Lebesgue Measure and Integration on Subsets of R^d
Article Sidebar
Page Numbers:
15-26
Digital Object Identifier:
10.58578/mjms.v3i1.3958
Viewed: 705 times
Downloaded: 230 times
LYAS Publisher cordially invites qualified professionals to serve as Editors or Reviewers. Your expertise will make an important contribution to maintaining and strengthening the academic quality of our publications. Interested applicants are kindly requested to complete the application form at the following link: Editors & Reviewers
Please feel free to contact us if you need any further information about the submission process or if you have any additional questions.
Authors:Nand Kishor Kumar1*, Chudamani Pokhrel2, Dipendra Prasad Yadav3
Copyright :Authors retain copyright and grant the journal right of first publication.
Main Article Content
Abstract
Henri Lebesgue, a French mathematician, discovered centuries ago that the Riemann Integral does not work well on unbounded functions. It prompts him to consider another way to integration known as Lebesgue Integral. This paper discusses the Riemann integral's shortcomings and introduce a more thorough concept of integration, the Lebesgue integral, repeated integration. There is also some debate about the Lebesgue measure, which determines the Lebesgue integral. Some examples are given, such as F_σ -set, G_δ -set and Cantor function. In this article, we first look into a unified theory for d-dimensional volume based on the concept of a measure, and then we will use that theory to build a stronger and more flexible theory for integration.
Citation Metrics:
Downloads
Citation Metrics & Similar Scopus Articles
-
Gangil G. (2027)Advanced optimization problems and solution methods for renewable energy integration - A bibliometric reviewUnconventional Resources, 17
-
Roy K.R. (2027)miRNA-mRNA multi-omics integration in breast cancer stagingAdvances in Biomarker Sciences and Technology, 9(1), 70-82
-
Sardar Khan S.K. (2027)MXene-based biosensors for biomedical diagnostics: From molecular recognition to wearable systemsAdvances in Biomarker Sciences and Technology, 9(1), 46-69
Article Details
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
References
The term volume is also used, more strictly, as a synonym of 3-dimensional volume
Lebesgue, H. (1902). Integral Longueur, Aire, Annali di Matematica Pura ed Applicata. 7: 231–359. doi:10.1007/BF02420592. S2CID 121256884.
Bartle, Robert G. (1995). The elements of integration and Lebesgue measure. Wiley Classics Library. New York: John Wiley & Sons Inc. xii+179. ISBN 0-471-04222-6. MR 1312157.
Lebesgue, H. (1904). Leçons sur l'intégration et la recherche des functions primitives". Lecons,2, Paris: Gauthier-Villars.
Heil, C. (2019). Introduction to Real Analysis, Springer, Cham, 17,1-480.
Stein. E.M. & Shakarchi, R. (2005). Real Analysis. NJ: Princeton University Press
Wheeden. R.L. & Zygmund, A. (1977). Measure and Integral. New York: Marcel Dekker.
Folland, G. B. (1999). Real Analysis, New York: Wiley, 2, 1-387.
Kesavan, S. (2019). Measure and Integration. Springer Singapore,1, 1-232.
William F. (2009). Trench. Introduction to real analysis.
Russell A. G. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock, 4. Ameri. Math. Soc.
Dennis D. W. (2008). William Mendenhall, and Richard L. Scheaffer. Mathematical statistics with applications. Brooks-Cole.
Interactive real analysis. http://www.mathcs.org/analysis/reals/. Accessed: 2017-05-10.