  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.
