Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Mackey-Zimmer theorem“. This paper is part of our longer term project to develop “uncountable” versions of various theorems in ergodic theory; see this previous paper of Asgar and myself for the first paper in this series (and another paper will appear shortly).
In this case the theorem in question is the Mackey-Zimmer theorem, previously discussed in this blog post. This theorem gives an important classification of group and homogeneous extensions of measure-preserving systems. Let us first work in the (classical) setting of concrete measure-preserving systems. Let be a measure-preserving system for some group
, thus
is a (concrete) probability space and
is a group homomorphism from
to the automorphism group
of the probability space. (Here we are abusing notation by using
to refer both to the measure-preserving system and to the underlying set. In the notation of the paper we would instead distinguish these two objects as
and
respectively, reflecting two of the (many) categories one might wish to view
as a member of, but for sake of this informal overview we will not maintain such precise distinctions.) If
is a compact group, we define a (concrete) cocycle to be a collection of measurable functions
for
that obey the cocycle equation
-
is the Cartesian product of
and
;
-
is the product measure of
and Haar probability measure on
; and
- The action
is given by the formula
This group skew-product comes with a factor map
and a coordinate map
, which by (2) are related to the action via the identities
We can now generalize the notion of group skew-product by just working with the maps , and weakening the requirement that
be measure-preserving. Namely, define a group extension of
by
to be a measure-preserving system
equipped with a measure-preserving map
obeying (3) and a measurable map
obeying (4) for some cocycle
, such that the
-algebra of
is generated by
. There is also a more general notion of a homogeneous extension in which
takes values in
rather than
. Then every group skew-product
is a group extension of
by
, but not conversely. Here are some key counterexamples:
- (i) If
is a closed subgroup of
, and
is a cocycle taking values in
, then
can be viewed as a group extension of
by
, taking
to be the vertical coordinate
(viewing
now as an element of
). This will not be a skew-product by
because
pushes forward to the wrong measure on
: it pushes forward to
rather than
.
- (ii) If one takes the same example as (i), but twists the vertical coordinate
to another vertical coordinate
for some measurable “gauge function”
, then
is still a group extension by
, but now with the cocycle
replaced by the cohomologous cocycle
Again, this will not be a skew product by, because
pushes forward to a twisted version of
that is supported (at least in the case where
is compact and the cocycle
is continuous) on the
-bundle
.
- (iii) With the situation as in (i), take
to be the union
for some
outside of
, where we continue to use the action (2) and the standard vertical coordinate
but now use the measure
.
As it turns out, group extensions and homogeneous extensions arise naturally in the Furstenberg-Zimmer structural theory of measure-preserving systems; roughly speaking, every compact extension of is an inverse limit of group extensions. It is then of interest to classify such extensions.
Examples such as (iii) are annoying, but they can be excluded by imposing the additional condition that the system is ergodic – all invariant (or essentially invariant) sets are of measure zero or measure one. (An essentially invariant set is a measurable subset
of
such that
is equal modulo null sets to
for all
.) For instance, the system in (iii) is non-ergodic because the set
(or
) is invariant but has measure
. We then have the following fundamental result of Mackey and Zimmer:
Theorem 1 (Countable Mackey Zimmer theorem) Letbe a group,
be a concrete measure-preserving system, and
be a compact Hausdorff group. Assume that
is at most countable,
is a standard Borel space, and
is metrizable. Then every (concrete) ergodic group extension of
is abstractly isomorphic to a group skew-product (by some closed subgroup
of
), and every (concrete) ergodic homogeneous extension of
is similarly abstractly isomorphic to a homogeneous skew-product.
We will not define precisely what “abstractly isomorphic” means here, but it roughly speaking means “isomorphic after quotienting out the null sets”. A proof of this theorem can be found for instance in .
The main result of this paper is to remove the “countability” hypotheses from the above theorem, at the cost of working with opposite probability algebra systems rather than concrete systems. (We will discuss opposite probability algebras in a subsequent blog post relating to another paper in this series.)
Theorem 2 (Uncountable Mackey Zimmer theorem) Letbe a group,
be an opposite probability algebra measure-preserving system, and
be a compact Hausdorff group. Then every (abstract) ergodic group extension of
is abstractly isomorphic to a group skew-product (by some closed subgroup
of
), and every (abstract) ergodic homogeneous extension of
is similarly abstractly isomorphic to a homogeneous skew-product.
We plan to use this result in future work to obtain uncountable versions of the Furstenberg-Zimmer and Host-Kra structure theorems.
As one might expect, one locates a proof of Theorem 2 by finding a proof of Theorem 1 that does not rely too strongly on “countable” tools, such as disintegration or measurable selection, so that all of those tools can be replaced by “uncountable” counterparts. The proof we use is based on the one given in this previous post, and begins by comparing the system with the group extension
. As the examples (i), (ii) show, these two systems need not be isomorphic even in the ergodic case, due to the different probability measures employed. However one can relate the two after performing an additional averaging in
. More precisely, there is a canonical factor map
given by the formula
7 comments
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1 October, 2020 at 8:23 pm
Anonymous
After theorem 1 says “A proof of this theorem can be found for instance in .” I think a citation is supposed to go there.
[Corrected, thanks – T.]
2 October, 2020 at 9:12 pm
Anonymous
Hmm, it still looks the same from here, through 2 different browsers. Maybe a caching issue? I’ll try again tomorrow.
2 October, 2020 at 2:16 am
Anonymous
I’m getting a lot of “Formula does not parse” for this article :/
2 October, 2020 at 5:11 am
Anonymous
[Corrected, thanks – T.]
2 October, 2020 at 7:09 am
Anonymous
Maybe a linebreak just before Defintion 1.1.
2 October, 2020 at 7:46 pm
Anonymous
The very last words of the blog “, and” should be cut off.
[Corrected, thanks – T.]
4 October, 2020 at 5:17 pm
Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration | What's new
[…] Asgar Jamneshan and I have just uploaded to the arXiv our paper “Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration“. This paper arose from our longer-term project to systematically develop “uncountable” ergodic theory – ergodic theory in which the groups acting are not required to be countable, the probability spaces one acts on are not required to be standard Borel, or Polish, and the compact groups that arise in the structural theory (e.g., the theory of group extensions) are not required to be separable. One of the motivations of doing this is to allow ergodic theory results to be applied to ultraproducts of finite dynamical systems, which can then hopefully be transferred to establish combinatorial results with good uniformity properties. An instance of this is the uncountable Mackey-Zimmer theorem, discussed in this companion blog post. […]