Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional expectation with respect to any sub-$\sigma$-algebra is the same as conditional expectation with respect to some random variable (valued in a R.-L. space)?



Sub-$\sigma$-algebras and conditional expectation
Yes - this is one of the key results of the Rokhlin theory. Namely, any complete sub-$\sigma$-algebra of a Lebesgue space can be realized as the preimage $\sigma$-algebra of a quotient map.


