Write $mathcal S$ for the set of probability measures on $[0, 1]$ that are non atomic and singular with respect to Lebesgue measure.

Two measures $mu$ and $nu$ in $mathcal S$ are said to be topologically conjugate if, denoting by $F_mu$ and $F_nu$ their respective distribution functions, there exist homeomorphisms $h$ and $g$ of $[0, 1]$ such that $h F_mu = F_nu

g$.

Topological conjugacy of measures thus forms an equivalence relationship on $mathcal S$. Intuitively, two measures are topologically conjugate if they differ only by a continuous change of coordinates in the domain and a continuous change of measure values.

Can we classify the measures in $mathcal S$ up to topological conjugacy?