Let $\{f_n\}$ be a sequence of nonnegative measurable function on $\mathbb{R}$ that converges pointwise on $\mathbb{R}$ to $f$ and $f$ is integrable over $\mathbb{R}$. Show that if $\int_\mathbb{R} f = \lim\limits_{n \to \infty} \int_\mathbb{R} f_n$, then $\int_E f = \lim\limits_{n \to \infty} \int_E f_n$ for any measurable set $E$.
While I was reading this problem my first reaction was Fatou's lemma, however we have "$=$" and not "$\leq$".
I'm not sure what heavy machinery I could throw at this.