Suppose we have a sequence of positive random variables which converges to 0 in probability, i.e. $X_n=o_P(1)$. I want to prove that $E[X_n]$ is bounded.
My idea: In particular $X_n$ is bounded in probability. Thus $\forall\epsilon\,\exists K$ such that $\forall n\in N$ $$P\{X_n>K\}\leq\epsilon.$$ Then we can write $$E[X_n]=E[X_n\,1_{\{X_n>K\}}]+E[X_n\,1_{\{X_n\leq K\}}]\leq K+E[X_n\,1_{\{X_n>K\}}].$$ Does it hold that $E[X_n\,1_{\{X_n>K\}}]$ is also bounded?
I would appreciate any idea (also different from mine). Thank you in advance.
No, this is not correct. Just consider $((0,1],\mathcal{B}((0,1]),\text{Leb})$ and $$X_n(\omega) := n^2 \cdot 1_{(0,1/n)}(\omega), \qquad \omega \in (0,1].$$ Then $X_n \to 0$ almost surely (hence, in particular, $X_n \to 0$ in probability), but $\mathbb{E}X_n = n$ is not bounded.
Another counterexample: $$X_n(\omega) := \frac{1}{\omega} 1_{(0,1/n)}(\omega).$$ Then $X_n \to 0$ almost surely, but $X_n$ is not even integrable.