Riemann Integral on Fractal Structures

José Fulgencio Gálvez-Rodríguez (), Cristina Martín-Aguado and Miguel Ángel Sánchez-Granero ()
Additional contact information
José Fulgencio Gálvez-Rodríguez: Department of Mathematics, University of Almería, 04120 Almería, Spain
Cristina Martín-Aguado: Department of Mathematics, University of Almería, 04120 Almería, Spain
Miguel Ángel Sánchez-Granero: Department of Mathematics, University of Almería, 04120 Almería, Spain

Mathematics, 2024, vol. 12, issue 2, 1-16

Abstract: In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ -algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ -measurable function is Riemann-integrable with respect to μ . Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of R n meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets.

Keywords: Riemann integral; Riemann-integrable; fractal structure; measure (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/2/310/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/2/310/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:2:p:310-:d:1321215

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().