On the Ranges of Baire and Borel Measures

Let X be a topological space, m a probability measure defined on the Baire s -field on X, and m ' a probability measure on teh Borel s -field which extends m. In the first part of the paper we deal with the relations existing between the ranges of m and m. In particular, we show cases in which these ranges coincide. In the second part we apply to comparative probability some of the techniques previously used. In this part it is proved that if a comparative probability relation defined on a Boolean algebra satisfies well known conditions, namely fineness and tightness (defined below), then the image of any probability measure that agreees with the relation is dense in the unity interval. This sharpens earlier results for comparative probability relations on power sets.