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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?

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    $\begingroup$ There is Ulam-Oxtoby theorem that says that if two measures $\mu$ and $\nu$ on $[0,1]$ are non-atomic, and every open set if of positive measure, then there is a homemorphism $h$ such that $\mu(A)=\nu(h(A))$. I cannot quite understand if there is any relation between this condition and topological conjugacy, but perhaps it is relevant $\endgroup$
    – erz
    Commented Apr 3, 2021 at 2:43
  • $\begingroup$ Thank you, that seems like it could be relevant indeed. Just to clarify, here A is an arbitrary measurable set, and the choice of h in general depends on A? $\endgroup$
    – Nate River
    Commented Apr 3, 2021 at 2:57
  • $\begingroup$ yes, $A$ is arbitrary measurable, but the whole point is that $h$ is independent of $A$. Check out Nishiura - Absolute Measurable Spaces, section 3.4 $\endgroup$
    – erz
    Commented Apr 3, 2021 at 4:06

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