Weak convergence and convergence almost everywhere If a bounded sequence $(u_n)$ converge weakly to $u$ in $W^{1,p}(\Omega)$ (where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ with $N>p$),

is it true that $u_n(x)$ converges to $u(x)$ for almost every $x\in \Omega$?

thank you 

Because $W_0^{1,p}$ is compactly embedded in $L_p$, the embedding operator is obviously compact so it maps weakly convergent sequences in strongly convergent, i.e a weakly convergent subsequence in $W_0^{1,p}$ is strongly convergent in $L_p$. Finally, you can extract a further subsequence, which is pointwise a.e convergent: see
Does convergence in $L^{p}$ implies convergence almost everywhere?
