Problem Statement:
Let $\{f_n\}$ be a sequence of nonnegative measurable functions converging pointwise to some measurable function $f$ on $\mathbb{R}$ and suppose that
$$\lim_{n \to \infty} \int_{\mathbb{R}}f_n dx = \int_{\mathbb{R}}fdx < \infty.$$
Show that for each measurable set $A,$
$$\lim_{n \to \infty} \int_A f_n dx = \int_A fdx < \infty.$$
My comments:
This problem is from an old real analysis qualifying exam. The book we are using is Real Analysis by Royden and Fitzpatrick (4th edition).
I'm thinking that maybe we need to use Fatou's Lemma...
I'd be happy with just a hint, Thanks!
Generalized Lebesgue Dominated Convergence Theorem: \begin{align*} \int_{A}f_{n}(x)dx&=\int_{{\bf{R}}}f_{n}(x)\chi_{A}(x)dx\\ \int_{A}f(x)dx&=\int_{{\bf{R}}}f(x)\chi_{A}(x)dx, \end{align*} and we see that $f_{n}(x)\chi_{A}(x)\leq f_{n}(x)$, $f_{n}(x)\chi_{A}(x)\rightarrow f(x)\chi_{A}(x)$ pointwise and $f_{n}(x)\rightarrow f(x)$ pointwise, and $\displaystyle\int_{{\bf{R}}}f_{n}(x)dx\rightarrow\int_{{\bf{R}}}f(x)dx$, so the result follows.