Orderable Set Functions and Continuity. II: Set Functions with Infinitely Many Null Points

Abstract

A set function (which is not necessarily additive) on a measurable space I is called orderable if for each measurable order $\mathcal{R}$ there is a measure $\varphi ^\mathcal{R} v$ on I such that for all initial segments J, $(\varphi ^\mathcal{R} v)(J) = v(J)$. Properties of orderable set functions v which have infinitely many null points are investigated in this paper. We show that such set functions are continuous and that a set A is v-null if and only if $| {\varphi ^\mathcal{R} v} |(A) = 0$ for all measurable orders $\mathcal{R}$ . A characterization of orderable nonatomic set functions as well as a characterization of weakly continuous set functions which have a mixing value are given. It is also shown that if a set function is weakly continuous with respect to a measure, then it is weakly equivalent to some measure.