I'm reading a proof and it contains the following inequality:
Suppose $u: E \rightarrow [0,\infty]$ is lower semicontinuous and let $u_t$ be a sequence of Lipschitz functions approaching $u$ from below. Let $\{\mu_n\} \rightarrow \mu$ in the weak-* sense. Taking $\phi \in C_c(E)$ where $0 \leq \phi \leq 1$ we get: $$\int_E \phi u_t \, d\mu = \lim_{n \rightarrow \infty}\int_E \phi u_t \, d\mu_n \leq \liminf_{n \rightarrow \infty} \int_E u_t \, d\mu_n$$
How do you get to the right hand side of that inequality?
Given the assumption that $u_t:E\to [0,\infty)$, we have $\phi u_t \le 1\cdot u_t$, hence $$ \int_E \phi u_t \ \mathrm d\mu_n\le \int_E u_t \ \mathrm d\mu_n. $$ We then take $\liminf_{n\to\infty}$ on both sides. Note that the LHS is actually convergent since $\phi(x) u_t(x)$ is a bounded continuous function and $\mu_n \to \mu$ weak-$*$ly. Hence, we get $$ \int_E \phi u_t \ \mathrm d\mu=\lim_{n\to\infty} \int_E \phi u_t \ \mathrm d\mu_n\le\liminf_{n\to\infty} \int_E u_t \ \mathrm d\mu_n. $$