Let $X, Y$ be conditionally independent given $Z$. Based on this, I am trying to get an intuition of what happens with the associated sigma-algebras and the conditional expectation. In particular, I want to calculate
$$ \mathbb{E}[B \vert X, Y,Z] = \mathbb{E}[B\vert \mathcal{F}]$$
where $B$ is an event and $\mathcal{F}$ is the sigma-algebra generated by $X,Y,Z$. Given the conditional independence stated above, I know that $\sigma(X)$ and $\sigma(Y)$ are conditionally independent given $\sigma(Z)$, but what does that mean for $\mathcal{F}$? Could I somehow partition this sigma-algebra into two independent components depending on $X,Z$ and $Y,Z$? In general, any intuition about the structure of $\mathcal{F}$ given conditional independence would be great.