Can someone please assist me with this question?
Let $f_n$ be a sequence of non-negative measurable functions on $(\Omega,\mathcal{A},\mu)$ that converge to $f$ pointwise. Suppose that $$\int_\Omega f \ \mathrm{d}\mu = \lim_{n\to\infty}\int_\Omega f_n \ \mathrm{d}\mu < \infty$$ Then show that for any $A \in \mathcal{A}$ $$\int_A f \ \mathrm{d}\mu = \lim_{n\to\infty}\int_A f_n \ \mathrm{d}\mu.$$ Trivially, I can use Fatou's lemma to show that $\int_A f \ \mathrm{d}\mu \leq \lim_{n\to\infty}\int_A f_n \ \mathrm{d}\mu$. But as for the reverse inequality, I'm not sure.