Integral of a lsc function is lsc?

Question

Let $f:\mathbb{R}^d\times \mathbb{R}^D\rightarrow \mathbb{R}$ be a lower semicontinuous function. Then is it true that $$ \begin{aligned} L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)&\rightarrow (-\infty,\infty] \\g&\mapsto \int_{\mathbb{R}^d} f(x,g(x)) \nu(dx) \end{aligned} $$ is a proper, lower-semicontinuous, and convex operator? Here $\nu$ is a $\sigma$-finite Borel measure on $\mathbb{R}^d$, and $L^2_{\nu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)$ is the corresponding Bochner-Lebesgue space.