How are conditional independence defined for various cases?
For one case, given two random variables $X$ and $Y$, and a subsigma algebra $\mathcal G$ of the underlying sigma algebra $\mathcal F$, or another random variable $Z$.
What is the definition of the conditional independence between $X$ and $Y$ given $\mathcal G$ or $Z$?
Is it related to the independence between $E(X\mid \mathcal G)$ and $E(Y \mid \mathcal G)$, and independence between $P(X\in A\mid \mathcal G)$ and $P(Y\in A\mid \mathcal G)$, for any $A \in \mathcal F$? I am trying to see if there is some relation between conditional independence and independence?
Thanks and regards!