All measures are Lebesgue.
$\forall n \in \mathbb{N}$, let $f_n: \mathbb{R} \rightarrow [0, \infty]$ be measurable and almost everywhere $f_n \rightarrow f$; moreover, suppose that $\int f_n dλ \rightarrow \int f dλ < +\infty$.
Show that, for any measurable set $A, \int_A f_n dλ \rightarrow \int_A f dλ$.
I really am not sure where to even start.
Assuming the $f_n$'s are integrable, we can apply the generalized dominated convergence theorem:
If $|f_n|\leq g_n$, $g_n$ are integrable, $g_n \rightarrow g \in L^1$ (integrable), $f_n \rightarrow f$, and $\int g_n \rightarrow \int g$, then $\int f_n \rightarrow \int f$.
The proof of this is very similar to the proof of the usual DCT.
Now just use $|f_n \chi_A |\leq |f_n|=f_n$ for any measurable $A$.