Distribution functions and the Daniell, Kolmogorov theorem

This thesis deals with the characterization of the distribution of random variables over a probability space. A characterization of distribution functions of one random variable is first given, then extended to the distribution functions of a finite sequence of random variables. The final topic, which includes the proof of the Daniell, Kolmogorov theorem, is the characterization of the distribution of a collection of random variables. Although this topic is the main theme of this thesis, the proof given for the Daniell, Kolmogorov theorem draws heavily on both the previous theorems and the specific proofs given for previous theorems. Apart from allowing the aforementioned characterization, the Daniell, Kolmogorov theorem is a major existence theorem in the theory of probability. It is used, for example, to establish the existence of a Brownian Motion. Such applications are excluded here due to lack of space. The major reference used is the book A Graduate Course in Probability by H. G. Tucker. The author gives his sincere thanks to his advisor, Dennis M. Rodriquez, for his hard work, which far exceeded the usual on a venture of this type, and for his continual encouragement.