Lebesgue measure and integration

Includes bibliographical references.The theory of Lebesgue integration is a more satisfactory theory than the theory of Riemann integration. It is more elegant aesthetically and more useable in advanced theories than the Riemann integral. Lebesgue theory must be preceded by measure theory and the theory of measurable functions. After the preliminaries, measure theory is presented in Chapter II. The axiomatic approach to outer measures, measurability, and measure functions is discussed briefly. Lebesgue measure is then presented from two distinct but equivalent points of view. The first point of view defines Lebesgue outer measure for all subsets of the real number system and then the restriction of this function to the class of measurable sets results in a measure function of the desired properties. The second point of view defines the Lebesgue measure function for intervals and then extends it to all subsets of real numbers by defining Lebesgue inner and outer measure functions in terms of the measure function for intervals. The second point of view is shown to be equivalent to the first and is not discussed at length. Next, the basic properties of measurable functions of a real variable are discussed. These include equivalent definitions, operations on measurable functions, and their relationship to continuous functions. Also presented here are the concept of "almost everywhere", sequences of measurable functions, some new, broader notions of convergence, and the approximation of measurable functions by simple functions, by continuous functions and by bounded measurable functions. The theory and defects of Riemann integration are examined at the beginning of Chapter IV and are followed by Lebesgue integration and the fundamental properties of the Lebesgue integral. The exact conditions for the existence of the Riemann integral are given and both types of integral are compared to show their relationship to each other and the absence in Lebesgue theory of those defects indigenous to Riemann theory.M.S. (Master of Science