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Jackson Duncan

Jackson Duncan

Answered question

2022-06-26

Let f n : R R be defined by f n ( ω ) = ω ( 1 + 1 n ) and let μ be the Lebesgue measure.
I have to show that f n does not converge in measure μ to ω.
Here's my attempt: By definition, f n converges in measure to ω if:
lim n μ ( ω : | f n ( ω ) ω | δ ) = 0.
Rewriting, it becomes that:
lim n μ ( ω : | ω | n δ ) = 0.
I realized that if δ = 1 for example, then the measure is surely 0 as n . I need to come up with a value of δ such that the measure is non-zero. I thought about letting δ = 1 n ,, but am unsure if it'll become indeterminate once I set n (because I basically get 1 ) .
Does this solution of δ = 1 n truly work as a counterexample? If not, what should I do?

Answer & Explanation

candelo6a

candelo6a

Beginner2022-06-27Added 24 answers

It doesn't matter what δ is. μ ( ω : | ω | n δ ) = for every n and every δ.
{ ω : | ω | n δ } = [ n δ , ) ( , n δ ]

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