TITLE: Furstenberg-Zimmer Theorem: non-invertible systems.
QUESTION [5 upvotes]: Questions

Is there a version of Furstenber-Zimmer Theorem for
non-invertible measure preserving systems?
Where can I find it?
What is the precise statement?

Background
In many works that reference the Furstenberg-Zimmer Theorem,
the theorem itself is not stated.
Authors usually cite the works of Furstenberg
(The structure of distal flows
and/or
Ergodic behavior of diagonal measures and a
theorem of Szemerédi on arithmetic progressions)
and Zimmer
(Extensions of ergodic group actions
and/or
Extensions of ergodic actions and generalized discrete spectrum).
The point is that in many places, the theorem is being used
for non-invertible systems.
This happens, for instance in
On Li-Yorke Pairs,
where the systems are assumed to be surjective, but not
necessarily invertible.
In this paper, for the proof of Theorem 2.1, the authors
use Furstenber-Zimmer Theorem.
As far as I understood, Zimmer's work deals with
group actions.
That is, invertible systems.
And for Furstenberg's Ergodic behaviour of diagonal measures [...],
he deals with regular measure preserving systems.
Unfortunately, Furstenberg and Zimmer (obviously) did not call their result
the Furstenberg-Zimmer Theorem.
In fact, it seems to me that
Furstenberg didn't even call it a theorem. :-P
I could find a precise statement of the theorem
for the invertible case at a
Terry Tao's post.
But I could not find any precise statement for the non-invertible case.

REPLY [3 votes]: Posted as requested - consult the book by Manfred Einsiedler and Tom Ward - "Ergodic Theory with a view towards number theory" - published in GTM, especially in ch 7.