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.