Given a probability space and two random variables defined on the space and taking values in , the subalgebras of generated by the random variables (e.g. preimages of the Borel sets) are noted and respectively.
the conditional expectation is a -measurable function. This implies:
.
Since the conditional expectation is constructed from ,is there any similar relation between the algebras generated by and the conditional expectation ? When can we say ?
Many thanks for your help.