I am looking for a reference of the following (true) fact:

Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a homeomorphism $h:2^\omega\to 2^\omega$ such that $h(\mathcal N_\mu)=\mathcal N_\lambda$.

Here by $\mathcal N_\mu=\{B\subset 2^\omega:\mu(B)=0\}$ we denote the $\sigma$-ideal of sets of $\mu$-measure zero in the Cantor cube; and $h(\mathcal N_\mu)=\{h(B):B\in\mathcal N_\mu\}$.

A measure $\mu$ on a topological space $X$ is called continuous if $\mu(\{x\})=0$ for all $x\in X$; and strictly positive if $\mu(U)>0$ for any non-empty open set $U\subset X$.

I believe that such a useful (and not very difficult) fact should be already known and published somewhere. Right?

Remark. Two measures on the Cantor cube need not be homeomorphic. For example, the Haar measures on the zero-dimensional compact topological groups $(\mathbb Z/2\mathbb Z)^\omega$ and $(\mathbb Z/3\mathbb Z)^\omega$ are not homeomorphic as they have different sets of values on closed-and-open subsets (and this set of values is a topological invariant of a measure).


1 Answer 1


Finally I have found a good reference to this theorem, which can be easily derived from the following Theorem 2.12 in this paper of Akin:

Theorem (Akin, 1999). For any strictly positive continuous measures $\mu,\nu$ on the Cantor cube $2^\omega$ and any $\varepsilon>0$ there exists a homeomorphism $h$ of $2^\omega$ such that $$(1-\varepsilon)\mu(B)\le \nu(h(B))\le(1+\varepsilon)\mu(B)$$for any Borel subset $B$ of $2^\omega$.


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