# Is the image of a metric automorphism under a measure-preserving mapping also a metric automorphism?

I'm having my first tiny little bits of ergodic theory so please forgive the probable naivete of the question. So, I'm looking at the first page of chapter 8 of Cornfeld, Fomin, Sinai's "Ergodic Theory". They state:

Now let us show that an arbitrary automorphism $$T'$$ of the measure space $$(M', \mathscr{F}, \mu)$$ naturally gives rise to stationary random processes.

Consider some countable partition $$\xi=(C_1,\cdots,C_k),1\leq k\leq\infty$$

For the state space choose $$Y=(1,2,\cdots,k)$$ and put $$M=\prod_{n=-\infty}^{\infty}Y^{(n)}$$ ; $$Y^{(n)}=Y$$. Consider the map $$\phi:M'\rightarrow M$$ defined as follows: the $$n$$th coordinate of the point $$\phi x'$$ equals $$j$$ if and only if $$(T')^nx'\in C_j$$. The map $$\phi$$ is measurable. It transforms the measure $$\mu'$$ on $$M'$$ onto a certain measure $$\mu$$ on $$M$$: $$\mu(A)\equiv\mu'(\phi^{-1}A), A\in\mathscr A$$, while the transformation $$T'$$ is the shift on $$M$$. From the fact that $$T'$$ is an automorphism it follows that the random process $$(M,\mathscr A, \mu)$$ is stationary (i.e. its measure is shift-ivariant (e.d.)).

Now, I have problems with the bold text: first, $$T'$$ does not act on $$M$$, so I guess they mean: "$$\phi T'\phi^{-1}$$ is the shift on $$M$$"; then, I can't see how the fact that $$T'$$ preserves the measure $$\mu'$$ of $$M'$$ should guarantee me that $$\phi T'\phi^{-1}$$ preserves the $$M$$-measure $$\mu$$.

I actually tried to sketch a proof: $$\mu(B) = \mu'(\phi^{-1}(B))= \mu'(T'\phi^{-1}(B))$$ And also: $$\mu(\phi T'\phi^{-1}B) = \mu'(\phi^{-1}\phi T'\phi^{-1}B)$$ Now, if I got things right I would want the two to be equal; yet, it seems to me that this would be the case only if $$\phi$$ itself was an isomorphism, so that $$\phi^{-1}\phi\equiv id_{M'}$$.

What am I missing here?

Take a finite partition $$\xi = {C_1 , C_2 , \cdots , C_r }$$ of $$M$$ . It generates a stationary random process of probability theory with values $$1, 2, \cdots , r$$ if one uses the formula:
$$w_k(x) = j\quad$$ if $$\quad x ∈ T^{−k} C_j , −∞ < k < ∞ .$$