Let $\mathcal{A}$ and $\mathcal{B}$ be two sub-$\sigma$-algebras in a measure space. To each one, there is a conditional expectation associated, respectively $E^\mathcal{A}$ and $E^\mathcal{B}$. Given the two $\sigma$-algebras, we can form a third one, $\sigma(\mathcal{A},\mathcal{B})$, generated by both, and consequently, its conditional expectation $E^{\mathcal{A},\mathcal{B}}$.

*My question is: knowing only the first two conditional expectations as projection operators in the measurable function space, can we obtain the third one as an algebraic expression of the first two, as the limit of a polynomial, for example?*

A first try was to think about them as geometrical projections and try to find a complementar conditional expectation and calculate it in a similar fashion to $A\cup B=(A^C\cap B^C)^C$, and define the intersection as the limit of $(E^\mathcal{A}E^\mathcal{B})^n$. But the expected complementar $1-E^\mathcal{A}+E$ fails to be a conditional expectation.

Thank you!