Suppose we have a measure space $(X,M,\mu)$, where $\mu(X)<+\infty$. Let ${f_n}$ be a sequence of non-negative functions, $f_n\in L_1,\ \forall n$ and $f_n$ converges to some $f$ pointwise. $f$ is not necessarily in $L_1$. Suppose now $\underset{X}{\int} f_nd\mu$ also converges to $\underset{X}{\int}fd\mu$, as $n\rightarrow \infty$. Is it true that $\underset{E}{\int} f_nd\mu\rightarrow\underset{E}{\int}fd\mu,\ \forall E\in M$? If not, can you give a counter-example?
I've encountered a similar problem, where I have the condition $f\in L_1(X,M,\mu)$ but don't have the condition $\mu(X)<+\infty$. In that case it is pretty easy to prove this problem. But here I don't have this condition, and $\underset{X}{\int}fd\mu = \infty$ can indeed happen, so how can I prove the problem now? I tried using Egoroff's theorem, but I failed to get it to work. Any suggestions? Thanks!