Convergence in mean for the sequence of positive random variables

This is a follow-up to this question.

Now let $(X_n)$ be a sequence of positive 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 $\lim_{n\rightarrow\infty}\mathbb{E}[|X_n-\mu|]=0$?

My previous question was for a general sequence of random variables, and Deven Ware showed that the above implication does not hold in the general case…

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