$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure
How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly.
We believe it is false. Since both convergences imply there is a subsequence that converges almost uniformly, but this does not mean the sequence is a.u. convergent.
But we cannot find an example.
First of all, $f_n\to f$ in $L^2$ implies $f_n\to f$ in measure, so the second assumption is redundant.
There is a standard counterexample to show that convergence in $L^p$, $1\le p<\infty$, does not imply convergence a.e., much less "almost uniformly". Namely, enumerate dyadic subintervals of $[0,1]$ as $I_1,I_2,\dots$ (order does not matter), and let $f_n = \chi_{I_n}$. This sequence converges to $0$ in $L^p[0,1]$ for any $p<\infty$. It does not converge a.e., or almost uniformly (which by Egoroff's theorem, is the same thing on $[0,1]$).