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

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for ${\mathrm{\Sigma }}_1^1$ 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.