Abstract
In spite of the Lebesgue density theorem, there is a positive \({\delta}\) such that, for every measurable set \({A \subset \mathbb{R}}\) with \({\lambda (A) > 0}\) and \({\lambda (\mathbb{R} \setminus A) > 0}\), there is a point at which both the lower densities of \({A}\) and of the complement of \({A}\) are at least \({\delta}\). The problem of determining the supremum of possible values of this \({\delta}\) was studied by V. I. Kolyada, A. Szenes and others. It seems that the authors considered this quantity a feature of density. We show that it is connected rather with a choice of a differentiation basis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Bruckner, Differentiation of Integrals, Amer. Math. Monthly, 78 (1971).
Csörnyei M., Grahl J., O’Neil T. C.: Points of middle density in the real line. Real Anal. Exchange, 37, 243–248 (2012)
Filipczak M., Hejduk J.: On topologies associated with the Lebesgue measure. Tatra Mt. Math. Publ., 28, 187–197 (2004)
M. Filipczak, T. Filipczak and J. Hejduk, On the comparison of the density type topologies, Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, LII (2004), 37–46.
V. I. Kolyada, On the metric Darboux property, Analysis Math., 9 (1983), 291–312 (in Russian).
Kurka O.: Optimal quality of exceptional points for the Lebesgue density theorem. Acta Math. Hungar., 134, 209–268 (2012)
Szenes A.: Exceptional points for Lebesgue density theorem on the real line. Adv. Math., 226, 764–778 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Filipczak, M., Filipczak, T., Horbaczewska, G. et al. Remarks on exceptional points and differentiation bases. Acta Math. Hungar. 148, 370–385 (2016). https://doi.org/10.1007/s10474-016-0588-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-016-0588-z