Can you help me on this proof. It is stated everywhere to show that the conditional expectation is uniform integrable. But the specific lemma is shown nowhere and I tried to proof it ( and failed )

Suppose we have a measeure space $\displaystyle $(\Omega, \mathbb{A}, \mathbb{P})$$ and $\displaystyle $\xi\in L^1$$.

$\displaystyle $\forall \epsilon>0 \exists \delta>0$$: $\displaystyle $A\in \mathbb{A}, \mathbb{P}(A)<\delta \ \Rightarrow \int_A \left |\xi\right |d\mathbb{P}<\epsilon$$.

I found that this follows with Borel Cantelli and Fatou, but how?

Thx