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

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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(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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