Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra/Corollary

Theorem

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

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

Let $X$ be a integrable random variable such that:

$\map \sigma X$ is independent of $\HH$

where $\map \sigma X$ is the $\sigma$-algebra generated by $X$.

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

Then:

$\expect {X \mid \HH} = \expect X$ almost surely.

Proof

Note that:

$\map \sigma X = \map \sigma {\set {\O, \Omega}, \map \sigma X}$

So by Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra, we have:

$\expect {X \mid \map \sigma {\set {\O, \Omega}, \HH} } = \expect {X \mid \set {\O, \Omega} }$

From Conditional Expectation Conditioned on Trivial Sigma-Algebra, we have:

$\expect {X \mid \set {\O, \Omega} } = \expect X$ almost surely.

Also, since:

$\set {\O, \Omega} \subseteq \HH$

we have that:

$\map \sigma {\set {\O, \Omega}, \HH} = \HH$

giving:

$\expect {X \mid \HH} = \expect X$ almost surely.

$\blacksquare$

Sources

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