Same limit for a different convergence How can you show that if a sequence $(x_n)$ converges in $L^2$ to $X$, and converges almost surely to $Y$, then $X$ and $Y$ are almost surely the same. Can you give me a hint?

As Daniele A said, the equality $X=Y$ a.s. follows from the  fact that a sequence that converges to $X$ in $L^2$ has a subsequence converging to $X$ a.s. Since the same subsequence converges to $Y$ a.s., the result follows. 

A more general statement can be found in  Pointwise a.e. convergence and weak convergence in Lp
