Open-multicommutativity of the Probability Measure Functor

It is well-known that the construction of space of probability measures P is functorial in the category Comp of compact Hausdorff spaces. The functor P is normal in the sense of E.V. Shchepin [1]. It is well-known that the functor P is open, i.e. it preserves the class of open surjective maps. This was first proved by Ditor and Eifler [2]. E.V. Shchepin [1] discovered tight relations between the properties of openness and bicommutativity. In particular, he proved that every open functor in Comp is bicommutative, i.e. preserves the class of the bicommutative diagrams in the sense of Kuratowski. In this paper we introduce the so-called open multi-commutativity property and show that this property is satisfied by the functor P .