Lipschitz one sets modulo sets of measure zero

Rok, strany: 2020, 567 - 584

We denote the local ``little" and ``big" Lipschitz functions of a function $f: \R\to \R$ by $ \lip f$ and $ \Lip f$. In this paper we continue our research concerning the following question. Given a set $E {\subset} \R$ is it possible to find a continuous function $f$ such that $ \lip f=\mathbf{1}_E$ or $ \Lip f=\mathbf{1}_E$? In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role. In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a $\Lip 1$ set. On the other hand, we prove that there exists a measurable SUDT set $E$ such that for any $G_δ$ set $\widetilde{E}$ satisfying $|EΔ\widetilde{E}|=0$ the set $\widetilde{E}$ does not have UDT. Combining these two results we obtain that there exist $\Lip 1$ sets not having UDT, that is, the converse of one of our earlier results does not hold.

ISO 690:

Buczolich, Z., Hanson, B., Maga, B., Vértesy, G. 2020. Lipschitz one sets modulo sets of measure zero. In Mathematica Slovaca, vol. 70, no.3, pp. 567-584. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0372

APA:

Buczolich, Z., Hanson, B., Maga, B., Vértesy, G. (2020). Lipschitz one sets modulo sets of measure zero. Mathematica Slovaca, 70(3), 567-584. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0372

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava