Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$
Now, this may be already solved in the following link:
Except for, there it says the measure space is $\sigma$-finite.
So, my question is, being $\sigma$-finite is completely necessary? Or this can be solved without that?