Given a probability space, a random variable $X$ and a sub sigma algebra $\mathcal G$. What is the sigma algebra generated by the conditional expectation $E (X\mid\mathcal G)$?
Is it the one generated by the intersection between the sigma algebra of $X$ and $\mathcal G$? Perhaps not.
It must be contained in $\mathcal G$, since $E (X\mid\mathcal G)$ is measurable wrt $\mathcal G$.
Thanks and regards!
Best Answer
Let $\cal H$ be the $\sigma$-algebra generated by ${\Bbb E} (X\mid\mathcal G)$. As you say, since ${\Bbb E} (X\mid\mathcal G)$ is $\mathcal G$-measurable, $\cal H$ is contained in $\mathcal G$. It doesn't have to be contained in $\sigma(X)$. For a simple counterexample, let the probability space be $\Omega=\{1,2,3,4\}$ with all subsets measurable and the uniform probability measure, and let $${\mathcal G}=\{\emptyset,\{1,2\},\{3\},\{4\},\{3,4\},\{1,2,3\},\{1,2,4\},\Omega\},$$ $$X(1)=1, X(2)=X(3)=X(4)=0.$$ Then $$ {\Bbb E} (X\mid\mathcal G)(1)={\Bbb E} (X\mid\mathcal G)(2)=\frac12, \ \ \ \ {\Bbb E} (X\mid\mathcal G)(3)={\Bbb E} (X\mid\mathcal G)(4)=0$$ so ${\cal H}=\{\emptyset,\{1,2\},\{3,4\},\Omega\}$ is not contained in $\sigma(X)=\{\emptyset,\{1\},\{2,3,4\},\Omega\}$.
Since $\cal H$ does not have to be contained in $\sigma(X)$, it need not equal ${\cal G}\cap\sigma(X)$. The example above shows that it also does not have to equal $\cal G$.
Addendum: As Byron Schmuland says below, if $X$ is an ${\cal F}/{\cal B}({\Bbb R})$-measurable real-valued function, then $\cal H$ is countably generated as it's the pullback of the countably generated $\sigma$-algebra ${\cal B}(\Bbb R)$ under ${\Bbb E} (X\mid\mathcal G)^ {-1}$. Also, any countably generated $\sigma$-algebra which is contained in $\cal G$ can occur as a value of $\cal H$ for some real-valued $X$. (${\Bbb E} (X\mid\mathcal G)$ is only defined up to equality on a set of measure $1$, so to be precise you should say that any countably generated $\sigma$-algebra which is contained in $\cal G$ can occur as a value of $\sigma({\Bbb E}(X\mid\mathcal G))$ for some real-valued $X$ and some version of ${\Bbb E}(X\mid \mathcal G)$.)