Does any measure preserving system have an invertible extension?

Question

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.

1. How does one show that $\tilde{X}$ is not empty in the first place?
2. If one can indeed show 1, why does $\tilde{\mu}$ define a measure on $\tilde{B}$?
3. 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?

Answer 1

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.

Answer 2

I have thought about this before in the past, it is a big technical. Since eventually we are interested in Borel probability spaces, I will stick to these (indeed, this book eventually only cares about these, and in some places actually makes assertions that only hold for these). Let me answer your questions. I will start with the first one.

How does one show that $\tilde{X}$ is not empty in the first place?

Of course if $T$ is surjective then we are done. I thought that every measure preserving system is isomorphic to a "surjective" (but on second thought I am not convinced). We know that $T(X)$ is measurable in the completion of $\mu$, and from this it is easy to show that there exists Borel $X_1 \subset T(X)$ such that $\mu(X_1)=1$. Thus any system is "almost" surjective, but I am unsure how to get that it is isomorphic to a surjective one. For the rest of this answer, let's assume that $T$ is surjective (you still get an interesting result).

If one can indeed show 1, why does μ~ define a measure on B~? And if indeed 1 and 2 hold, how do we show that if T~−1 i.e. the right shift, is still measure preserving?

Of course the authors havn't defined the measure on the whole $\sigma$-algebra. The point is there exists a unique probability measure that satisfies those properties. If you have seen the construction of the product measure on $[0,1]^{\mathbb{N}}$, then this is an analogous sort of thing: you define the measure first on basic sets and then show it can be extended. To show the existence of such a measure, I suggest you use your favourite extension theorem, a good example is the one here http://en.wikipedia.org/wiki/Hahn-Kolmogorov_theorem . So apply this theorem to an easy algebra that generates $\tilde{\mathcal{B}}$, it is a bit involved to check all the details, but doable (just like in the case of infinite products).