As the title says. So let $\{f_n\}_n$ be a sequence of Lebesgue measurable functions that converge $\lambda-$a.e. to a real valued function $f$ and let $E$ be Lebesgue measurable such that $\lambda(E)< \infty$ where $\lambda$ denotes Lebesgue measure. Suppose that there is an $M>0$ such that $|f_n(x)| \leq M \hspace{2mm}\forall x \in E$ and $n\in \mathbb{N}$. Show that $f$ is Lebesgue integrable on $E$ and use the dominated convergence theorem to prove that $$\int_E fd\lambda = \lim_{n\rightarrow \infty}\int_E f_n d\lambda$$
Heres my attempt: Since we have that $\{f_n\}_n$ is a sequence of Lebesgue measurable functions that converge $\lambda-$a.e. to $f$, there is Lebesgue measurable set $F$ such that $\lambda(F)=0$ and that $f_n(x) \rightarrow f(x)$ for $x \in F^c$. Since $M>0$ we can set $g(x) = M$ for all $x\in E$, and so for all $n$ we have $|f_n(x)|\leq g$. So now we can apply the DCT to show that $f$ is Lebesgue integrable and that the integral equation holds, but my only concern is that $f_n(x)\rightarrow f(x)$ on $F^c$. What do I do to relate $F^c$ and $E$ because that seems to be the hiccup here.