Topological spaces whose Baire measure admits a regular Borel extension

Abstract:

A completely regular, Hausdorff space $X$ is called a Măík space if every Baire measure on $X$ admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence ${F_n} \downarrow \emptyset$ of regular closed sets such that ${ \cap _{n \in \omega }}{B_n} \ne \emptyset$ whenever ${B_n}$’s are Baire sets with ${F_n} \subset {B_n}$; a countably paracompact, non-Măík space; a locally compact, non-Măík space $X$ such that the absolute $E(X)$ is a Măík space; and a locally compact, Măík space $X$ for which $E(X)$ is not. It is also proved that Michael’s product space is not weakly Baire-dominated.