On continuous image averaging of Borel measures

Abstract

Let X, T be topological spaces and M( X) the space of all non-negative and finite Borel measures on X, endowed with the weak topology. Given a non-negative and finite Borel measure λ ≠ 0 on T, denote the space of λ-integrable, continuous kernels Φ : T → M( X), endowed with the compact-open topology, by I( T, X, λ). We show the map I( T, X, λ) ∋ Φ → λΦϵ M( X) to be open if T is completely regular and prove that the assumption ‘ T regular’ is not sufficient.