Reconciliation of probability measures

We discuss the reconciliation problem between probability measures: given n 2 probability spaces (Ω, F1, P1),. .. , (Ω, Fn, Pn) with a common sample space, does there exist an overall probability measure P on F = σ(F1,. .. , Fn) such that, for all i, the restriction of P to Fi coincides with Pi ? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given.