Existence of joint conditional expectation when varying background measure Let $X$ be a random variable defined on $(\Omega, \mathbb{F})$ which has two probability measures $\mathbb{P}_1$ and $\mathbb{P}_2$ inducing two distributions for $X$. Assume that $X$ has finite first moment under both probability measures. For each of the measures we can find the conditional expectation of $X$ given the sub-$\sigma$-algebra $\mathbb{D}$ of $\mathbb{F}$, $\mathbb{E}_1(X \ | \ \mathbb{D})$ and $\mathbb{E}_2(X \ | \ \mathbb{D})$ respectively.
Is it possible to find $Y$ on $(\Omega, \mathbb{F})$ such that $Y$ is equal to $\mathbb{E}_1(X \ | \ \mathbb{D})$ under $\mathbb{P}_1$ and equal to $\mathbb{E}_2(X \ | \ \mathbb{D})$ under $\mathbb{P}_2$?
 A: Without further assumptions, this is not possible in general. I will prove this by giving a counter example.
Consider $(\Omega, \mathbb{F}) = (\mathbb{R}^2,\mathcal{B}(\mathbb{R}^2))$ and let $X$ and $Y$ be defined as $X(x,y) = x$ and $Y(x,y) = y$ for $(x,y)\in \mathbb{R}^2$. Consider two distinct distributions $\mu_1$ and $\mu_2$ (both with finite mean) and define $\mathbb{P}_1$ and $\mathbb{P}_2$ by
$$\mathbb{P}_1 = \mu_1 \otimes \mu_2$$
and
$$\mathbb{P_2}(A\times B) = \int_B \delta_y(A) \: \mu_2(dy)$$
where $\delta_y$ is a dirac measure. Now as our sub-$\sigma$-algebra we consider $\mathbb{D} = \sigma(Y)$, such that $\mathbb{E}_i[X \: |\: \mathbb{D}] = \mathbb{E}_i[X\: |\:Y]$ for $i=1,2$. It is not too hard to see that $X$ and $Y$ are independent under $\mathbb{P}_1$ and therefore
$$\mathbb{E}_1[X \: | \: Y] = \mathbb{E}_1[X]=\int_\mathbb{R}x \:\mu_1(dx)$$
in particular $\mathbb{E}_1[X \: | \: Y]$ is constant. Now to find $\mathbb{E}_2[X \: | \: Y]$ we note that
\begin{align*}
\mathbb{P}_2(X=Y) = \int_\mathbb{R} \delta_{y}(\{y\}) \: \mu_2(dy) = \int_\mathbb{R} \mu_2(dy)=1
\end{align*}
which means that $\mathbb{E}_2[X \: | \: Y]=Y$, and that both $X$ and $Y$ have distribution $\mu_2$ under $\mathbb{P}_2$. But if $\mu_2$ is a non-trivial distribution, then $Y$ will not be constant under $\mathbb{P}_2$.
