Can one have almost everywhere convergence in the Sobolev approximation theorems?

Question

Here are two theorems regarding the Sobolev functions from Evans's Partial Differential Equations:

Here is my question:

Is there any handy theorem showing that the modes of convergence in the above theorems can also be "almost everywhere" convergence?

Answer 1

Thanks to MaoWao's comment, one should consider the following theorem:

Every $L^p$ convergent sequence has an almost everywhere convergent subsequence.