Kurt Gödel Research Center

2017 seminar talk: The density function: some remarks, results, and open problems

Talk held by Riccardo Camerlo (Politecnico di Torino, Italy) at the KGRC seminar on 2017-01-12.

Abstract

Given a Radon metric space $(X,d,\mu )$ and a measurable $A\subseteq X$, the density function associated with $A$ is the (partial) function on $X$ defined by \[ \mathcal D_A(x)=\lim_{\varepsilon\rightarrow 0^+} \frac{\mu (A\cap \mathcal B_{\varepsilon}(x))}{\mu ( \mathcal B_{\varepsilon }(x))} \] where $ \mathcal B_{\varepsilon }(x)$ is the open ball centered at $x$ of radius $\varepsilon $.

I will discuss properties of this function relevant to descriptive set theory, especially for Cantor space and the real line, together with some open questions.

Most results are joint work with A. Andretta and C. Costantini.