QUASI-MEASURES ON COMPLETELY REGULAR SPACES

JOHN P. BOARDMAN

Abstract:

Let X be a completely regular space. We give definitions of a Baire quasi-measure on X and a quasi-state on C_b(X), the space of bounded, real-valued continuous functions on X. A representation theorem is developed for quasi-states on C_b(X) in terms of Baire quasi-measures on X. We also define various notions of smoothness for quasi-measures and quasi-states, and then we furnish examples which demonstrate the different types of smoothness. Finally, by considering the space X to be embedded in its Stone-Cech compactification $\beta X$, the smoothness of a Baire quasi-measure $\mu$ on X is characterized by the behavior of $\bar \mu$, the corresponding Baire quasi-measure on $\beta X$.