7710d97e267f%2Fq7d52f_processed.jpeg&w=3840&q=75)
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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