A Historical Research On Definitions Of Lebesgue Integral

Lebesgue integral is the landmark of the development of mathematics, the watershed between classical analysis and modern analysis, the important part of the structural mathematics in the 20th century. This creation should owe to Henri Leon Lebesgue, a French mathematician, originally. He came up with several different definitions of the integral in succession. However, Lebesgue was not the only one who created the new theory. For Williom Henri Young, an English mathematician, put forward a new integral in a different way independently which is equivalent with Lebesgue’s shortly. Subsequently, mathematicians developed a further study on Lebesgue’s theory of integration and put forward dozens of definitions within a few decades. Moreover, With the development of modern mathematics, the integrals other definitions are springing up.This dissertation is based on the original literature and supplemented by relevant research documents. Chronologically, through analyzing and comparing, it studies the definitions of Lebesgue integral historically, with exploring the new theory’s historical background, all sorts of its definitions, its application and influence to modern mathe-matics, etc. The main contents are as follows:1. The dissertation summarizes the development of the integral and measure theories before the 20th century fully, and analyses the creating background of Lebesgue integral from the two perspectives.2. The dissertation studies the definitions of Lebesgue integral in detail. First, it discusses the integral’s geometrical definition, analytical definition, axiomatic definition and the definition based on differential which wrere proposed by Lebesgue. Second, it analyses Young’s two new manners of defining:generalizing the definition of Darboux integral and using monotone sequences. As to its other definitions, the paper gives several specific approaches which can permit the new theory to be included in the classical framework of the theory of integration. At last, it classifies all these definitions and compares the two classes of methods.3. The dissertation explores Lebesgue integral’s application and influence system-atically. Examining the theory’s value of application in solving the problems of trigono-metric series and the primitive and discussing its effect to the functional analysis and the probability theory, this dissertation evaluates the importance of Lebesgue’s theory of integration in modern mathematics highly.