Definition:Conditional Expectation/General Case/Random Variable

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Definition

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

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\SS$ be a set of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.


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

$\expect {X \mid \SS} = \expect {X \mid \map \sigma \SS}$

where:

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


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

$\expect {X \mid \SS} = \expect {X \mid X_1, X_2, \ldots}$


Sources