prove a function is $L^{1}$

Question

If we have a function $F:[0,T] \to H^{1}_{0}(\Omega)\times L^{2}(\Omega)$, how exactly would we show that $F\in L^{1}([0,T],H^{1}_{0}(\Omega)\times L^{2}(\Omega))$? Is it enough to prove that $\int_{0}^{T}|F(t)|dt<\infty$?

I have seen this notation in several books, but I do not know what to make of it.

Answer 1

Have a look for Bochner-Lebesgue-spaces. You need to show that $$ F : [0,T] \to H_0^1(\Omega) \times L^2(\Omega) $$ is Bochner-measurable (i.e. the limit of simple measurable functions) and $$ \int_0^T \|F(t)\|_{H_0^1(\Omega) \times L^2(\Omega)} \, \mathrm{d}t < \infty.$$