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?