Lebesgue density and exceptional points

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 ${\Sigma }_3^0$-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 ${\Pi }_3^0$-complete; the set of all $[K]$ with $K$ compact is ${\Pi }_3^0$-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.

References in corpus (3)

• Local homogeneity and dimensions of measures
• Analytic sets of reals and the density function in the Cantor space
• Optimal quality of exceptional points for the Lebesgue density theorem

Cited by in corpus (1)

• Lebesgue's Density Theorem and definable selectors for ideals