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.