How can I express this random variable in the form of a conditional expectation?

Question

Let $X,Y:\Omega \to \mathbb{R}$ be integrable random variables, and suppose that they are independent. Define $Z:\Omega \to \mathbb{R}$ by $$Z(\omega)=E[Y|X(\omega)\geq Y].$$ Can we write $Z=E[\tilde Y|X]$ for some $\tilde Y$? What is $\tilde Y$ explicitly? Thanks for any comment!

Answer 1

Let $B(x) =\{Y\le x\}$. Then for a fixed $\omega$, $$ Z(\omega) ={E[ Y\cdot 1_{(-\infty,X(\omega)]}(Y)]\over P[Y\le X(\omega)]}. $$ On the other hand, if $\tilde Y$ takes the form $h(X,Y)$, then $E[Y|X]=g(X)$, where $g(x):=\int_\Omega h(Y,x)\,dP$. (Here $(\Omega,\mathcal F,P)$ is the background probability space.) To make things match up it looks like you need to take $h(x,y) =y{1_{\{y\le x\}}\over P[Y\le x]}$.