Measurability of $t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$

Question

Let $\Omega$ be a bounded domain and let $u = L^2(0,T;L^2(\Omega))$. Is the map $$t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$$ measurable for a constant $C$? Here $\chi$ is the indicator function.

I am not sure how the measurability of the set in the indicator function comes into it, since we are concerned about measurability over $t$.