1. bookVolume 29 (2021): Issue 4 (December 2021)
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
access type Open Access

Relationship between the Riemann and Lebesgue Integrals

Published Online: 09 Jul 2022
Volume & Issue: Volume 29 (2021) - Issue 4 (December 2021)
Page range: 185 - 199
Accepted: 30 Sep 2021
Journal Details
License
Format
Journal
eISSN
1898-9934
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English
Summary

The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous article [5], we constructed a one-dimensional Lebesgue measure. The one-dimensional Lebesgue measure provides a measure of any intervals, which can be used to prove the well-known relationship [6] between the Riemann and Lebesgue integrals [1]. We also proved the relationship between the integral of a given measure and that of its complete measure. As the result of this work, the Lebesgue integral of a bounded real valued function in the Mizar system [2], [3] can be calculated by the Riemann integral.

Keywords

MSC

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[5] Noboru Endou. Reconstruction of the one-dimensional Lebesgue measure. Formalized Mathematics, 28(1):93–104, 2020. doi:10.2478/forma-2020-0008. Open DOISearch in Google Scholar

[6] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2nd edition, 1999. Search in Google Scholar

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