How to show this dual space pairing is measurable

Question

If $f \in L^2(0,T;H^{-1})$ and $g \in L^2(0,T;H^1)$, how to show that $\langle f(t), g(t) \rangle_{H^{-1},H^1}$ is measurable over [0,T]?

If it's measurable, it's clearly integrable. But how to show that it is measurable at all?

Answer 1

By definition, there are simple functions $f_n$, $g_n$, which converge towards $f$ and $g$ pointwise a.e., respectively. There duality-product converges towards $\langle f, g\rangle$ pointwise a.e. Hence, it is measurable.