Conditional Expectation of Constant
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $c \in \R$.
Define $X : \Omega \to \R$ by $\map X \omega = c$ for each $\omega \in \Omega$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
- $\expect {X \mid \GG} = c$ almost everywhere.
Proof
From Constant Function is Measurable, $X$ is a real-valued random variable.
The result then follows immediately from Conditional Expectation of Measurable Random Variable.
$\blacksquare$