Convergence in mean This is a very basic question, however, I can't find a definitive answer for it.
Let $(X_n)$ be a sequence of random variables.  Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=\mu$.  Does that imply that $(X_n)$ converges to $\mu$ in mean, i.e., that $\lim_{n\rightarrow\infty}\mathbb{E}[|X_n-\mu|]=0$?
Intuitively it seems so (by the identity of indiscernibles), but I've learned to be careful with absolute values of random variables, so hence the question to the community...

This is not true. Take $X_n$ to be i.i.d with $P(X_1 = 1) = 1/2$ and $P(X_1 = -1) = 1/2$.
Then $EX_n$ is always 0, but $E \lvert X_n\rvert = 1$  for every $n$.
