How do you prove that if X converges in probability and expectation that this implies convergence in mean? I think I have to use Chebyshev's Inequality, but am not sure how to incorporate the expectation convergence. Thanks
Here is an example which might shed some light on this question. Assume that $X_n=n$, $X_n=-n$ or $X_n=0$ with respective probabilities $a_n$, $a_n$ and $1-2a_n$. Then:
In particular, $X_n\to0$ in probability and $\mathbb E(X_n)=0$ for every $n$ do not imply that $X_n\to0$ in $L^1$.