The question considered in this paper is which sequences of $p$-integrable random variables can be represented as conditional expectations of a fixed random variable with respect to a given sequence of sigma-fields. For finite families of sigma-fields, explicit inequality equivalent to solvability is stated; sufficient conditions are given for finite and infinite families of sigma-fields, and explicit expansions are presented.
"Conditional Moment Representations for Dependent Random Variables." Electron. J. Probab. 1 1 - 14, 1996. https://doi.org/10.1214/EJP.v1-7