Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra

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 $\map \sigma X$ be the $\sigma$-algebra generated by $X$.

Let $\HH \subseteq \Sigma$ be a sub-$\sigma$-algebra that is independent of $\map \sigma {\map \sigma X, \GG}$, the $\sigma$-algebra generated by $\map \sigma X \cup \GG$

Then:


 * $\expect {X \mid \map \sigma {\GG, \HH} } = \expect {X \mid \GG}$ almost surely.