On the regularity of measures on locally compact spaces

Abstract:

The purpose of this paper is to present the two following theorems: (1) Every Baire measure on the $\sigma$-algebra ${\mathcal {B}_a}$ generated by the compact ${\mathcal {G}_\delta }$ subsets of a paracompact, locally compact space is outer regular; (2) in a paracompact, locally compact space, any Baire measure on ${\mathcal {B}_a}$ can be extended to an outer regular Borel measure on the $\sigma$-algebra generated by the closed subsets. In addition, this paper contains an example which shows that neither of these two theorems is true for all arbitrary locally compact Hausdorff spaces.