Sufficient conditions for the graph measurability of a multivalued function

Question

I am currently working on a problem related to the measurability of multi-functions in the context of mathematical economics. Specifically, I am searching for sufficient conditions regarding the graph measurability of a multi-function.

Formally speaking, the assumptions are as follows.

• $(T, \sum)$ is a measurable space
• $X$ is a separable Banach space
• $B:T\twoheadrightarrow X$ is a multi-function such that $B(t)$ is a nonempty weakly compact subset of $X$ for all $t \in T.$ And, its graph is $\sum \otimes \mathcal{B}(X)$-measurable,where $\mathcal{B}(X)$ denotes the Borel $\sigma$-algebra of $X$.
• $P:T\times X \twoheadrightarrow X$ is a multi-function.

Then, define $D:T\twoheadrightarrow X$ as $D(t) = \{x \in B(t): P(t,x) \cap B(t) = \emptyset\}$.

I am searching for a sufficient condition on $P$ for the graph measurability of $D$. If the graph of $P$ is $\sum \otimes \mathcal{B}(X)\otimes \mathcal{B}(X)$-measurable, then does $D$ have a $\sum \otimes \mathcal{B}(X)$-measurable graph?

I would greatly appreciate insights or suggestions.

Thank you.

Answer 1

Graph measurability of $P$ is not sufficient. Let $E\subseteq[0,1]^2$ be a Borel set whose projection $\pi(E)$ onto the first coordinate is not Borel. Let $X=\mathbb{R}$ and let $B$ have the constant value $[0,1]$. Let $P$ have graph $T\times E$. Then $D$ has graph $T\times [0,1]\setminus\pi(E)$ and is, therefore, not graph measurable.