Idempotent probability measures, I

TL;DR

This paper studies the functorial properties of idempotent probability measures on compact spaces, establishing its normality, monad structure, and behavior with respect to open surjective maps, highlighting differences from classical probability measures.

Contribution

It introduces the functor of idempotent probability measures, proves its normality and monad structure, and analyzes its properties, including the preservation of open surjective maps and the relationship with hyperspace monads.

Findings

The functor is normal in the sense of E. Shchepin.

It forms a monad containing the hyperspace monad as a submonad.

It preserves the class of open surjective maps, unlike the classical case.

Abstract

The set of all idempotent probability measures (Maslov measures) on a compact Hausdorff space endowed with the weak* topology determines is functorial on the category $\comp$ of compact Hausdorff spaces. We prove that the obtained functor is normal in the sense of E. Shchepin. Also, this functor is the functorial part of a monad on $\comp$. We prove that the idempotent probability measure monad contains the hyperspace monad as its submonad. A counterpart of the notion of Milyutin map is defined for the idempotent probability measures. Using the fact of existence of Milyutin maps we prove that the functor of idempotent probability measures preserves the class of open surjective maps. Unlikely to the case of probability measures, the correspondence assigning to every pair of idempotent probability measures on the factors the set of measures on the product with these marginals, is not open.