Given a probability space $\displaystyle (\Omega,\mathcal{F},\mathbb{P}) $ and two random variables $\displaystyle f,g$ defined on the space and taking values in $\displaystyle \mathbb{R}$, the subalgebras of $\displaystyle \mathcal{F}$ generated by the random variables (e.g. preimages of the Borel sets) are noted $\displaystyle \sigma(f)$ and $\displaystyle \sigma(g)$ respectively.

the conditional expectation $\displaystyle \mathbb{E}(f|g)$ is a $\displaystyle \sigma(g)$-measurable function. This implies:

$\displaystyle \sigma(\mathbb{E}(f|g)) \subset \sigma(g)$.

Since the conditional expectation is constructed from $\displaystyle f$,is there any similar relation between the algebras generated by $\displaystyle f$ and the conditional expectation $\displaystyle \mathbb{E}(f|g)$? When can we say $\displaystyle \sigma(\mathbb{E}(f|g)) \subset \sigma(f)$?

Many thanks for your help.