# pr.probability – Classifying non atomic singular measures up to topological conjugacy

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?