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2) Prove that if lim m({fn - f>0}) 84x measure. = 0, then (fn)n converges to fin

2) Prove that if lim m({fn - f>0}) 84x measure. = 0, then (fn)n converges to fin

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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(2) Prove that if lim m({fn-f|>0}) = 0, then (fn)n converges to fin n-x 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 & > 0, {\ƒ − g | > e} = {\ƒ − fn\ > } U {\9 - £₂ | > ² })
Transcribed Image Text:(2) Prove that if lim m({fn-f|>0}) = 0, then (fn)n converges to fin n-x 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 & > 0, {\ƒ − g | > e} = {\ƒ − fn\ > } U {\9 - £₂ | > ² })
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