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When $\mu(X)=\infty$, Can anyone give an example for $f_n$ converge a.e, but not converge in measure.

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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 $f_n \rightarrow 0$.