Lower semicontinuity of a Bochner integral of a convex function

Question

I'm looking for the following result:

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $f$. The map $$u \mapsto \int_0^T \int_{\Omega} f(u(t))$$ is lower semicontinuous for $u \in L^2(0,T;L^2)$ where $f:\mathbb R \to \mathbb R$ is convex.

Does anyone know how to prove this, or a reference for this result? Thank you.