Given $(X,Y)$ and $(X',Y')$ pairs of rvs on different probability spaces $\Omega_{1},\Omega_{2}$ respectively but with equal joint distribution show that $E[X|\sigma(Y)]\stackrel{d}{=}E[X'|\sigma(Y')]$.
Also, we have $X,X'\in \mathbb{R}$, $E[|X|],E[|X'|]<\infty$ and $Y,Y'\in S$, where $(S,\mathcal{B})$ is some measure space with Borel sigma algebra.
only hints.
Attempt
1)I need a way to express $E[X|\sigma(Y)]$ in terms of the joint distribution $(X,Y)$. Assuming they have joint density I am done since $E[X|\sigma(Y)]=\int x f(x,Y)dx\frac{1}{\int f(x,Y)dx}$ and $Y\stackrel{d}{=}Y'$ by marginalizing.
For finite sigma algebras we can write $E[X|G]=\sum E[X|A_{j}]1_{A_{j}}$ where $G=\sigma(\{A_{j}\}^{n})$ and $A_{j}$ partition the probability space $\Omega$.
So I was trying to relate this to the possibly infinite sigma algebra $\sigma(Y)$. If we can partition space S into generating sets $\{B_{j}\}_{\mathbb{N}}$ for $\mathcal{B}$, then $\sigma(Y)=\sigma(\{Y^{-1}(B_{j})\}_{\mathbb{N}})$ by measurability and so $E[X|\sigma(Y)]=\sum E[X|Y^{-1}(B_{j})]1_{Y^{-1}(B_{j})}$.