实变函数的关键部分是勒贝格测度和积分理论，是对黎曼积分的一次深刻的变革。黎曼积分的诞生，始于创建一系列易于计算的区域，将给定的函数根据其定义域划分为小区间，解决了许多问题，并且开创了新的思维方式。但是，黎曼积分也有一些缺点，勒贝格的测度和积分理论，在一定程度上克服了黎曼积分的这些局限性，使得微积分理论的发展更加完善。勒贝格积分的诞生，成功地解决了许多以往在积分领域无法处理的难题。从黎曼积分过渡到勒贝格积分，对于两者综合起来的看待，会更有助于学习理解。本文旨在通过两种积分的定义、基本性质、极限运算、存在条件等方面进行分析，并且通过学习一些实际的运算例子对黎曼积分的不足之处和勒贝格积分针对性的改动进行讨论，从而进一步了解到黎曼积分和勒贝格积分的区别与联系。

The emergence of Riemann integrals started with constructing a series of easily computable regions, dividing a given function into small intervals according to its defined domain, solving many problems and opening up new ideas. However, Riemann integration also has limitations, and Lebesgue integration overcomes the shortcomings of Riemann integration, making the operation of differentiation and integration more perfect. The emergence of Lebesgue integral solved many problems that could not be solved in previous integrals. Transitioning from Riemann integrals to Lebesgue integrals, taking a comprehensive view of both, will be more helpful for learning and understanding. This article aims to analyze the definitions, basic properties, limit operations, and existence conditions of two types of integrals, and discuss the shortcomings of Riemann integrals and targeted modifications to Lebesgue integrals through some practical operation examples, in order to further understand the comparison between Riemann integrals and Lebesgue integrals, and gain a deeper understanding of integrals. When writing this article, the literature research method will be used for analysis.