广义Riemann积分中的Lebesgue方法

Lebesgue Methods In Generalized Riemann Integral

在实际问题和数学分析后续课程(如概率论)中,经常出现广义Riemann积分。但是我们发现,现有教科书上对此类积分的研究都是基于定积分的思想方法,要求被积函数有一定的光滑性,这大大限制了广义积分的研究范围。该文研究Lebesgue积分方法在广义Riemann积分的收敛性判别和计算以及含参量广义Riemann积分性质等问题中的应用。通过理论与实例结合,充分说明了Lebesgue方法的简便与灵活。因此,我们在学习广义Riemann积分时,不应拘泥于教科书上的现有知识和方法,应该拓宽思路,合理结合其他的课程。

Generalized Riemann integral is very useful in practical problems and subsequent courses of mathematical analysis（such as Probability theory）.However,the study of such points in our textbooks is based on the methods of definite integral,requiring the functions with certain smoothness,which greatly limits the research scope of generalized integral.In this paper,we discuss some applications of Lebesgue methods in generalized Riemann integral about convergence and calculation as well as generalized Riemann integral with parameters.Combined theory with examples,we show the simplicity and flexibility of Lebesgue methods. When we study the generalized Riemann integral,therefore,should not be constrained by the existing knowledge and method of textbook.We should broaden the train of thought and combine with other courses appropriately.