A b s t r a c t. Measures on epimorphic images of the coproduct of a non-empty family of measurable spaces are shown to be equivalent (in some precise sense) to measures on Boolean algebras. We obtain two sets of sufficient conditions for the existence of probability measures on factor spaces, from which a given measure $\mu$ on (an epimorphic image of) the coproduct can be retrieved. The results are applied to probability logic - to prove, reformulate and generalize Los representation theorem.