On the approximation of a random variable by a conditional expectation of another random variable

Let X and Y be R-valued random variables on a non-atomic probability space (Ω, F, P). We give conditions under which Y can be approximated by a conditional expectation of X. In particular, we prove the following theorem: Let X be an R-valued random variable such that EX+ = EX− = ∞. Then for each random variable Y and arbitrary ϵ > 0 there exist B ∈ F and a sub-σ-field S of F such that P (B) ≤ ϵ and E (X|S) = Y a.s. on Bc. We also review some facts on the conditional expectation of unintegrable random variables.