#### Title

The Theory of the Lebesgue Integral and an Investigation of Its Usefulness in the Solution of Random Variable Problems

#### Publication Year

1962

#### Keywords

mathematics, random variable problems, Lebesgue integral, probability, functions

#### Disciplines

Applied Mathematics | Mathematics | Physical Sciences and Mathematics

#### Abstract

The research represented in this paper was motivated by one sentence in the text which was used when I studied probability theory as a junior, *Modern Probability Theory and its Applications* by Emmanuel Parzen. After introducing the basic concepts and tools of probability such as distribution functions and probability density functions, Parzen stated he would restrict himself to functions which could be integrated only in the sense of the Riemann integral and that functions integrable in the sense of Lebesgue would be omitted. Parzen gave no indication of how relevant the Lebesgue integral is to the study of probability and proceeded to develop his theory for functions which were Riemann integrable. The purpose of this paper, then, is to discuss the nature of the Lebesgue integral, to contrast it with the more familiar Riemann integral, and to investigate its usefulness in the theory of probability. More precisely, the purpose is to examine the applicability of the Lebesgue integral to the solution of random variable problems.

#### Department 1 Awarding Honors Status

Mathematics

#### Recommended Citation

Yost, F.
(1962).
*The Theory of the Lebesgue Integral and an Investigation of Its Usefulness in the Solution of Random Variable Problems* (Undergraduate honors thesis, University of Redlands).
Retrieved from https://inspire.redlands.edu/cas_honors/944