Expectation of Conditional Expectation

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

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

Let $X$ be a integrable random variable.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Then:


 * $\expect {\expect {X \mid \GG} } = \expect X$

Proof
We have: