Let $X_1, X_2,...\in L^p$ be a sequence of random variables defined on $(\Omega, \mathcal{F}, P)$ for some $p\geq 1$. Suppose $X_n\to X$ in probability (or almost surely), where $X$ is a random variable defined on the same space. Question: Is X in $L^p$?
I think X might not necessary be in $L^p$ but I cannot find a counterexample. Can someone help?
On $(0,1)$ witt Lebesgue measure define$X_n(x)=\frac 1 x$ if $x >\frac 1 n$ and $0$ otherwise. Let $X=\frac 1 x$ at every point. Then $X_n \to X$at every point and each $X_n \in L^{1}$ but $X \notin L^{1}$.