Let $f_n,g_n,f,g \in L^1$, $f_n \to f$ and $g_n \to g$ a.e, $|f_n|\leq g_n$ and $\int g_n \to \int g$.
Then $\int f_n \to \int f$
I think I should use the DCT but I need to find a function in $L^1$ dominating the $f_n$'s. I thought that since for each $x$ (except maybe for a null set) the sequence $f_n(x)$ is bounded, I could use the function that takes each $x$ to the infimum of the (essential) bounds of $f_n(x)$ but I'm not sure that function is in $L^1$ or how to prove it.
Another thought was to use $\sup g_n$ to dominate the $f_n$'s, but again I need to justify it is in $L^1$. I feel quite lost on how to use the fact that $\int g_n \to \int g$, any hint or suggestion is very welcome.