TITLE: Questions about convergence in Lp
QUESTION [6 upvotes]: If $X_n$ converges to $X$ in $L^p$, do we have $X_n^p$ converges to $X^p$ in $L^1$?
We can prove that it is true when p=1,2 easily. I am curious whether this is true for all $p>0$.

REPLY [3 votes]: It's not correct for $0 < p < 1$. Take $X_n: [0,1] \to \mathbb{R}$ as $X_n(x) = n \cdot I_{[0,1/n)}(x)$, where $I_A$ = indicator function of a set $A$. Then $\|X_n\|_p = n^{1-1/p}$ and therefore $X_n \to 0$ in $L^p$ for $p < 1$, but $X_n$ does not converge in $L^1$.