从新视角看Lebesgue积分与Riemann积分的关系

The Relationship Between Lebesgue Integral and Riemann Integral from a New Perspective

证明了:任何一个非负Lebesgue可积函数的Lebesgue积分都可以表示成一个单调递减函数的Riemann积分(含Riemann瑕积分、Riemann无穷区间积分);任何一个Lebesgue可积函数的积分都可以表示成两个单调递减函数之差在(0,+∞)上的Riemann积分,或一个在(-∞,0)和(0,+∞)上单调递减函数的Riemann积分.

It has been proved in this paper that the Lebesgue integral of any non-negative Lebesgue integrable function can be expressed as the Riemann integral of a monotone decreasing function（including the Riemann improper integral and the Riemann infinite interval integral）.The integral of any Lebesgue integrable function can be expressed as either the Riemann integral of the difference between two monotone decreasing functions defined in（0,＋∞）or the Riemann integral of a monotone decreasing function respectively in（-∞,0） and（0,＋∞）.