|dc.description.abstract||This thesis presents an axiomatic approach to the theory of inner
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
Finally, in Chapter IV, some relation between a sequence of
contracted inner measures and the associated measure spaces is established