Conditional Expectation of Constant

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.

$\blacksquare$