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 $\expect {X \mid \GG}$ 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.

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

Let $A \in \HH$.

Then:

$\ds \int_A \expect {X \mid \HH} \rd \Pr$

$=$

$\ds \int_A X \rd \Pr$

$\ds $

$=$

$\ds \int_A \expect {X \mid \GG} \rd \Pr$

Definition of Conditional Expectation on Sigma-Algebra, since $\HH \subseteq \GG$

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

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

$\blacksquare$

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