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I'm working on a project where I define a new sort of measure defined as the following: $$m_\phi(E)=m(\phi^{-1}(E))\;,$$ where $E \subset C$ for the ternary Cantor set $C$. Mind you that $$\phi: \text{binary sequences in}\ [0,1] \rightarrow C\;.$$ This measure is defined for the following $\sigma$-algebra:
$$\mathcal{M_\phi}= \{E \subset C \mid \phi^{-1}(E) \in \mathcal{M}\}\;,$$ where $\mathcal{M}$ is the $\sigma$-algebra of Lebesgue measurable sets. Keeping the ternary Cantor set $C$ as my "universe," I built a measure space $(C, \mathcal{M_\phi}, m_\phi)$ from which I concluded that $m_\phi(C)= 1$.

So this means that the measure space I built is in fact a probability measure space. My question is how could I construct a measure-preserving transformation for my measure space? I did read and got some basic ideas relating to what measure-preserving transformations are in general, but so far it seems I'm getting almost nowhere. Could this transformation perhaps be some sort of fractional linear map or group action? I did read somewhere that $C$ could be taken as the set of $2$-adic integers, so perhaps I could relate my measure to $2$-adic measure, but how do we account for convergence when it comes to the set of $2$-adic integers?

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Do you mean that $\phi$ is the map that takes $\sum_ne_n2^{-n}$ to $\sum_n2e_n3^{-n}$, where each $e_n\in\{0,1\}$? – Brian M. Scott Feb 22 '12 at 0:08
@BrianM.Scott: You got it! – Libertron Feb 22 '12 at 0:10
There are many such maps. Try for instance $x \to 3x [1]$, that's probably the simplest (beside the identity). – D. Thomine Feb 22 '12 at 0:13
up vote 1 down vote accepted

Two straightforward possibilities:

  1. For any $n\ge 0$ send $x$ to $3^nx$ reduced mod $1$; the case $n=1$ was D. Thomine’s suggestion in the comments.

  2. Let $F$ be any finite set of positive integers, and let $$\varphi_F:\{0,2\}\to\{0,2\}:d\mapsto\begin{cases}2-d,&\text{if }d\in F\\d,&\text{if }d\notin F\;.\end{cases}$$ If $$x=\sum_{n\ge 1}\frac{d_n}{3^n}\in C\;,$$ where each $d_n\in\{0,2\}$, let $$f_F(x)=\sum_{n\ge 1}\frac{\varphi_F(d_n)}{3^n}\;.$$ These functions simply shuffle some of the ‘blocks’ of $C$. For example, $f_{\{1\}}$ swaps the left and right halves of $C$ while preserving the order within each half.

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