Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

H. L. Smith


Dr. H. L. Smith in a paper which has not been published as yet 3hows that by starting with a general function satisfying Caratheoacry's first two postulates on an outer measure function, it is possible to construct a function which satisfies all four postulates. In this dissertation we have stuuied some of the characteristics o of this.function, principally those which are of use in deriving our‘theorems on density. We have set up three general density functions, and nave succeeded in showing- tnat for the most general of these, it is true thac the Smith measure of the set at which the upper density is less tnan 1, and the one at which the lower density is greater than 1, is zero. We have also established tnat under certain circumstances there is a definite relation between this function and the density function defined by Besicovitch. Considerable attention nas been devotee to certain fundamental geometric theorems, which have led us to a generalized form of tne Vitali tneorem. We have also uerived a set of sufficient conditions for the validity of this theorem, including the one for the Smith function.