When $\mu(X)=\infty$, Can anyone give an example for $f_n$ converge a.e, but not converge in measure.
When $\mu(X)=\infty$, example for $f_n$ converge a.e but not converge in measure.
3
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real-analysis
convergence-divergence
1 Answers
4
Yes, you can do this in $\mathbb{R}$ with Lebesgue measure.
$f_n = 1_{[n,n+1]}$
Then $\mu(\{ x : \left| f_n(x) \right| \geq 1\}) = 1$ for every $n$; but $f_n \rightarrow 0$.