I will ask a similar question to the one from yesterday. If we have
$$\displaystyle\int_Ef(x,t)|^{\tau_2}_{\tau_1}\,dx$$
where $f$ is a Sobolev function (or maybe even just some $L^p$ function) (not neccessarily continuous) on a space-time cylinder. Is this pointwise evaluation in $\tau_2$ and $\tau_1$ without the extra assumption $f\in C([\tau_1,\tau_2])$ actually well defined/possible? I saw this yesterday and am confused now because Sobolev (and $L^p$) functions are merely defined a.e....or is this alright since we are safe as long we are "under" the integral where it's anyways always just a.e. of matter?
Maybe I am misunderstanding something here. Thanks in advance and br!