Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.

I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How can I show that $u\in L^{p}((0,T);W^{1,p}(\Omega))$?

It is easy to show that $\displaystyle\int_{0}^T \Vert u(t,\cdot)\Vert_{W^{1,p}(\Omega)}^p\ dt<\infty$. I'm having problems in showing that $u:(0,T)\to W^{1,p}(\Omega)$ is Bochner measurable.

So we need to find a sequence $u_n:(0,T)\to W^{1,p}(\Omega)$ of simple functions such that $\Vert u_n(t)-u(t)\Vert_{W^{1,p}(\Omega)}\longrightarrow 0$ for a.a. $t\in (0,T)$.

How can we do that?

Simple functions that are obtained as sum of characteristic functions of cartesian products $A\times B$ with $A\subset (0,T)$ and $B\subset\Omega$ are dense in $L^{p}((0,T)\times\Omega)$ (see here Simple functions on a product measure space)...I am stucked here.