Approximation in $L^2(\Omega)$

Question

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, but how can I dominate this function? I would appreciate any idea. Thank you in advance.

Answer 1

Liu Gang already explained the various reasons why this does not work, but here is a concrete example: $$f_n(x) = \max(0,1-n|x|),\qquad f(x)\equiv 0; \qquad X\equiv 0$$ Here $f_n\to f$ in $L^2$. On the other hand, $f_n(X)$ does not converge to $f(X)$ in any sense, because $f_n(X)\equiv 1$ and $f(X)\equiv 0$.