- 期刊
Comparative Study of Riemann and Lebesgue Integrals
摘要
The Riemann integral can be seen as making a division in the domain of definition of the integral, while the Lebesgue integral is making a division in the domain of values. When performing the division, the Riemann integral may result in huge amplitudes and, as a consequence, there is not Riemann integrability. This shortcoming is circumvented by the Lebesgue integral, which divides the value domain in such a way that the amplitude is satisfactory. In this paper, we introduce the concept of interval measure of the domain of definition of a function, and use the countable and additive properties of the Lebesgue integral to solve the Riemann integral problem for partially discontinuous functions. If the Riemann-producible functions are not closed to the limit operation, then it is possible to calculate the columns of those functions by exchanging the symbols for the integral and the limit. However, the conditions that must be met in order for integral and limit operations to be exchangeable are fairly stringent. The Lebesgue integral improves the solution to this problem by easing some of the conditions that must be met for exchangeability.