$X_n$ converges to $X$ in $L^1$, then $\limsup_H|EX_n1_H-EX1_H|=0$. I want to prove it, is the following proof right?
$$\lim|EX_n1_H-EX1_H|=\lim|E(X_n-X)1_H|=|\lim E(X_n-X)1_H|\\=|E\lim(X_n-X)1_H|=0$$ It's true for all $H$. So, $\limsup|EX_n1_H-EX1_H|=0$
And I also confuse how I can get $|=|\lim E(X_n-X)1_H|=|E\lim(X_n-X)1_H|=0$ by dominated convergence theorem.