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).



A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
Finally I have found a good reference to this theorem, which can be easily derived from the following Theorem 2.12 in [this paper](http://topology.auburn.edu/tp/reprints/v24/tp24201.pdf) 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 $2^\omega$.


