Dirichlet And Legesgue, Henri Lebesgue's Contribution To Mathematics

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Abstract—The idea of Riemann integration is now popular and very helpful in mathematics. In this paper we explore the background of mathematicians Henri Lebesgue, Peter Gustav Lejeune-Dirichlet and Bernhard Riemann. We discover some of their contributions to mathematics and how they all contributed to this famous concept of Riemann integration.

Index Terms— Bernhard Riemann, Henri Lebesgue, Peter Gustav Lejeune-Dirichlet, Riemann integration

I. INTRODUCTION
Even though the idea of calculus was already in the air, the clear definition of it or its concepts could still be improved. One of these important concepts was that of integration. Riemann integration came to be thanks to the contributions of many throughout time. Even though its name
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1. Henri Lebesgue [8]

Lebesgue is credited for many amazing discoveries to different areas of mathematics. In the area of topology, Lebesgue is known for his covering theorem which is used for finding the dimensions of a set. He is also credited for his work on the Fourier series. He was able to demonstrate that using term by term integration of a series that were Lebesgue integrable functions was always valid and therefore, gave validation to Fourier’s proof of his series.
What is now considered to be one of Lebesgue’s greatest contribution was his work with the Riemann integral. Before his work, the Riemann method was only applicable to continuous functions with a few exceptions. Lebesgue’s work is considered of great importance because it led to the generalization of this integral. In other words, he was able to formulate a new method of measure and gave a new definition to the definite integral. Thanks to his work, we now have the Lebesgue integral which is now considered to be one of the greatest contributions of modern real analysis. [5]

III. PETER GUSTAV
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It’s important to know and understand that many great minds took part in the creation of this concept. German mathematician Peter Gustav Lejeune Dirichlet is considered to be the founder of the theory of the Fourier series. Also German mathematician, Bernhard Riemann then became student of Dirichlet and when he introduced the beginnings of the concept of Riemann integration, credited him for being the first to write about the subject. [11] Among the many contributions by German mathematician Henri Lebesgue was that of the generalization of the Riemann integral. His new definition for the integral, allowed it to be a more useful tool for problems of greater difficulty and even inspired him to continue making even more contributions to the field of

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