On the Splicing of Measures

Abstract

Given probabilities $\mu$ and $\nu$ on $(X, \mathscr{A})$ and $(X, \mathscr{B})$ respectively, a probability $\eta$ on $(X, \mathscr{A} \vee \mathscr{B})$ is called a splicing of $\mu$ and $\nu$ if $\eta(A \cap B) = \mu(A) \nu(B)$ for all $A \in \mathscr{A}, B \in \mathscr{B}$. Using a result of Marczewski we give an elementary proof of Stroock's result on the existence of splicing. We also discuss the splicing problem when $\mu$ and $\nu$ are compact measures.