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
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.2$