TITLE: Definitions of conditional independence
QUESTION [5 upvotes]: 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!

REPLY [5 votes]: The random variables $X$ and $Y$ are independent conditionally on $\mathcal G$ if and only if, for every measurable $A$ and $B$, one has $\mathbb P((X,Y)\in A\times B\mid\mathcal G)=\mathbb P(X\in A\mid\mathcal G)\cdot\mathbb P(Y\in B\mid\mathcal G)$ almost surely.
Some random variables are independent conditionally on the random variable $Z$ if and only if they are independent conditionally on $\sigma(Z)$.