Analytic sets of reals and the density function in the Cantor space

Abstract

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact and $r \in ( 0 ; 1 )$ being the density of some point with respect to $K$. This result yields that the set of all $K$ such that the range of its density function is $S \cup \{ 0 , 1 \}$, for some fixed uncountable analytic set $S \subseteq ( 0 ; 1 )$, is $\Pi^{1}_{2}$-complete.