How $\varphi(X, y)$ is integrable and the map $\psi:y \mapsto \mathbb E (\varphi(X, y))$ is Borel?

Question

I'm reading a proposition in this note. I'm trying to understand how related objects are well-defined.

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $X,Y:\Omega \to \mathbb R$ real-valued random variables. Let $\varphi:\mathbb R \times \mathbb R \to \mathbb R$ be Borel such that $\varphi (X,Y)$ is integrable. Clearly, $\varphi(X, y)$ is Borel for all $y \in \mathbb R$.

Could you explain how $\varphi(X, y)$ is integrable and the map $\psi:y \mapsto \mathbb E (\varphi(X, y))$ is Borel?

Answer 1

This is false as stated. Take $\phi (x,y)=x-y$ and $Y=X$. Then $\phi (X,Y)=0$ is integrable but $\phi(X,y)$ is not integrable for any $y$ if $X$ is not integrable.

If $X$ and $Y$ are independent then $\phi (X,y)$ is measurable for almost all $y$ and it is integrable also,by Fubini's Theorem.