Let $(Z,d)$ be a Polish space, let $\mathcal{B}(Z)$ denote its Borel $\sigma$-algebra, and let $(\alpha_n)_{n \in \mathbb{N}}$ and $\alpha$ be Borel probability measures on $Z$. Suppose that $\alpha_n \rightharpoonup \alpha$ weakly in $\mathcal{P}(Z)$. If $f: Z \to (-\infty,\infty]$ is lower semicontinuous and bounded from below, meaning that there exists $m \in \mathbb{R}$ such that $f(z) \geq m$ for every $z \in Z$, then the extended Lebesgue integrals satisfy

\begin{align*} \int_Z f(z)\,d\alpha(z) \leq \liminf_{n \to \infty} \int_Z f(z)\,d\alpha_n(z). \end{align*}