I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim:
Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and $(W,Z)$ have same joint distribution, then is it true that for any Borel measurable set $B$, we have $\int_{X^{-1}(B)}Y=\int_{W^{-1}(B)}Z$?
Thanks.
Note that $$ \int_{X^{-1}(B)}Y\,\mathrm dP=\int_\Omega Y\mathbf{1}_B(X)\,\mathrm dP={\rm E}[ f(X,Y)] $$ with $f(x,y)=y\mathbf{1}_B(x)$ being Borel measurable from $\mathbb{R}^2$ to $\mathbb{R}$. Since ${\rm E}[f(X,Y)]$ only depends on the distribution of $(X,Y)$ we conclude that ${\rm E}[f(X,Y)]={\rm E}[f(W,Z)]$ or in other words $$ \int_{X^{-1}(B)}Y\,\mathrm dP=\int_{W^{-1}(B)}Z\,\mathrm dP. $$