Does the measurability of $x\mapsto\operatorname P_x[A]$ imply the measurability of $x\mapsto\operatorname E_x[X]$?

Question

Let

• $(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
• $(\operatorname P_x)_{x\in E}$ be a family of probability measures on $(\Omega,\mathcal A)$ such that $$E\ni x\mapsto\operatorname P_x[A]$$ is $\mathcal E$-measurable, for all $A\in\mathcal A$

Let $X$ be a $\mathcal A$-measurable random variable. Can we show, that $$E\ni x\mapsto\operatorname E_x[X]$$ is $\mathcal E$-measurable, too?

Answer 1

In addition to the "approximation by simple functions" approach, one can use the monotone class theorem for functions, as found for example here. The conditions of the theorem quoted there are met by taking the $\pi$-system to be your ${\mathcal A}$ and the vector space ${\mathcal H}$ to be the class of bounded ${\mathcal A}$-measurable functions $X:\Omega\to{\Bbb R}$ with the property that $x\mapsto{\Bbb E}_x[X]$ is ${\mathcal E}$ measurable.

This shows that the asserted measurability holds for all bounded ${\mathcal A}$-measurable $X$. The boundedness assumption can be relaxed by truncation arguments.