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)?

Many thanks for your help.