Definition:Conditional Expectation/General Case/Random Variable

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\mathcal S$ be a set of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.

Then we define the conditional expectation of $X$ given $\mathcal S$:


 * $\expect {X \mid \mathcal S} = \expect {X \mid \map \sigma {\mathcal S} }$

where:


 * $\map \sigma {\mathcal S}$ denotes the $\sigma$-algebra generated by $\mathcal S$
 * $\expect {X \mid \map \sigma {\mathcal S} }$ denotes the conditional expectation of $X$ given $\map \sigma {\mathcal S}$
 * $=$ is understood to mean almost-sure equality.

If $\mathcal S$ is countable set, say $\mathcal S = \set {X_n : n \in \N} = \set {X_1, X_2, \ldots}$, we may write:


 * $\expect {X \mid \mathcal S} = \expect {X \mid X_1, X_2, \ldots}$