If $f \in L^1(M)$, is it true that $f(x) < \infty$ for almost all $x$? If $M$ is a measurable space (eg. $M$ is a Riemannian manifold which is compact) and if $f \in L^1(M)$, is it true that $|f(x)| < \infty$ for almost all $x$?
I am trying to figgur out if $u \in H^1((0,T)\times \Omega)$ implies that $u(x,\cdot) \in H^1(0,T)$ for a.a. $x$!!!

The answer is yes. By definition, a $L^p$ function takes finite values (in $\mathbb R$ or $\mathbb C$) on a set of full measure (if you are in doubt, check your go-to reference for functional analysis and re read carefully the definition).
