If $u \in L^2(0,T;L^2(\Omega))$ with $u(t) \in L^\infty(\Omega)$ uniformly, is $u \in L^\infty(0,T;L^\infty(\Omega))$?

Question

Suppose I have a function $u \in L^2(0,T;L^2(\Omega))$. If I know that for almost all $t$, $\lVert{u(t)}\rVert_{L^\infty(\Omega)} \leq C$ for some constant, does it follow that $u \in L^\infty(0,T;L^\infty(\Omega))$?

The issue may be the measurability of $u\colon (0,T) \to L^\infty(\Omega)$.