## Conditional probabilities

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

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

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