Let X be a random variable defined on the probability space $(\Omega,\mathbf{F},P )$. If $E|X|<+\infty$, How do I prove that $\lim_{n\to \infty}\int_{\left(|X|>n\right)} X \ dP=0 \,?$
If E|X|<+\infty then \int_{\left(|X|>n\right)}X \;dP\to 0
-
0Oops, my mistake. Rolling back. – 2011-10-28
2 Answers
Let $X_n=|X|\cdot[|X|\geqslant n]$. Since $\lim\limits_{n\to\infty} X_n=0$ almost surely, you ask why $\lim\limits_{n\to\infty}\mathrm E(X_n)=\mathrm E\left(\lim\limits_{n\to\infty}X_n\right)$.
Hint: Use Lebesgue's dominated convergence theorem with the domination condition $|X_n|\leqslant|X|$.
Put $A_k:=\left\{\omega\in\Omega,k\leq |X(\omega)|