Inaug.-disc.--Leipzig. / Lebenslauf. "Bibliographie": p. [117]-125.

"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.

This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space.

The purpose of the thesis is to investigate an approach to Lebesgue-Stieltjes measure and integration.

This thesis attempts to prove the Lebesgue measure is a concrete realization of measure.

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

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática e Computação Científica. / Made available in DSpace on 2012-10-18T19:41:50Z (GMT). No. of bitstreams: 1 189800.pdf: 259053 bytes, checksum: 32ee78247ba20b891a2ebcaf3acde334 (MD5) / Neste trabalho, estudamos uma modificação da Integral de Riemann, a Integral Henstock-Kurzweil, ou Integral de Riemann Generalizada, ou ainda, Integral "Gauge"(denotamos Integral R*). Mostramos o Teorema de Hake e o Lema de Saks-Henstock, e estes servem como ferramentas na aplicação da Integral R*. Esta integral contrasta com outras integrais, em particular com respeito a formulação do Teorema Fundamental do Cálculo, e sua respectiva classe de funções integráveis. Nós provamos que a Integral R* permite um elegante Teorema Fundamental e concluímos que integra uma classe maior de funções que a integral de Lebesgue, a qual generaliza.

This paper discusses the concept of a general definition of measure, and shows that the Lebesgue measure satisfies the requirements set forth for the ideal definition.

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.