Lebesgue density and exceptional points

TL;DR

This paper investigates the descriptive set-theoretic complexity of density points and exceptional points in the measure algebra of the Lebesgue measure on the Cantor space, revealing high complexity levels and constructing special sets with unique density properties.

Contribution

It establishes the complexity classifications of sets of points with undefined or special densities and demonstrates measure-preserving embeddings of Cantor space into Polish spaces.

Findings

Set of points with undefined density is $oldsymbol{ m ext{Σ}}^{0}_{3}$-complete.

Set of points where density exists and differs from 0 or 1 is $oldsymbol{ m ext{Π}}^{0}_{3}$-complete.

Existence of a set in $oldsymbol{ m ext{R}}$ with density either 0, 1, or undefined at every point.

Abstract

Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points $x$ such that the density of $x$ at $K$ exists and it is different from $0$ or $1$ is $\Pi^{0}_{3}$-complete; the set of all $[K]$ with $K$ compact is $\Pi^{0}_{3}$-complete. There is a set (which can be taken to be open or closed) in $\mathbb R$ such that the density of any point is either $0$ or $1$, or else undefined. Conversely, if a subset of $\mathbb R^n$ is such that the density exists at every point, then the value $1/2$ is always attained. On the route to this result we show that Cantor space can be embedded in a measured Polish space in a measure-preserving fashion.