Let $(\Omega,\mathcal{F},P)$ be a probability space.

Conditional Independence Given an Event

1. Two events $A$ and $B$ in $\mathcal{F}$ are said to be conditionally independent given $C$ if we have the following equality of conditional probabilities: $$P(A\cap B|C)=P(A|C)P(B|C).$$
2. Two sub sigma algebras $\mathcal{F}_1,\mathcal{F}_2$ of $\mathcal{F}$ are conditionally independent given $C$ if any two events $A\in \mathcal{F}_1$ and $B\in \mathcal{F}_2$ are conditionally independent given $C$ .
3. Two real random variables $X,Y:\Omega \to \mathbb{R}$ are conditionally independent given event $C$ if $\mathcal{F}_X$ and $\mathcal{F}_Y$ , the sub sigma algebras generated by $X$ and $Y$ are conditionally independent given $C$ .

Conditional Independence Given a Sigma Algebra

1. Two events $A$ and $B$ in $\mathcal{F}$ are said to be conditionally independent given $\mathcal{G}$ if we have the following equality of conditional probabilities (as random variables): $$P(A\cap B|\mathcal{G})=P(A|\mathcal{G})P(B|\mathcal{G}).$$
2. Two sub sigma algebras $\mathcal{F}_1,\mathcal{F}_2$ of $\mathcal{F}$ are conditionally independent given $\mathcal{G}$ if any two events $A\in \mathcal{F}_1$ and $B\in \mathcal{F}_2$ are conditionally independent given $\mathcal{G}$ .
3. Two real random variables $X,Y:\Omega \to \mathbb{R}$ are conditionally independent given event $\mathcal{G}$ if $\mathcal{F}_X$ and $\mathcal{F}_Y$ , the sub sigma algebras generated by $X$ and $Y$ are conditionally independent given $\mathcal{G}$ .
4. Finally, we can define conditional idependence given a random variable, say $Z:\Omega\to \mathbb{R}$ in each of the above three items by setting $\mathcal{G}=\mathcal{F}_Z$ .