We have:
- three random variables $X_1,X_2,X_3$,
- three $\sigma$-fields $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3$,
- three random variables $Y_{1,2}$, $Y_{2,3}$, $Y_{3,1}$, such that:
- $X_1=\mathbb{E}(Y_{1,2}|\mathcal{G_1})$ and $X_2=\mathbb{E}(Y_{1,2}|\mathcal{G_2})$,
- $X_2=\mathbb{E}(Y_{2,3}|\mathcal{G_2})$ and $X_3=\mathbb{E}(Y_{2,3}|\mathcal{G_3})$,
- $X_3=\mathbb{E}(Y_{3,1}|\mathcal{G_3})$ and $X_1=\mathbb{E}(Y_{3,1}|\mathcal{G_1})$.
(all the above hold a.s.) My question is:
does this imply, that there exists a random variable $Z$, such that
$$X_i=\mathbb{E}(Z|\mathcal{G}_i),$$ for $i=1,2,3$?
Let us assume that there exists a random variable $U\sim \mathcal{U}(0,1)$ which is independent from all $X_i, Y_{i,j}$.
I have been thinking about this a bit, and I am rather convinced, that this is not true. Even though, I do not have a good idea on how to construct a counterexample. I will be glad for any help.