Is there some version of DCT for $p=\infty$. That is, is it true that if there is a sequence of measurable functions defined on an open set $\Omega$ in $\mathbb{R}^n$, $f_n$ converging pointwise to a function $f$ such that there exists $g\in L^{\infty}(\Omega)$ with $|f_n(x)|\leq|g(x)|$, then $f_n,f\in L^{\infty}(\Omega)$ and $||f_n-f||_{\infty}\to 0$?
If this is indeed true, could someone point out the proof?