Let $\xi , \xi_1, \xi_2, ... $ be nonnegative integrable functions in a measure space $(\Omega,\mathcal{A},\mu)$ with $\mu(\Omega)=1$, such that $\int \xi_n d\mu \rightarrow \int\xi d\mu$ and $\mu(\xi-\xi_n >\epsilon)\rightarrow 0$ as $n\rightarrow\infty$, for any $\epsilon > 0$. Show that then $\int | \xi_n - \xi | d\mu \rightarrow 0$, $n \rightarrow \infty$.
I've tried splitting the integral into one over $C=\{x\in\Omega | \xi(x)-\xi_n(x) > \epsilon \}$ and the other over the complement of this set, but I don't know how to effectively bound these parts
