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2012•Journal of Southwest China Normal UniversityRequires access

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

Bulin Zhang

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Abstract

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,+∞).

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What this paper is about

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,+∞).

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Available abstract

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,+∞).

Key concepts: Lebesgue integration, Riemann integral, Lebesgue–Stieltjes integration, Mathematics, Daniell integral, Riemann–Stieltjes integral, Monotone polygon, Pure mathematics

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