LIPSCHITZ ONE SETS MODULO SETS OF MEASURE ZERO

Buczolich, Zoltán and Hanson, Bruce and Maga, Balázs and Vértesy, Gáspár (2020) LIPSCHITZ ONE SETS MODULO SETS OF MEASURE ZERO. MATHEMATICA SLOVACA, 70 (3). pp. 567-584. ISSN 0139-9918

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Abstract

We denote the local "little" and "big" Lipschitz functions of a function f : R -> R by lip f and Lip f. In this paper we continue our research concerning the following question. Given a set E subset of R is it possible to find a continuous function f such that lip f = 1(E) or Lip f = 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 Lip1 set. On the other hand, we prove that there exists a measurable SUDT set E such that for any G(delta) set (E) over tilde satisfying vertical bar E Delta(E) over tilde vertical bar = 0 the set (E) over tilde does not have UDT. Combining these two results we obtain that there exist Lip1 sets not having UDT, that is, the converse of one of our earlier results does not hold. (C) 2020 Mathematical Institute Slovak Academy of Sciences

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