Tower Property of Conditional Expectation

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

Let $\HH \subseteq \GG$ be sub-$\sigma$-algebras of $\Sigma$.

Let $X$ be a integrable random variable.

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

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

Then:


 * $\expect {\expect {X \mid \GG} \mid \HH} = \expect {X \mid \HH}$ almost everywhere.

Proof
We show that $\expect {X \mid \HH}$ is a version of $\expect {\expect {X \mid \GG} \mid \HH}$.

Let $A \in \HH$.

Then:

Since $\expect {X \mid \HH}$ is $\HH$-measurable, we have that $\expect {X \mid \HH}$ is a version of $\expect {\expect {X \mid \GG} \mid \HH}$.

So by Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra:


 * $\expect {\expect {X \mid \GG} \mid \HH} = \expect {X \mid \HH}$ almost everywhere.