Conditional Expectation Bounded

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 (Worried))

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

Re: Conditional Expectation Bounded

okey, I solved it by myself!

thx anyway