Caratheodory''s general outer measure

The present thesis is essentially an exposition of Caratheodory's general theories of outer measure of sets and set measurability; however some relations between measurability and additivity in regard to classes of point sets and functions defined over such classes have been briefly noted. The behavior, in the limit, of sequences of sets from additive classes has also been investigated superficially. In gathering data, it was found helpful to obtain a great deal of information on various specific systems of outer measure. While not referred to explicitely in the thesis such additional data facilitated a presentation of the general theory as a system which lends itself implicitely to a classification of specific measures according to their generating functions, While considerable work has been done toward linking measure theory to algebraic topology, discussions of these developments have been excluded for reasons of brevity and unity of approach. It is proved in the thesis that several broad classes of sets are measurable for every set function which satisfies Caratheodory's definition of an outer measure function. It is further shown that measurability produces certain additivity conditions in sequences of measurable sets, and that monotonic sequences of sets taken from additive classes have definite additivity properties in the limit.