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:

\(\ds \expect {\expect {X \mid \GG} }\)

\(=\)

\(\ds \int_\Omega \expect {X \mid \GG} \rd \Pr\)

Definition of Expectation

\(\ds \)

\(=\)

\(\ds \int_\Omega X \rd \Pr\)

Definition of Conditional Expectation on Sigma-Algebra

\(\ds \)

\(=\)

\(\ds \expect X\)

Definition of Expectation

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list