If $f_n$ converges in measure to $f$ on a the set $E$,and $\int_E{|f_n|} \leq M$ for all $n$. Claim: $f$ is integrable.
I think by Riesz theorem there is a subsequence that convergence pointwise to $f$, but then can I apply Bounded convergence theorem as integrals of $f_n$ bounded then $f_n$ is bounded as well?
Note that, convergence in measure $\implies$ $\{f_n\}$ is cauchy in measure. Therefore, it has a subsequence $\{f_{n_k}\}$ such that $f_{n_k} \to f$ almost uniformly. Now, apply Fatou's to $\{f_{n_k}\}$ to show that $f$ is integrable.