Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d random variables with distribution $F$. Then, for each $x \in R$, do the random variables
$$ F_n(x):=\frac{1}{n}\sum_{i=1}^n 1_{[X_k \le x]} $$
converges to $F(x)$ in $L^1$? By this I mean, $\mathbb{E}[|F_n-F|] \to 0$ as $n\to \infty$. I could prove that it converges $\mathbb{P}-$a.e via the law of large numbers, but I can't see how to prove the convergence in mean, in fact, I do not know if does.