Let $\mu_n, \mu$ be probability measures on a Polish space $X$. We say that $\mu_n$ converges weakly to $\mu$ if
$$
\int f d\mu_n \to \int f d\mu \:\:\;\;\;\; \forall f \in C_b(X)
$$
Show that it is equivalent to ask
$$
\int g d\mu \leq \liminf \int g d\mu_n \:\:\;\;\;\; \mbox{for all g lower semicontinuous bounded from below}
$$
This should be standard but unfortunately I could not find any reference
Best Answer
I do not know a reference, but you can argue as follows: by the Skorokhod representation theorem we may find random variables $X_n,X$ (changing the probability space if necessary) such that the law of $X_n$ is $\mu_n$, the law of $X$ is $\mu$ and such that $X_n \to X$ almost surely as $n \to \infty$. Letting $g$ be lower semicontinuous and bounded from below, we want to show that $$\mathbb{E}(g(X)) \leq \liminf_{n\to\infty} \mathbb{E}(g(X_n)).$$ By lower semicontinuity, as $X_n \to X$ we almost surely have $g(X) \leq \liminf_{n\to\infty}g(X_n)$ and therefore Fatou's lemma shows $$\mathbb{E}(g(X)) \leq \mathbb{E}(\liminf_{N\to\infty}g(X_n)) \leq \liminf_{n\to\infty}\mathbb{E}(g(X_n)).$$