Some Characterizations of Absolute Continuity

Abstract

Suppose now that F is a nondecreasing function of a real variable. Then we may use F to construct a Caratheodory outer measure F* and a family L(F) of measurable sets. F* will be a measure on the completely additive class L(F). (Here and in all that follows we shall assume that all nondecreasing functions F referred to are normalized so that they are continuous from the right and so that F(0) =0. This does not affect the family L(F) nor the measure F*.) If we have two such functions F1 and F2, we may talk about the absolute continuity of Fj* with respect to F2*, in the sense of Definition 1, It is frequently not pointed out that in this context Definition 1 involves certain relations between the classes L(F1) and L(F2). It is the purpose of this note to point out some of these relations. For example, if F1 is continuous, then the relation L(F2) CL(F1) is equivalent to the absolute continuity of Fj* with respect to F2* in the sense of Definition 1. We first observe that in Definition 1, we consider the measures u and v on a common completely additive class X. If we are to say that Fj* is absolutely continuous with respect to F2* in the sense of Definition 1, what should we take as X? Since BCL(F) for any nondecreasing function F (here B stands for the Borel sets), it seems natural to take X to be B. Definition 1 would then be expressed as