On the $L_1$ distance between the empirical distribution and the population distribution

Question

Let $X_1, \dots, X_n$ be i.i.d. random variables with a cumulative distribution function $F$. Denote the empirical distribution by $$F_n(a) = \frac{1}{n} \sum \limits_{i=1}^n 1(X_i \leq a).$$ It's well known (Kolmogorov-Smirnov) that $D_n = \sup \limits_{x \in \mathbb{R}} \left| F_n(x) - F(x) \right|$ converges to zero and $D_n = O_P(1/\sqrt{n})$. Do we know anything about $$\int_{\mathbb{R}}\left| F_n(x) - F(x) \right| \, \mathrm{d}x?$$ I suspect this is also $O_P(1/\sqrt{n})$, but I can't come up with a proof. I might be missing something very elementary here.