If $(X, \mathbb{X}, \mu)$ space of finite measure. $f_n\to f$ in measure implies $f_n\to f$ almost uniformly.

Question

If $(X, \mathbb{X}, \mu)$ space of finite measure. Is true or false the following: $f_n\to f$ in measure implies $f_n\to f$ almost uniformly.

I know that convergence in measure implies that some subsequence of $ f_n $ converges almost evenly, but I am not able to prove this result, nor do I find a counterexample. Can someone give a tip

Answer 1

False. There is a well known example of sequence that converges in measure but does not converge at any point. [Indicator functions of dyadic intervasl $(\frac {i-1} {2^{n}},\frac i {2^{n}})$ arranged in sequence with increasing order of $n$ is such a sequence].