A comparative study of the Riemann and Lebesgue integrals.

摘要

Integration theory has undergone a long and interesting history. Many mathematicians have contributed to a gradual and fruitful development of a theory that today enjoys an important place in applicative branches of mathematics like probability, quantum physics and hydromechanics. Research and major developments seem to address themselves to bringing forth a theory whose generalizability adheres and comprehensively addresses itself to the existing mathematical concepts and theories. More often, mathematicians have had to make improvements on existing theories in a bid to address themselves to the limitations of previously existing theories. For example, Bernard Riemann proposed an integral that addressed itself to the previous theories of integration which were only limited to a certain class of functions. Lebesgue build upon Riemann's work to come up with a more general definition that catered in a great way for the deficiencies and limitations of the Riemann integral. This paper aims at discussing, comparing and contrasting the works of two mathematicians, Riemann and Lebesgue. Their contributions are viewed by most mathematicians as the major building blocks of today's theory of integration. Though not exhaustive, I hope I will be able to appreciatively point out the contributions of each intellect's involvement in the development of the modern integration theory.