An introduction to fractal Lebesgue integral

Kalita H., Golmankhanehand A. K.

Georgian Mathematical Journal, vol.33, no.2, pp.277-287, 2026 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 33 Issue: 2
  • Publication Date: 2026
  • Doi Number: 10.1515/gmj-2025-2067
  • Journal Name: Georgian Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, zbMATH
  • Page Numbers: pp.277-287
  • Keywords: fractal calculus, fractal functions, Fractal set, generalized fractal measure
  • Van Yüzüncü Yıl University Affiliated: No

Abstract

This manuscript explores various characteristics of generalized fractal measures. We expand the concept of fractal integrals in relation to step functions and examine their numerous properties. Notably, since all step functions are classified as simple functions, we apply the aforementioned generalized measure to introduce Lebesgue-type integrals, referred to as FL-integrals. Additionally, we demonstrate that all F α {F^{\alpha}} -integrable functions are FL-integrals. Lastly, we address the bounded convergence theorem within the context of fractals.