Upon replacing $\mu_{j}$ with $(\int\mu_{j})^{-1}\mu_{j}$ we may assume that the $\mu_{j}$’s are probability measures. Fix any $\varepsilon>0$. By Lemma 2.24 there exists a continuous function $g\geq f$ on $X$ such that $\int g\mu<\int f\mu+\varepsilon$. By Dini’s lemma, we have $f_{j}<g+\varepsilon$ for all $j\gg 1$, hence

$$\limsup_{j}\int f_{j}\mu_{j}\leq\limsup_{j}\int g\mu_{j}+\varepsilon=\int g\mu+\varepsilon\leq\int f\mu+2\varepsilon.$$

since $\int g\mu_{j}\to\int g\mu$ by the definition of weak convergence. The result follows. ∎