On distributions of conditional expectations (Q2772054)

The following result is proved. Let \(F\) and \(G\) be arbitrary distribution functions on the real line \({\mathbb R}\). Then there exists a random variable \(X\) and a \({\sigma}\)-field \({\mathcal U}\) satisfying conditions \(P(X<a)=F(a)\) and \(P(E(X\mid {\mathcal U})<a)=G(a)\) if and only if the inequalities NEWLINE\[NEWLINE\int_a^{\infty} (F(t)-G(t)) dt \leq 0 \leq \int_{-\infty}^a (F(t)-G(t)) dtNEWLINE\]NEWLINE are satisfied for any \(a \in {\mathbb R}\). The proof of this result is based on rather elementary arguments.