Properties of measures on totally ordered spaces (Q1263691)

Measures on compact totally ordered spaces share a lot of properties with measures on compact metric spaces. The aim of this paper is to study the properties of measures on totally ordered spaces which are not compact. It is shown that if the cardinal of every closed discrete subset of such a space is of measure zero then every non-atomic measure on it is \(\tau\)- additive. It is also proved that a continuous Borel measure on it is Radon if and only if it is perfect, \(\tau\)-additive and the support of the measure is almost a totally ordered space. The relation between the uniform regularity and \(\tau\)-additivity of measures is examined too.