I have seen the following LLN result on empirical measures:
(Statement 1) Let $X, X_1, X_2, \cdots, X_n$ be i.i.d. random variables taking values in $[0, 1]$. Then $$ \mathbb{E}\sup_{f \in \mathcal{F}}\left| \frac{1}{n}\sum_{i = 1} ^n f(X_i) - \mathbb{E}f(X) \right| \leq \frac{CL}{\sqrt{n}}, $$ where $\mathcal{F}$ is the class of $L$-Lipschitz function.
Recently I saw the following result used in a paper:
(Statement 2) For $\mu_n$ that is an empirical probability measure and $\mu$ the original measure, we have by "LLN for empirical measures (Glivenko-Cantelli)" that $\mu_n$ converges to $\mu$ weakly a.s.
I see the similarities in the two statements above, but I am having trouble to recover statement 2 from statement 1. It seems like the best I could get from statement 1 is that, along a subsequence $\mu_{n_j} \to \mu$ weakly a.s. instead of the entire sequence $\mu_n$. This is due to the fact that $L^1$ convergence implies subsequential a.s. pointwise convergence.
I am curious if my logic so far is correct and if one could recover entirely the second statement from the first. If not, this means the first statement is strictly weaker than the second statement and I wonder if there is a good counterexample for this?
