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