# Lipschitz one sets modulo sets of measure zero

## TL;DR

This paper investigates the structure of Lipschitz sets modulo measure zero, showing that almost all measurable sets are Lipschitz 1 sets and exploring the properties of special sets related to uniform density types.

## Contribution

It proves that modulo measure zero, any measurable set coincides with a Lipschitz 1 set and constructs examples of Lipschitz 1 sets lacking uniform density type properties.

## Key findings

- Almost all measurable sets are Lipschitz 1 sets modulo measure zero.
- Existence of Lipschitz 1 sets without uniform density type.
- Counterexample to the converse of a previous result.

## Abstract

We denote the local "little" and "big" Lipschitz functions of a function $f: {{\mathbb R}}\to {{\mathbb R}}$ by $ {\mathrm {lip}}f$ and $ {\mathrm {Lip}}f$. In this paper we continue our research concerning the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}}f=\mathbf{1}_E$ or $ {\mathrm {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 ${\mathrm {Lip}} 1$ set. On the other hand, we prove that there exists a measurable SUDT set $E$ such that for any $G_\delta$ set $\widetilde{E}$ satisfying $|E\Delta\widetilde{E}|=0$ the set $\widetilde{E}$ does not have UDT. Combining these two results we obtain that there exists ${\mathrm {Lip}} 1$ sets not having UDT, that is, the converse of one of our earlier results does not hold.