<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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.</p>
</abstract>

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.

Relationship between the Riemann and Lebesgue Integrals

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"Relationship between the Riemann and Lebesgue Integrals"}

The goal of this article is to clarify the relationship between Riemann and Lebesgue integrals. In previous article [5], we constructed a one-dimensional...

Relationship between the Riemann and Lebesgue Integrals

Published Online: Jul 09, 2022

Page range: 185 - 199

Accepted: Sep 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0018

Keywords
Riemann integrals, Lebesgue integrals

© 2022 Noboru Endou, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Relationship between the Riemann and Lebesgue Integrals

Published Online: Jul 09, 2022

Page range: 185 - 199

Accepted: Sep 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0018

KeywordsRiemann integrals, Lebesgue integrals

© 2022 Noboru Endou, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
Riemann integrals, Lebesgue integrals