"The purpose of this paper is to develop the fundamental concepts, especially those of measure, that are necessary to define the Lebesgue Integral. A rigorous definition of definite integral for a continuous function was first formulated by Cauchy at the beginning of the 19th century. This was then generalized to discontinuous functions by Riemann in the middle of the 19th century. But difficulties were encountered in the Riemann theory such as limits of Riemann-integrable functions (or even of continuous functions) may fail to be Riemann integrable. It was Lebesgue who first generalized the old process of integration of Cauchy-Riemann using the theory of measure, which has now almost eliminated the difficulty mentioned above, since limits of measurable functions are measurable"--Introduction. / Typescript. / "June, 1959." / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: H. C. Griffith, Professor Directing Paper. / Includes bibliographical references (leaf 30).

It is the purpose of this paper to define a Lebesgue integral over a measurable set, the integration being performed with respect to a monotone non-decreasing function as in the Stieltjes integral, and to develop a few of the fundamental properties of such an integral.

The problem with which this paper is concerned is that of defining the Riemann-Complete Integral and comparing it with the Riemann and the Lebesgue Integrals.

The purpose of this thesis is to restate the definition of the integral as given by O. Perron, to establish some of the fundamental properties of the Perron integral, and to prove the equivalence between the Perron and Lebesgue integrals in the bounded case.

Thesis (Masters of Science for Teachers)--University of Missouri-Columbia, 2008. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on August 25, 2008) Vita. Includes bibliographical references.

The object of this paper is to consider necessary and sufficient conditions in order for certain important inequalities, which are frequently used in analysis, to reduce to equalities.

The purpose of this thesis is to present a study of the Lebesgue definite integral, defined in four different ways.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 65-68). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 65-68). Also available on the Internet.

In this thesis we shall examine the role of measurability in the theory of Lebesgue Integration. This shall be done in the context of the real line where we define the notion of an integral of a bouuded real-valued function over a set of bounded outer measure without a prior assumption of measurability concerning the function and the domain of integration