Theory of inner measures : an axiomatic approach

Abstract

This thesis presents an axiomatic approach to the theory of inner measures. In Chapter I, we recognize that an axiomatic approach analogous to that for the theory of outer measures is not appropriate: such an approach involves only a finiteness concept, and consequently, as soon as countable collections are involved, it fails. It is then natural for us to speculate that our model should permit the change of limits. This idea leads us to the definition of an inner measure. In Chapter II, we consider a space of finite measure and characterize inner measurability. We also prove that the definitions of inner measurability given by Young and Caratheodory are equivalent, and that Lebesgue’s definition of measurability is equivalent to those given by Young and Caratheodory. Then, the assumption that the space has finite measure is dropped, and we study the inner measures induced by a measure. In Chapter III, inner measures are contracted so as to guarantee that they are countably additive over their classes of inner measurable sets, and so that they always generate an outer measure. The last part of this chapter deals with conditions under which the set function [symbol] generated by an inner measure [symbol] is a regular outer measure. Finally, in Chapter IV, some relation between a sequence of contracted inner measures and the associated measure spaces is established .