Absolute Continuity and the Fine Topology

Abstract

One of the basic contrasts between the classical and axiomatic theories on the one hand and their probabilistic analogues on the other is that many of the underlying hypotheses of the former are topological, and of the latter, measure-theoretical. A case in point is the regularity of excessive functions, which is assured in the classical and axiomatic settings by assuming lower semi-continuity, and in the probabilistic setting by assuming much weaker conditions such as the absolute continuity condition (hypothesis (L) of Meyer).