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(1) Prove that a sequence (fn)n converges to f in measure if and only if for every k N, lim m ({\fn - f\ > }) = 0. n→∞

(1) Prove that a sequence (fn)n converges to f in measure if and only if for every k N, lim m ({\fn - f\ > }) = 0. n→∞

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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Question 4 (1) Prove that a sequence (fn)n converges to f in measure if and only if for every k N, lim m ({\fn - f\ > }) = 0. n→∞0 (2) Prove that if lim m ({\fn - ƒ| >0}) = 0, then (fn)n converges to fin n→∞ measure. (3) Prove that if a sequence (fn)n converges to f in measure, then the limit f is unique, almost everywhere. (Hint: first show that for every e > 0,
Transcribed Image Text:Question 4 (1) Prove that a sequence (fn)n converges to f in measure if and only if for every k N, lim m ({\fn - f\ > }) = 0. n→∞0 (2) Prove that if lim m ({\fn - ƒ| >0}) = 0, then (fn)n converges to fin n→∞ measure. (3) Prove that if a sequence (fn)n converges to f in measure, then the limit f is unique, almost everywhere. (Hint: first show that for every e > 0,
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