TITLE: Does any measure preserving system have an invertible extension?
QUESTION [8 upvotes]: Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure
preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$
with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ is called a
$T$-invariant sub-$\sigma$-algebra. Show that the system
$\tilde{\mathsf{X}}=\{ \tilde{X},\tilde{B},\tilde{\mu},\tilde{T} \}$
defined by

$\tilde{X}=\{ x\in X^{\mathbb{Z}}\,|\, x_{k+1}=T(x_k)\,\text{for all } k\in \mathbb{Z} \}$;
$(\tilde{T}(x))_k = x_{k+1}$ for all $\,k\in \mathbb{Z}$ and $x\in \tilde{X}$;
$\tilde{\mu}(\{ x\in \tilde{X}\,|\,x_0\in A \}) =\mu(A)$ for any $A\in \mathcal{B}$, and $\tilde{\mu}$ is invariant under $\tilde{T}$;
$\tilde{B}$ is the smallest $\tilde{T}$-invariant $\sigma$-algebra
for which the map $\pi:x\mapsto x_0$ from $\tilde{X}$ to $X$ is
measurable;

is an invertible measure-preserving system, and that the map $\pi:x\mapsto x_0$
is a factor map. The system $\tilde{\mathsf{X}}$ is called the
invertible extension of $\mathsf{X}$.
This is exercise 2.1.7 of Ergodic theory--with a view towards number theory by Manfred Einsiedler and Thomas Ward (GTM 259). We have made the convention that all measure space considered is a probability measure. But I have some doubts about this construction.

How does one show that $\tilde{X}$ is not empty in the first place?
If one can indeed show 1, why does $\tilde{\mu}$ define a measure on $\tilde{B}$?
And if indeed 1 and 2 hold, how do we show that if $\tilde{T}^{-1}$ i.e. the right shift, is still measure preserving?

If some of these indeed can not be done, then is it really true that any measure preserving system has an invertible extension ? And if so, how can we construct one?

REPLY [2 votes]: Let me offer a different answer: I thought about it just now, and indeed I think checking that the Hahn-Kolmogorov consistency theorem applies is actually non-trivial, so let me give a reference to a nice exposition (which addresses you question entirely):
See the first three sections of chapter 5 of Parthasarathy, K. R. Probability measures on metric spaces. In particular Theorem 3.2 is what you want.
It constructs what is known as an inverse limit of a sequence $$X_0 \leftarrow X_1 \leftarrow \ldots $$ of surjective measure preserving maps between Borel probability spaces. In particular the invertible extension is the case where each $X_i=X$ and the mappings are $T$.  
I'm not sure how much we can relax the surjectivity requirement.