On spaces of σ-additive probability measures

Abstract

The functor P σ of σ-additive probability measures on the category of Tychonoff spaces is investigated. It is shown that the space P σ ( X) is Hewitt complete for every Tychonoff space X; and P σ ( X) is an AE-space of weight ⩽ ω 1 iff P σ ( X) is an AE(0)-space of weight ⩽ ω 1 iff the Hewitt completion of X is an AE(0)-space of weight ⩽ ω 1. It is shown that for every separable metrizable absolute Borel space X the space P σ ( X ω 1 ) is homeomorphic to P σ ( X) ω 1 . In particular, P σ( R ω 1 ) is homeomorphic to R ω 1 . We find conditions on a Tychonoff (uniform) space X under which P σ ( X) is a Hewitt completion (respectively is naturally homeomorphic to the completion) of the (uniform) spaces P R ( X) and P τ ( X) of Radon and τ-additive probability measures on X.