Smoothness properties of quasi-measures (Q1011014)

A Baire quasi-measure on a completely regular space \(X\) is a \([0,+\infty[-\)valued function on the system of zero sets and cozero sets of \(X\) which is monotone, additive and inner regular. \textit{J. Boardman} [Rocky Mountain J. Math. 27, 447--470 (1997; Zbl 0913.46042)] introduced several smoothness conditions for quasi-measures analogously to conditions introduced by \textit{V. S. Varadarajan} [Am. Math. Soc., Transl., II. Ser. 48, 161--228 (1965); translation from Mat. Sb., n. Ser. 55(97), 35--100 (1961; Zbl 0152.04202)] for measures. E.g. a quasi-measure \(\mu\) is \(\sigma-\)smooth if \(\mu (Z_n)\rightarrow 0\) whenever \(Z_n\) is a sequence of zero sets with \(Z_n\downarrow \emptyset\). The author presents a series of examples clarifying the relationship between the various smoothness conditions. Moreover, smoothness conditions of Baire quasi-measures on \(X\) are characterized with the aid of the induced Borel quasi-measure on \(\beta X\). Finally, it is shown that particular properties of \(X\) or of \(\beta X\) guarantee some smoothness conditions for Baire quasi-measures on \(X\).