Let $f_n \geq 0$ be a sequence of Lebesgue measurable functions on $(a,b)\subset\mathbb{R}$ such that $f_n(x) \to f(x)$ a.e. on $(a,b)$. Let $F(x) = \int_a^x f(y)dy$ and $F_n(x) = \int_a^x f_n(y)dy$, where $x \in [a,b]$.
Show $\int_a^b \left[f(x)+F(x)\right]dx \leq \liminf \int_a^b \left[f_n(x)+F_n(x)\right]dx$.
My attempt:
Define $g_n(x) = f_n(x) + F_n(x)$. By Fatou's Lemma, $\int_a^b \liminf g_n \leq \liminf \int_a^b\left[f_n(x)+F_n(x)\right]dx$. Since $f_n \to f$, $\liminf f_n = f$. However, I'm stuck on showing that $F_n \to F$. This seems like a situation where dominated convergence would be useful, but how would I bound the sequence $(f_n)$?
