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

for each

and all

. (One could weaken this requirement by only demanding the cocycle equation to hold for almost all

, rather than all

; we will effectively do so later in the post, when we move to opposite probability algebra systems.) Any such cocycle generates a
group skew-product 
of

, which is another measure-preserving system

where
The cocycle equation
(1) guarantees that

is a homomorphism, and the (left) invariance of Haar measure and Fubini’s theorem guarantees that the

remain measure preserving. There is also the more general notion of a
homogeneous skew-product 
in which the group

is replaced by the homogeneous space

for some closed subgroup of

, noting that

still comes with a left-action of

and a Haar measure. Group skew-products are very “explicit” ways to extend a system

, as everything is described by the cocycle

which is a relatively tractable object to manipulate. (This is not to say that the cohomology of measure-preserving systems is trivial, but at least there are many tools one can use to study them, such as the
Moore-Schmidt theorem discussed in this previous post.)
This group skew-product
comes with a factor map
and a coordinate map
, which by (2) are related to the action via the identities

and

where in
(4) we are implicitly working in the group of (concretely) measurable functions from

to

. Furthermore, the combined map

is measure-preserving (using the product measure on

), indeed the way we have constructed things this map is just the identity map.
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) Let
be 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) Let
be 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

This is a factor map not only of

-systems, but actually of

-systems, where the
opposite group 
to

acts (on the left) by right-multiplication of the second coordinate (this reversal of order is why we need to use the opposite group here). The key point is that the ergodicity properties of the system

are closely tied the group

that is “secretly” controlling the group extension. Indeed, in example (i), the invariant functions on

take the form

for some measurable

, while in example (ii), the invariant functions on

take the form

. In either case, the invariant factor is isomorphic to

, and can be viewed as a factor of the invariant factor of

, which is isomorphic to

. Pursuing this reasoning (using an abstract ergodic theorem of Alaoglu and Birkhoff, as discussed in
the previous post) one obtains the
Mackey range 
, and also obtains the quotient

of

to

in this process. The main remaining task is to lift the quotient

back up to a map

that stays measurable, in order to “untwist” a system that looks like (ii) to make it into one that looks like (i). In countable settings this is where a “
measurable selection theorem” would ordinarily be invoked, but in the uncountable setting such theorems are not available for concrete maps. However it turns out that they still remain available for abstract maps: any abstractly measurable map

from

to

has an abstractly measurable lift from

to

. To prove this we first use a canonical model for opposite probability algebras (which we will discuss in a companion post to this one, to appear shortly) to work with continuous maps (on a Stone space) rather than abstractly measurable maps. The measurable map

then induces a probability measure on

, formed by pushing forward

by the graphing map

. This measure in turn has several lifts up to a probability measure on

; for instance, one can construct such a measure

via the Riesz representation theorem by demanding

for all continuous functions

. This measure does not come from a graph of any single lift

, but is in some sense an “average” of the entire ensemble of these lifts. But it turns out one can invoke the
Krein-Milman theorem to pass to an extremal lifting measure which
does come from an (abstract) lift

, and this can be used as a substitute for a measurable selection theorem. A variant of this Krein-Milman argument can also be used to express any homogeneous extension as a quotient of a group extension, giving the second part of the Mackey-Zimmer theorem.
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