A random variable Y is defined on probability space $(\Omega , F,P)$. A new probability measure is defined $P_o$ on $(\Omega , F)$ by $P_o:=\int_A YdP $ .Take another random variable X on this new space and G:=a sub-$\sigma$ field of $F$. How do we define $E[XY|G]$ in this case?
And is it equal to $E_o[X|G]*E[Y|G]$ ?
I know that, \begin{align} \int_A YdP &= \int_A E(Y|\mathcal{G})dP \end{align} But this is w.r.t a single probability measure.How will I write this in terms of $XY$?