Category:Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra

This category contains pages concerning Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra:

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 $\map \sigma X \cup \GG$.

Let $\map \sigma {\GG, \HH}$ be the $\sigma$-algebra generated by $\GG \cup \HH$.

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

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

Then:

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