Let $(X, \Sigma)$ be a measurable space. $ f: X \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a $\Sigma \otimes \mathcal{B}\left(\mathbb{R}^{n}\right)$-measurable function and $\forall x \in X, y \mapsto f(x, y)$ is a convex function. Could we prove that there exists a map $g: X \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ such that $g$ is $\Sigma \otimes \mathcal{B}\left(\mathbb{R}^{n}\right)$-measurable and for every $(x, y)$, $g(x, y) \in \partial f(x, \cdot)(y)$?
thanks for any help!
