An introduction to fractal Lebesgue integral
-
Hemanta Kalita
and
Alireza K. Golmankhanehand
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
Acknowledgements
The authors would like to thank the reviewer’s for reading the manuscript carefully and making valuable suggestions that significantly improve the presentation of the paper.
References
[1] D. Bongiorno, Riemann-type definition of the improper integrals, Czechoslovak Math. J. 54(129) (2004), no. 3, 717–725. 10.1007/s10587-004-6420-xSearch in Google Scholar
[2] D. Bongiorno, Derivatives not first return integrable on a fractal set, Ric. Mat. 67 (2018), no. 2, 597–604. 10.1007/s11587-018-0390-zSearch in Google Scholar
[3] D. Bongiorno and G. Corrao, On the fundamental theorem of calculus for fractal sets, Fractals 23 (2015), no. 2, Article ID 1550008. 10.1142/S0218348X15500085Search in Google Scholar
[4] F. Burk, Lebesgue Measure and Integration, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1998. Search in Google Scholar
[5] A. M. Caetano, S. N. Chandler-Wilde, A. Gibbs, D. P. Hewett and A. Moiola, A Hausdorff-measure boundary element method for acoustic scattering by fractal screens, Numer. Math. 156 (2024), no. 2, 463–532. 10.1007/s00211-024-01399-7Search in Google Scholar
[6] M. Csörnyei, U. B. Darji, M. J. Evans and P. D. Humke, First-return integrals, J. Math. Anal. Appl. 305 (2005), no. 2, 546–559. 10.1016/j.jmaa.2004.11.051Search in Google Scholar
[7] A. Gibbs, D. P. Hewett and B. Major, Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets, Numer. Algorithms 97 (2024), no. 1, 311–343. 10.1007/s11075-023-01705-8Search in Google Scholar
[8]
M. Giona,
Fractal calculus on
[9]
A. K. Golmankhaneh,
Fractal Calculus and its Applications—
[10] A. K. Golmankhaneh and D. Baleanu, On a new measure on fractals, J. Inequal. Appl. 2013 (2013), Paper No. 522. 10.1186/1029-242X-2013-522Search in Google Scholar
[11] A. K. Golmankhaneh and D. Baleanu, Fractal calculus involving gauge function, Commun. Nonlinear Sci. Numer. Simul. 37 (2016), 125–130. 10.1016/j.cnsns.2016.01.007Search in Google Scholar
[12] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Grad. Stud. Math. 4, American Mathematical Society, Providence, 1994. 10.1090/gsm/004Search in Google Scholar
[13] R. Henstock, Definitions of Riemann type of the variational integrals, Proc. Lond. Math. Soc. (3) 11 (1961), 402–418. 10.1112/plms/s3-11.1.402Search in Google Scholar
[14] H. Jiang and W. Su, Some fundamental results of calculus on fractal sets, Commun. Nonlinear Sci. Numer. Simul. 3 (1998), no. 1, 22–26. 10.1016/S1007-5704(98)90054-5Search in Google Scholar
[15] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7(82) (1957), 418–449. 10.21136/CMJ.1957.100258Search in Google Scholar
[16] H. Lebesgue, Sur l’intégration des fonctions discontinues, Ann. Sci. Éc. Norm. Supér. (3) 27 (1910), 361–450. 10.24033/asens.624Search in Google Scholar
[17] A. Parvate and A. D. Gangal, Calculus on fractal subsets of real line. I. Formulation, Fractals 17 (2009), no. 1, 53–81. 10.1142/S0218348X09004181Search in Google Scholar
[18] S. Schwabik and G. Ye, Topics in Banach Space Integration, Ser. Real Anal. 10, World Scientific, Hackensack, 2005. 10.1142/9789812703286Search in Google Scholar
[19] C. Swartz, Introduction to Gauge Integrals, World Scientific, Singapore, 2001. 10.1142/9789812810656Search in Google Scholar
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