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 $\Sigma^{0}_{3}$-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.