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Asgar Jamneshan and myself have just uploaded to the arXiv our preprint “The inverse theorem for the ${U^3}$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches“. This paper, which is a companion to another recent paper of ourselves and Or Shalom, studies the inverse theory for the third Gowers uniformity norm

$\displaystyle \| f \|_{U^3(G)}^8 = {\bf E}_{h_1,h_2,h_3,x \in G} \Delta_{h_1} \Delta_{h_2} \Delta_{h_3} f(x)$

on an arbitrary finite abelian group ${G}$, where ${\Delta_h f(x) := f(x+h) \overline{f(x)}}$ is the multiplicative derivative. Our main result is as follows:

Theorem 1 (Inverse theorem for ${U^3(G)}$) Let ${G}$ be a finite abelian group, and let ${f: G \rightarrow {\bf C}}$ be a ${1}$-bounded function with ${\|f\|_{U^3(G)} \geq \eta}$ for some ${0 < \eta \leq 1/2}$. Then:
• (i) (Correlation with locally quadratic phase) There exists a regular Bohr set ${B(S,\rho) \subset G}$ with ${|S| \ll \eta^{-O(1)}}$ and ${\exp(-\eta^{-O(1)}) \ll \rho \leq 1/2}$, a locally quadratic function ${\phi: B(S,\rho) \rightarrow {\bf R}/{\bf Z}}$, and a function ${\xi: G \rightarrow \hat G}$ such that

$\displaystyle {\bf E}_{x \in G} |{\bf E}_{h \in B(S,\rho)} f(x+h) e(-\phi(h)-\xi(x) \cdot h)| \gg \eta^{O(1)}.$

• (ii) (Correlation with nilsequence) There exists an explicit degree two filtered nilmanifold ${H/\Lambda}$ of dimension ${O(\eta^{-O(1)})}$, a polynomial map ${g: G \rightarrow H/\Lambda}$, and a Lipschitz function ${F: H/\Lambda \rightarrow {\bf C}}$ of constant ${O(\exp(\eta^{-O(1)}))}$ such that

$\displaystyle |{\bf E}_{x \in G} f(x) \overline{F}(g(x))| \gg \exp(-\eta^{-O(1)}).$

Such a theorem was proven by Ben Green and myself in the case when ${|G|}$ was odd, and by Samorodnitsky in the ${2}$-torsion case ${G = {\bf F}_2^n}$. In all cases one uses the “higher order Fourier analysis” techniques introduced by Gowers. After some now-standard manipulations (using for instance what is now known as the Balog-Szemerédi-Gowers lemma), one arrives (for arbitrary ${G}$) at an estimate that is roughly of the form

$\displaystyle |{\bf E}_{x \in G} {\bf E}_{h,k \in B(S,\rho)} f(x+h+k) b(x,k) b(x,h) e(-B(h,k))| \gg \eta^{O(1)}$

where ${b}$ denotes various ${1}$-bounded functions whose exact values are not too important, and ${B: B(S,\rho) \times B(S,\rho) \rightarrow {\bf R}/{\bf Z}}$ is a symmetric locally bilinear form. The idea is then to “integrate” this form by expressing it in the form

$\displaystyle B(h,k) = \phi(h+k) - \phi(h) - \phi(k) \ \ \ \ \ (1)$

for some locally quadratic ${\phi: B(S,\rho) \rightarrow {\bf C}}$; this then allows us to write the above correlation as

$\displaystyle |{\bf E}_{x \in G} {\bf E}_{h,k \in B(S,\rho)} f(x+h+k) e(-\phi(h+k)) b(x,k) b(x,h)| \gg \eta^{O(1)}$

(after adjusting the ${b}$ functions suitably), and one can now conclude part (i) of the above theorem using some linear Fourier analysis. Part (ii) follows by encoding locally quadratic phase functions as nilsequences; for this we adapt an algebraic construction of Manners.

So the key step is to obtain a representation of the form (1), possibly after shrinking the Bohr set ${B(S,\rho)}$ a little if needed. This has been done in the literature in two ways:

• When ${|G|}$ is odd, one has the ability to divide by ${2}$, and on the set ${2 \cdot B(S,\frac{\rho}{10}) = \{ 2x: x \in B(S,\frac{\rho}{10})\}}$ one can establish (1) with ${\phi(h) := B(\frac{1}{2} h, h)}$. (This is similar to how in single variable calculus the function ${x \mapsto \frac{1}{2} x^2}$ is a function whose second derivative is equal to ${1}$.)
• When ${G = {\bf F}_2^n}$, then after a change of basis one can take the Bohr set ${B(S,\rho)}$ to be ${{\bf F}_2^m}$ for some ${m}$, and the bilinear form can be written in coordinates as

$\displaystyle B(h,k) = \sum_{1 \leq i,j \leq m} a_{ij} h_i k_j / 2 \hbox{ mod } 1$

for some ${a_{ij} \in {\bf F}_2}$ with ${a_{ij}=a_{ji}}$. The diagonal terms ${a_{ii}}$ cause a problem, but by subtracting off the rank one form ${(\sum_{i=1}^m a_{ii} h_i) ((\sum_{i=1}^m a_{ii} k_i) / 2}$ one can write

$\displaystyle B(h,k) = \sum_{1 \leq i,j \leq m} b_{ij} h_i k_j / 2 \hbox{ mod } 1$

on the orthogonal complement of ${(a_{11},\dots,a_{mm})}$ for some coefficients ${b_{ij}=b_{ji}}$ which now vanish on the diagonal: ${b_{ii}=0}$. One can now obtain (1) on this complement by taking

$\displaystyle \phi(h) := \sum_{1 \leq i < j \leq m} b_{ij} h_i h_k / 2 \hbox{ mod } 1.$

In our paper we can now treat the case of arbitrary finite abelian groups ${G}$, by means of the following two new ingredients:

• (i) Using some geometry of numbers, we can lift the group ${G}$ to a larger (possibly infinite, but still finitely generated) abelian group ${G_S}$ with a projection map ${\pi: G_S \rightarrow G}$, and find a globally bilinear map ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ on the latter group, such that one has a representation

$\displaystyle B(\pi(x), \pi(y)) = \tilde B(x,y) \ \ \ \ \ (2)$

of the locally bilinear form ${B}$ by the globally bilinear form ${\tilde B}$ when ${x,y}$ are close enough to the origin.
• (ii) Using an explicit construction, one can show that every globally bilinear map ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ has a representation of the form (1) for some globally quadratic function ${\tilde \phi: G_S \rightarrow {\bf R}/{\bf Z}}$.

To illustrate (i), consider the Bohr set ${B(S,1/10) = \{ x \in {\bf Z}/N{\bf Z}: \|x/N\|_{{\bf R}/{\bf Z}} < 1/10\}}$ in ${G = {\bf Z}/N{\bf Z}}$ (where ${\|\|_{{\bf R}/{\bf Z}}}$ denotes the distance to the nearest integer), and consider a locally bilinear form ${B: B(S,1/10) \times B(S,1/10) \rightarrow {\bf R}/{\bf Z}}$ of the form ${B(x,y) = \alpha x y \hbox{ mod } 1}$ for some real number ${\alpha}$ and all integers ${x,y \in (-N/10,N/10)}$ (which we identify with elements of ${G}$. For generic ${\alpha}$, this form cannot be extended to a globally bilinear form on ${G}$; however if one lifts ${G}$ to the finitely generated abelian group

$\displaystyle G_S := \{ (x,\theta) \in {\bf Z}/N{\bf Z} \times {\bf R}: \theta = x/N \hbox{ mod } 1 \}$

(with projection map ${\pi: (x,\theta) \mapsto x}$) and introduces the globally bilinear form ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ by the formula

$\displaystyle \tilde B((x,\theta),(y,\sigma)) = N^2 \alpha \theta \sigma \hbox{ mod } 1$

then one has (2) when ${\theta,\sigma}$ lie in the interval ${(-1/10,1/10)}$. A similar construction works for higher rank Bohr sets.

To illustrate (ii), the key case turns out to be when ${G_S}$ is a cyclic group ${{\bf Z}/N{\bf Z}}$, in which case ${\tilde B}$ will take the form

$\displaystyle \tilde B(x,y) = \frac{axy}{N} \hbox{ mod } 1$

for some integer ${a}$. One can then check by direct construction that (1) will be obeyed with

$\displaystyle \tilde \phi(x) = \frac{a \binom{x}{2}}{N} - \frac{a x \binom{N}{2}}{N^2} \hbox{ mod } 1$

regardless of whether ${N}$ is even or odd. A variant of this construction also works for ${{\bf Z}}$, and the general case follows from a short calculation verifying that the claim (ii) for any two groups ${G_S, G'_S}$ implies the corresponding claim (ii) for the product ${G_S \times G'_S}$.

This concludes the Fourier-analytic proof of Theorem 1. In this paper we also give an ergodic theory proof of (a qualitative version of) Theorem 1(ii), using a correspondence principle argument adapted from this previous paper of Ziegler, and myself. Basically, the idea is to randomly generate a dynamical system on the group ${G}$, by selecting an infinite number of random shifts ${g_1, g_2, \dots \in G}$, which induces an action of the infinitely generated free abelian group ${{\bf Z}^\omega = \bigcup_{n=1}^\infty {\bf Z}^n}$ on ${G}$ by the formula

$\displaystyle T^h x := x + \sum_{i=1}^\infty h_i g_i.$

Much as the law of large numbers ensures the almost sure convergence of Monte Carlo integration, one can show that this action is almost surely ergodic (after passing to a suitable Furstenberg-type limit ${X}$ where the size of ${G}$ goes to infinity), and that the dynamical Host-Kra-Gowers seminorms of that system coincide with the combinatorial Gowers norms of the original functions. One is then well placed to apply an inverse theorem for the third Host-Kra-Gowers seminorm ${U^3(X)}$ for ${{\bf Z}^\omega}$-actions, which was accomplished in the companion paper to this one. After doing so, one almost gets the desired conclusion of Theorem 1(ii), except that after undoing the application of the Furstenberg correspondence principle, the map ${g: G \rightarrow H/\Lambda}$ is merely an almost polynomial rather than a polynomial, which roughly speaking means that instead of certain derivatives of ${g}$ vanishing, they instead are merely very small outside of a small exceptional set. To conclude we need to invoke a “stability of polynomials” result, which at this level of generality was first established by Candela and Szegedy (though we also provide an independent proof here in an appendix), which roughly speaking asserts that every approximate polynomial is close in measure to an actual polynomial. (This general strategy is also employed in the Candela-Szegedy paper, though in the absence of the ergodic inverse theorem input that we rely upon here, the conclusion is weaker in that the filtered nilmanifold ${H/\Lambda}$ is replaced with a general space known as a “CFR nilspace”.)

This transference principle approach seems to work well for the higher step cases (for instance, the stability of polynomials result is known in arbitrary degree); the main difficulty is to establish a suitable higher step inverse theorem in the ergodic theory setting, which we hope to do in future research.

Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group ${\Gamma}$ acting in a measure-preserving fashion on a probability space ${(X,\mu)}$, thus each group element ${\gamma \in \Gamma}$ gives rise to a measure-preserving map ${T^\gamma: X \rightarrow X}$. Define the third Gowers-Host-Kra seminorm ${\|f\|_{U^3(X)}}$ of a function ${f \in L^\infty(X)}$ via the formula

$\displaystyle \|f\|_{U^3(X)}^8 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2,h_3 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2,\omega_3 \in \{0,1\}}$

$\displaystyle {\mathcal C}^{\omega_1+\omega_2+\omega_3} f(T^{\omega_1 h_1 + \omega_2 h_2 + \omega_3 h_3} x)\ d\mu(x)$

where ${\Phi_n}$ is a Folner sequence for ${\Gamma}$ and ${{\mathcal C}: z \mapsto \overline{z}}$ is the complex conjugation map. One can show that this limit exists and is independent of the choice of Folner sequence, and that the ${\| \|_{U^3(X)}}$ seminorm is indeed a seminorm. A Conze-Lesigne system is an ergodic measure-preserving system in which the ${U^3(X)}$ seminorm is in fact a norm, thus ${\|f\|_{U^3(X)}>0}$ whenever ${f \in L^\infty(X)}$ is non-zero. Informally, this means that when one considers a generic parallelepiped in a Conze–Lesigne system ${X}$, the location of any vertex of that parallelepiped is more or less determined by the location of the other seven vertices. These are the important systems to understand in order to study “complexity two” patterns, such as arithmetic progressions of length four. While not all systems ${X}$ are Conze-Lesigne systems, it turns out that they always have a maximal factor ${Z^2(X)}$ that is a Conze-Lesigne system, known as the Conze-Lesigne factor or the second Host-Kra-Ziegler factor of the system, and this factor controls all the complexity two recurrence properties of the system.

The analogous theory in complexity one is well understood. Here, one replaces the ${U^3(X)}$ norm by the ${U^2(X)}$ norm

$\displaystyle \|f\|_{U^2(X)}^4 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2 \in \{0,1\}} {\mathcal C}^{\omega_1+\omega_2} f(T^{\omega_1 h_1 + \omega_2 h_2} x)\ d\mu(x)$

and the ergodic systems for which ${U^2}$ is a norm are called Kronecker systems. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group ${Z}$ equipped with Haar probability measure and a translation action ${T^\gamma \colon z \mapsto z + \phi(\gamma)}$ for some homomorphism ${\phi: \Gamma \rightarrow Z}$ with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).

We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) nilsystems, in which the space ${X}$ is a quotient ${G/\Lambda}$ of a two-step nilpotent Lie group ${G}$ by a lattice ${\Lambda}$ (equipped with Haar probability measure), and the action is given by a translation ${T^\gamma x = \phi(\gamma) x}$ for some group homomorphism ${\phi: \Gamma \rightarrow G}$. For instance, the Heisenberg ${{\bf Z}}$-nilsystem

$\displaystyle \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix}$

with a shift of the form

$\displaystyle Tx = \begin{pmatrix} 1 & \alpha & 0 \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end{pmatrix} x$

for ${\alpha,\beta}$ two real numbers with ${1,\alpha,\beta}$ linearly independent over ${{\bf Q}}$, is a Conze-Lesigne system. As the base case of a well known result of Host and Kra, it is shown in fact that all Conze-Lesigne ${{\bf Z}}$-systems are inverse limits of nilsystems (previous results in this direction were obtained by Conze-Lesigne, Furstenberg-Weiss, and others). Similar results are known for ${\Gamma}$-systems when ${\Gamma}$ is finitely generated, thanks to the thesis work of Griesmer (with further proofs by Gutman-Lian and Candela-Szegedy). However, this is not the case once ${\Gamma}$ is not finitely generated; as a recent example of Shalom shows, Conze-Lesigne systems need not be the inverse limit of nilsystems in this case.

Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which ${G}$ is a locally compact Polish group rather than a Lie group:

Theorem 1 (Classification of Conze-Lesigne systems) Let ${\Gamma}$ be a countable abelian group, and ${X}$ an ergodic measure-preserving ${\Gamma}$-system. Then ${X}$ is a Conze-Lesigne system if and only if it is the inverse limit of translational systems ${G/\Lambda}$, where ${G}$ is a nilpotent locally compact Polish group of nilpotency class two, and ${\Lambda}$ is a lattice in ${G}$ (and also a lattice in the commutator group ${[G,G]}$), with ${G/\Lambda}$ equipped with the Haar probability measure and a translation action ${T^\gamma x = \phi(\gamma) x}$ for some homomorphism ${\phi: \Gamma \rightarrow G}$.

In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm ${U^3(G)}$ for an arbitrary finite abelian group ${G}$ (with no restrictions on the order of ${G}$, in particular our result handles the case of even and odd ${|G|}$ in a unified fashion). In principle, having a higher order version of this theorem will similarly allow us to derive inverse theorems for ${U^{s+1}(G)}$ norms for arbitrary ${s}$ and finite abelian ${G}$; we hope to investigate this further in future work.

We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension ${Z \rtimes_\rho K}$, where ${Z}$ is a Kronecker system (a rotational system on a compact abelian group ${Z = (Z,+)}$ and translation action ${\phi: \Gamma \rightarrow Z}$), ${K = (K,+)}$ is another compact abelian group, and the cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is a collection of measurable maps ${\rho_\gamma: Z \rightarrow K}$ obeying the cocycle equation

$\displaystyle \rho_{\gamma_1+\gamma_2}(x) = \rho_{\gamma_1}(T^{\gamma_2} x) + \rho_{\gamma_2}(x) \ \ \ \ \ (1)$

for almost all ${x \in Z}$. Furthermore, ${\rho}$ is of “type two”, which means in this concrete setting that it obeys an additional equation

$\displaystyle \rho_\gamma(x + z_1 + z_2) - \rho_\gamma(x+z_1) - \rho_\gamma(x+z_2) + \rho_\gamma(x) \ \ \ \ \ (2)$

$\displaystyle = F(x + \phi(\gamma), z_1, z_2) - F(x,z_1,z_2)$

for all ${\gamma \in \Gamma}$ and almost all ${x,z_1,z_2 \in Z}$, and some measurable function ${F: Z^3 \rightarrow K}$; roughly speaking this asserts that ${\phi_\gamma}$ is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where ${Z}$ is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group ${K}$ is a circle ${{\bf T} = {\bf R}/{\bf Z}}$, one can use some Fourier analysis to convert (2) into the more tractable Conze-Lesigne equation

$\displaystyle \rho_\gamma(x+z) - \rho_\gamma(x) = F_z(x+\phi(\gamma)) - F_z(x) + c_z(\gamma) \ \ \ \ \ (3)$

for all ${\gamma \in \Gamma}$, all ${z \in Z}$, and almost all ${x \in Z}$, where for each ${z}$, ${F_z: Z \rightarrow K}$ is a measurable function, and ${c_z: \Gamma \rightarrow K}$ is a homomorphism. (For technical reasons it is often also convenient to enforce that ${F_z, c_z}$ depend in a measurable fashion on ${z}$; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group ${K}$ (as long as one has the measurability in ${z}$ mentioned previously), but the converse turns out to fail for some groups ${K}$, such as solenoid groups (e.g., inverse limits of ${{\bf R}/2^n{\bf Z}}$ as ${n \rightarrow \infty}$), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$. Putting together the ${K={\bf T}}$ and ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$ cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian Lie groups ${K}$ (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system ${X}$ as a translational system ${G/\Lambda}$, where the Host-Kra group ${G}$ is the set of all pairs ${(z, F_z)}$ that solve an equation of the form (3) (with these pairs acting on ${X \equiv Z \rtimes_\rho K}$ by the law ${(z,F_z) \cdot (x,k) := (x+z, k+F_z(x))}$), and ${\Lambda}$ is the stabiliser of a point in this system. This then establishes the theorem in the case when ${K}$ is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of ${X}$ form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)

There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of ${G}$ on ${X}$ is technically only a near-action rather than a genuine action, and as such one cannot directly define ${\Lambda}$ to be the stabiliser of a point without running into multiple problems. To fix this, one has to pass to a topological model of ${X}$ in which the action becomes continuous, and the stabilizer becomes well defined, although one then has to work a little more to check that the action is still transitive. This can be done via Gelfand duality; we proceed using a mixture of a construction from this book of Host and Kra, and the machinery in this recent paper of Asgar and myself.

Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$. As this group embeds into the torus ${{\bf T}}$, it is easy to use existing methods obtain (3) but with the homomorphism ${c_z}$ and the function ${F_z}$ taking values in ${{\bf R}/{\bf Z}}$ rather than in ${\frac{1}{N}{\bf Z}/{\bf Z}}$. The main task is then to fix up the homomorphism ${c_z}$ so that it takes values in ${\frac{1}{N}{\bf Z}/{\bf Z}}$, that is to say that ${Nc_z}$ vanishes. This only needs to be done locally near the origin, because the claim is easy when ${z}$ lies in the dense subgroup ${\phi(\Gamma)}$ of ${Z}$, and also because the claim can be shown to be additive in ${z}$. Near the origin one can leverage the Steinhaus lemma to make ${c_z}$ depend linearly (or more precisely, homomorphically) on ${z}$, and because the cocycle ${\rho}$ already takes values in ${\frac{1}{N}{\bf Z}/{\bf Z}}$, ${N\rho}$ vanishes and ${Nc_z}$ must be an eigenvalue of the system ${Z}$. But as ${Z}$ was assumed to be separable, there are only countably many eigenvalues, and by another application of Steinhaus and linearity one can then make ${Nc_z}$ vanish on an open neighborhood of the identity, giving the claim.

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.

In the course of this project, we ran into the obstacle that many foundational results, such as the Riesz representation theorem, often require one or more of these countability hypotheses when encountered in textbooks. Other technical issues also arise in the uncountable setting, such as the need to distinguish the Borel ${\sigma}$-algebra from the (two different types of) Baire ${\sigma}$-algebra. As such we needed to spend some time reviewing and synthesizing the known literature on some foundational results of “uncountable” measure theory, which led to this paper. As such, most of the results of this paper are already in the literature, either explicitly or implicitly, in one form or another (with perhaps the exception of the canonical disintegration, which we discuss below); we view the main contribution of this paper as presenting the results in a coherent and unified fashion. In particular we found that the language of category theory was invaluable in clarifying and organizing all the different results. In subsequent work we (and some other authors) will use the results in this paper for various applications in uncountable ergodic theory.

The foundational results covered in this paper can be divided into a number of subtopics (Gelfand duality, Baire ${\sigma}$-algebras and Riesz representation, canonical models, and canonical disintegration), which we discuss further below the fold.

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 ${Y = (Y, \mu_Y, T_Y)}$ be a measure-preserving system for some group ${\Gamma}$, thus ${(Y,\mu_Y)}$ is a (concrete) probability space and ${T_Y : \gamma \rightarrow T_Y^\gamma}$ is a group homomorphism from ${\Gamma}$ to the automorphism group ${\mathrm{Aut}(Y,\mu_Y)}$ of the probability space. (Here we are abusing notation by using ${Y}$ 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 ${Y_{\mathbf{ConcPrb}_\Gamma}}$ and ${Y_{\mathbf{Set}}}$ respectively, reflecting two of the (many) categories one might wish to view ${Y}$ as a member of, but for sake of this informal overview we will not maintain such precise distinctions.) If ${K}$ is a compact group, we define a (concrete) cocycle to be a collection of measurable functions ${\rho_\gamma : Y \rightarrow K}$ for ${\gamma \in \Gamma}$ that obey the cocycle equation

$\displaystyle \rho_{\gamma \gamma'}(y) = \rho_\gamma(T_Y^{\gamma'} y) \rho_{\gamma'}(y) \ \ \ \ \ (1)$

for each ${\gamma,\gamma' \in \Gamma}$ and all ${y \in Y}$. (One could weaken this requirement by only demanding the cocycle equation to hold for almost all ${y}$, rather than all ${y}$; 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 ${X = Y \rtimes_\rho K}$ of ${Y}$, which is another measure-preserving system ${(X, \mu_X, T_X)}$ where
• ${X = Y \times K}$ is the Cartesian product of ${Y}$ and ${K}$;
• ${\mu_X = \mu_Y \times \mathrm{Haar}_K}$ is the product measure of ${\mu_Y}$ and Haar probability measure on ${K}$; and
• The action ${T_X: \gamma \rightarrow }$ is given by the formula

$\displaystyle T_X^\gamma(y,k) := (T_Y^\gamma y, \rho_\gamma(y) k). \ \ \ \ \ (2)$

The cocycle equation (1) guarantees that ${T_X}$ is a homomorphism, and the (left) invariance of Haar measure and Fubini’s theorem guarantees that the ${T_X^\gamma}$ remain measure preserving. There is also the more general notion of a homogeneous skew-product ${X \times Y \times_\rho K/L}$ in which the group ${K}$ is replaced by the homogeneous space ${K/L}$ for some closed subgroup of ${L}$, noting that ${K/L}$ still comes with a left-action of ${K}$ and a Haar measure. Group skew-products are very “explicit” ways to extend a system ${Y}$, as everything is described by the cocycle ${\rho}$ 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 ${X}$ comes with a factor map ${\pi: X \rightarrow Y}$ and a coordinate map ${\theta: X \rightarrow K}$, which by (2) are related to the action via the identities

$\displaystyle \pi \circ T_X^\gamma = T_Y^\gamma \circ \pi \ \ \ \ \ (3)$

and

$\displaystyle \theta \circ T_X^\gamma = (\rho_\gamma \circ \pi) \theta \ \ \ \ \ (4)$

where in (4) we are implicitly working in the group of (concretely) measurable functions from ${Y}$ to ${K}$. Furthermore, the combined map ${(\pi,\theta): X \rightarrow Y \times K}$ is measure-preserving (using the product measure on ${Y \times K}$), 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 ${\pi, \theta}$, and weakening the requirement that ${(\pi,\theta)}$ be measure-preserving. Namely, define a group extension of ${Y}$ by ${K}$ to be a measure-preserving system ${(X,\mu_X, T_X)}$ equipped with a measure-preserving map ${\pi: X \rightarrow Y}$ obeying (3) and a measurable map ${\theta: X \rightarrow K}$ obeying (4) for some cocycle ${\rho}$, such that the ${\sigma}$-algebra of ${X}$ is generated by ${\pi,\theta}$. There is also a more general notion of a homogeneous extension in which ${\theta}$ takes values in ${K/L}$ rather than ${K}$. Then every group skew-product ${Y \rtimes_\rho K}$ is a group extension of ${Y}$ by ${K}$, but not conversely. Here are some key counterexamples:

• (i) If ${H}$ is a closed subgroup of ${K}$, and ${\rho}$ is a cocycle taking values in ${H}$, then ${Y \rtimes_\rho H}$ can be viewed as a group extension of ${Y}$ by ${K}$, taking ${\theta: Y \rtimes_\rho H \rightarrow K}$ to be the vertical coordinate ${\theta(y,h) = h}$ (viewing ${h}$ now as an element of ${K}$). This will not be a skew-product by ${K}$ because ${(\theta,\pi)}$ pushes forward to the wrong measure on ${Y \times K}$: it pushes forward to ${\mu_Y \times \mathrm{Haar}_H}$ rather than ${\mu_Y \times \mathrm{Haar}_K}$.
• (ii) If one takes the same example as (i), but twists the vertical coordinate ${\theta}$ to another vertical coordinate ${\tilde \theta(y,h) := \Phi(y) \theta(y,h)}$ for some measurable “gauge function” ${\Phi: Y \rightarrow K}$, then ${Y \rtimes_\rho H}$ is still a group extension by ${K}$, but now with the cocycle ${\rho}$ replaced by the cohomologous cocycle

$\displaystyle \tilde \rho_\gamma(y) := \Phi(T_Y^\gamma y) \rho_\gamma \Phi(y)^{-1}.$

Again, this will not be a skew product by ${K}$, because ${(\theta,\pi)}$ pushes forward to a twisted version of ${\mu_Y \times \mathrm{Haar}_H}$ that is supported (at least in the case where ${Y}$ is compact and the cocycle ${\rho}$ is continuous) on the ${H}$-bundle ${\bigcup_{y \in Y} \{y\} \times \Phi(y) H}$.
• (iii) With the situation as in (i), take ${X}$ to be the union ${X = Y \rtimes_\rho H \uplus Y \rtimes_\rho Hk \subset Y \times K}$ for some ${k \in K}$ outside of ${H}$, where we continue to use the action (2) and the standard vertical coordinate ${\theta: (y,k) \mapsto k}$ but now use the measure ${\mu_Y \times (\frac{1}{2} \mathrm{Haar}_H + \frac{1}{2} \mathrm{Haar}_{Hk})}$.

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 ${Y}$ 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 ${(X,\mu_X,T_X)}$ is ergodic – all invariant (or essentially invariant) sets are of measure zero or measure one. (An essentially invariant set is a measurable subset ${E}$ of ${X}$ such that ${T^\gamma E}$ is equal modulo null sets to ${E}$ for all ${\gamma \in \Gamma}$.) For instance, the system in (iii) is non-ergodic because the set ${Y \times H}$ (or ${Y \times Hk}$) is invariant but has measure ${1/2}$. We then have the following fundamental result of Mackey and Zimmer:

Theorem 1 (Countable Mackey Zimmer theorem) Let ${\Gamma}$ be a group, ${Y}$ be a concrete measure-preserving system, and ${K}$ be a compact Hausdorff group. Assume that ${\Gamma}$ is at most countable, ${Y}$ is a standard Borel space, and ${K}$ is metrizable. Then every (concrete) ergodic group extension of ${Y}$ is abstractly isomorphic to a group skew-product (by some closed subgroup ${H}$ of ${K}$), and every (concrete) ergodic homogeneous extension of ${Y}$ 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 ${\Gamma}$ be a group, ${Y}$ be an opposite probability algebra measure-preserving system, and ${K}$ be a compact Hausdorff group. Then every (abstract) ergodic group extension of ${Y}$ is abstractly isomorphic to a group skew-product (by some closed subgroup ${H}$ of ${K}$), and every (abstract) ergodic homogeneous extension of ${Y}$ 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 ${X}$ with the group extension ${Y \rtimes_\rho K}$. 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 ${K}$. More precisely, there is a canonical factor map ${\Pi: X \rtimes_1 K \rightarrow Y \times_\rho K}$ given by the formula

$\displaystyle \Pi(x, k) := (\pi(x), \theta(x) k).$

This is a factor map not only of ${\Gamma}$-systems, but actually of ${\Gamma \times K^{op}}$-systems, where the opposite group ${K^{op}}$ to ${K}$ 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 ${Y \times_\rho K}$ are closely tied the group ${H}$ that is “secretly” controlling the group extension. Indeed, in example (i), the invariant functions on ${Y \times_\rho K}$ take the form ${(y,k) \mapsto f(Hk)}$ for some measurable ${f: H \backslash K \rightarrow {\bf C}}$, while in example (ii), the invariant functions on ${Y \times_{\tilde \rho} K}$ take the form ${(y,k) \mapsto f(H \Phi(y)^{-1} k)}$. In either case, the invariant factor is isomorphic to ${H \backslash K}$, and can be viewed as a factor of the invariant factor of ${X \rtimes_1 K}$, which is isomorphic to ${K}$. Pursuing this reasoning (using an abstract ergodic theorem of Alaoglu and Birkhoff, as discussed in the previous post) one obtains the Mackey range ${H}$, and also obtains the quotient ${\tilde \Phi: Y \rightarrow K/H}$ of ${\Phi: Y \rightarrow K}$ to ${K/H}$ in this process. The main remaining task is to lift the quotient ${\tilde \Phi}$ back up to a map ${\Phi: Y \rightarrow K}$ 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 ${\tilde \Phi}$ from ${Y}$ to ${K/H}$ has an abstractly measurable lift from ${Y}$ to ${K}$. 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 ${\tilde \Phi}$ then induces a probability measure on ${Y \times K/H}$, formed by pushing forward ${\mu_Y}$ by the graphing map ${y \mapsto (y,\tilde \Phi(y))}$. This measure in turn has several lifts up to a probability measure on ${Y \times K}$; for instance, one can construct such a measure ${\overline{\mu}}$ via the Riesz representation theorem by demanding

$\displaystyle \int_{Y \times K} f(y,k) \overline{\mu}(y,k) := \int_Y (\int_{\tilde \Phi(y) H} f(y,k)\ d\mathrm{Haar}_{\tilde \Phi(y) H})\ d\mu_Y(y)$

for all continuous functions ${f}$. This measure does not come from a graph of any single lift ${\Phi: Y \rightarrow K}$, 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 ${\Phi}$, 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.

Asgar Jamneshan and I have just uploaded to the arXiv our paper “An uncountable Moore-Schmidt theorem“. This paper revisits a classical theorem of Moore and Schmidt in measurable cohomology of measure-preserving systems. To state the theorem, let ${X = (X,{\mathcal X},\mu)}$ be a probability space, and ${\mathrm{Aut}(X, {\mathcal X}, \mu)}$ be the group of measure-preserving automorphisms of this space, that is to say the invertible bimeasurable maps ${T: X \rightarrow X}$ that preserve the measure ${\mu}$: ${T_* \mu = \mu}$. To avoid some ambiguity later in this post when we introduce abstract analogues of measure theory, we will refer to measurable maps as concrete measurable maps, and measurable spaces as concrete measurable spaces. (One could also call ${X = (X,{\mathcal X}, \mu)}$ a concrete probability space, but we will not need to do so here as we will not be working explicitly with abstract probability spaces.)

Let ${\Gamma = (\Gamma,\cdot)}$ be a discrete group. A (concrete) measure-preserving action of ${\Gamma}$ on ${X}$ is a group homomorphism ${\gamma \mapsto T^\gamma}$ from ${\Gamma}$ to ${\mathrm{Aut}(X, {\mathcal X}, \mu)}$, thus ${T^1}$ is the identity map and ${T^{\gamma_1} \circ T^{\gamma_2} = T^{\gamma_1 \gamma_2}}$ for all ${\gamma_1,\gamma_2 \in \Gamma}$. A large portion of ergodic theory is concerned with the study of such measure-preserving actions, especially in the classical case when ${\Gamma}$ is the integers (with the additive group law).

Let ${K = (K,+)}$ be a compact Hausdorff abelian group, which we can endow with the Borel ${\sigma}$-algebra ${{\mathcal B}(K)}$. A (concrete measurable) ${K}$cocycle is a collection ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ of concrete measurable maps ${\rho_\gamma: X \rightarrow K}$ obeying the cocycle equation

$\displaystyle \rho_{\gamma_1 \gamma_2}(x) = \rho_{\gamma_1} \circ T^{\gamma_2}(x) + \rho_{\gamma_2}(x) \ \ \ \ \ (1)$

for ${\mu}$-almost every ${x \in X}$. (Here we are glossing over a measure-theoretic subtlety that we will return to later in this post – see if you can spot it before then!) Cocycles arise naturally in the theory of group extensions of dynamical systems; in particular (and ignoring the aforementioned subtlety), each cocycle induces a measure-preserving action ${\gamma \mapsto S^\gamma}$ on ${X \times K}$ (which we endow with the product of ${\mu}$ with Haar probability measure on ${K}$), defined by

$\displaystyle S^\gamma( x, k ) := (T^\gamma x, k + \rho_\gamma(x) ).$

This connection with group extensions was the original motivation for our study of measurable cohomology, but is not the focus of the current paper.

A special case of a ${K}$-valued cocycle is a (concrete measurable) ${K}$-valued coboundary, in which ${\rho_\gamma}$ for each ${\gamma \in \Gamma}$ takes the special form

$\displaystyle \rho_\gamma(x) = F \circ T^\gamma(x) - F(x)$

for ${\mu}$-almost every ${x \in X}$, where ${F: X \rightarrow K}$ is some measurable function; note that (ignoring the aforementioned subtlety), every function of this form is automatically a concrete measurable ${K}$-valued cocycle. One of the first basic questions in measurable cohomology is to try to characterize which ${K}$-valued cocycles are in fact ${K}$-valued coboundaries. This is a difficult question in general. However, there is a general result of Moore and Schmidt that at least allows one to reduce to the model case when ${K}$ is the unit circle ${\mathbf{T} = {\bf R}/{\bf Z}}$, by taking advantage of the Pontryagin dual group ${\hat K}$ of characters ${\hat k: K \rightarrow \mathbf{T}}$, that is to say the collection of continuous homomorphisms ${\hat k: k \mapsto \langle \hat k, k \rangle}$ to the unit circle. More precisely, we have

Theorem 1 (Countable Moore-Schmidt theorem) Let ${\Gamma}$ be a discrete group acting in a concrete measure-preserving fashion on a probability space ${X}$. Let ${K}$ be a compact Hausdorff abelian group. Assume the following additional hypotheses:

• (i) ${\Gamma}$ is at most countable.
• (ii) ${X}$ is a standard Borel space.
• (iii) ${K}$ is metrisable.

Then a ${K}$-valued concrete measurable cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is a concrete coboundary if and only if for each character ${\hat k \in \hat K}$, the ${\mathbf{T}}$-valued cocycles ${\langle \hat k, \rho \rangle = ( \langle \hat k, \rho_\gamma \rangle )_{\gamma \in \Gamma}}$ are concrete coboundaries.

The hypotheses (i), (ii), (iii) are saying in some sense that the data ${\Gamma, X, K}$ are not too “large”; in all three cases they are saying in some sense that the data are only “countably complicated”. For instance, (iii) is equivalent to ${K}$ being second countable, and (ii) is equivalent to ${X}$ being modeled by a complete separable metric space. It is because of this restriction that we refer to this result as a “countable” Moore-Schmidt theorem. This theorem is a useful tool in several other applications, such as the Host-Kra structure theorem for ergodic systems; I hope to return to these subsequent applications in a future post.

Let us very briefly sketch the main ideas of the proof of Theorem 1. Ignore for now issues of measurability, and pretend that something that holds almost everywhere in fact holds everywhere. The hard direction is to show that if each ${\langle \hat k, \rho \rangle}$ is a coboundary, then so is ${\rho}$. By hypothesis, we then have an equation of the form

$\displaystyle \langle \hat k, \rho_\gamma(x) \rangle = \alpha_{\hat k} \circ T^\gamma(x) - \alpha_{\hat k}(x) \ \ \ \ \ (2)$

for all ${\hat k, \gamma, x}$ and some functions ${\alpha_{\hat k}: X \rightarrow {\mathbf T}}$, and our task is then to produce a function ${F: X \rightarrow K}$ for which

$\displaystyle \rho_\gamma(x) = F \circ T^\gamma(x) - F(x)$

for all ${\gamma,x}$.

Comparing the two equations, the task would be easy if we could find an ${F: X \rightarrow K}$ for which

$\displaystyle \langle \hat k, F(x) \rangle = \alpha_{\hat k}(x) \ \ \ \ \ (3)$

for all ${\hat k, x}$. However there is an obstruction to this: the left-hand side of (3) is additive in ${\hat k}$, so the right-hand side would have to be also in order to obtain such a representation. In other words, for this strategy to work, one would have to first establish the identity

$\displaystyle \alpha_{\hat k_1 + \hat k_2}(x) - \alpha_{\hat k_1}(x) - \alpha_{\hat k_2}(x) = 0 \ \ \ \ \ (4)$

for all ${\hat k_1, \hat k_2, x}$. On the other hand, the good news is that if we somehow manage to obtain the equation, then we can obtain a function ${F}$ obeying (3), thanks to Pontryagin duality, which gives a one-to-one correspondence between ${K}$ and the homomorphisms of the (discrete) group ${\hat K}$ to ${\mathbf{T}}$.

Now, it turns out that one cannot derive the equation (4) directly from the given information (2). However, the left-hand side of (2) is additive in ${\hat k}$, so the right-hand side must be also. Manipulating this fact, we eventually arrive at

$\displaystyle (\alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2}) \circ T^\gamma(x) = (\alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2})(x).$

In other words, we don’t get to show that the left-hand side of (4) vanishes, but we do at least get to show that it is ${\Gamma}$-invariant. Now let us assume for sake of argument that the action of ${\Gamma}$ is ergodic, which (ignoring issues about sets of measure zero) basically asserts that the only ${\Gamma}$-invariant functions are constant. So now we get a weaker version of (4), namely

$\displaystyle \alpha_{\hat k_1 + \hat k_2}(x) - \alpha_{\hat k_1}(x) - \alpha_{\hat k_2}(x) = c_{\hat k_1, \hat k_2} \ \ \ \ \ (5)$

for some constants ${c_{\hat k_1, \hat k_2} \in \mathbf{T}}$.

Now we need to eliminate the constants. This can be done by the following group-theoretic projection. Let ${L^0({\bf X} \rightarrow {\bf T})}$ denote the space of concrete measurable maps ${\alpha}$ from ${{\bf X}}$ to ${{\bf T}}$, up to almost everywhere equivalence; this is an abelian group where the various terms in (5) naturally live. Inside this group we have the subgroup ${{\bf T}}$ of constant functions (up to almost everywhere equivalence); this is where the right-hand side of (5) lives. Because ${{\bf T}}$ is a divisible group, there is an application of Zorn’s lemma (a good exercise for those who are not acquainted with these things) to show that there exists a retraction ${w: L^0({\bf X} \rightarrow {\bf T}) \rightarrow {\bf T}}$, that is to say a group homomorphism that is the identity on the subgroup ${{\bf T}}$. We can use this retraction, or more precisely the complement ${\alpha \mapsto \alpha - w(\alpha)}$, to eliminate the constant in (5). Indeed, if we set

$\displaystyle \tilde \alpha_{\hat k}(x) := \alpha_{\hat k}(x) - w(\alpha_{\hat k})$

then from (5) we see that

$\displaystyle \tilde \alpha_{\hat k_1 + \hat k_2}(x) - \tilde \alpha_{\hat k_1}(x) - \tilde \alpha_{\hat k_2}(x) = 0$

while from (2) one has

$\displaystyle \langle \hat k, \rho_\gamma(x) \rangle = \tilde \alpha_{\hat k} \circ T^\gamma(x) - \tilde \alpha_{\hat k}(x)$

and now the previous strategy works with ${\alpha_{\hat k}}$ replaced by ${\tilde \alpha_{\hat k}}$. This concludes the sketch of proof of Theorem 1.

In making the above argument rigorous, the hypotheses (i)-(iii) are used in several places. For instance, to reduce to the ergodic case one relies on the ergodic decomposition, which requires the hypothesis (ii). Also, most of the above equations only hold outside of a set of measure zero, and the hypothesis (i) and the hypothesis (iii) (which is equivalent to ${\hat K}$ being at most countable) to avoid the problem that an uncountable union of sets of measure zero could have positive measure (or fail to be measurable at all).

My co-author Asgar Jamneshan and I are working on a long-term project to extend many results in ergodic theory (such as the aforementioned Host-Kra structure theorem) to “uncountable” settings in which hypotheses analogous to (i)-(iii) are omitted; thus we wish to consider actions on uncountable groups, on spaces that are not standard Borel, and cocycles taking values in groups that are not metrisable. Such uncountable contexts naturally arise when trying to apply ergodic theory techniques to combinatorial problems (such as the inverse conjecture for the Gowers norms), as one often relies on the ultraproduct construction (or something similar) to generate an ergodic theory translation of these problems, and these constructions usually give “uncountable” objects rather than “countable” ones. (For instance, the ultraproduct of finite groups is a hyperfinite group, which is usually uncountable.). This paper marks the first step in this project by extending the Moore-Schmidt theorem to the uncountable setting.

If one simply drops the hypotheses (i)-(iii) and tries to prove the Moore-Schmidt theorem, several serious difficulties arise. We have already mentioned the loss of the ergodic decomposition and the possibility that one has to control an uncountable union of null sets. But there is in fact a more basic problem when one deletes (iii): the addition operation ${+: K \times K \rightarrow K}$, while still continuous, can fail to be measurable as a map from ${(K \times K, {\mathcal B}(K) \otimes {\mathcal B}(K))}$ to ${(K, {\mathcal B}(K))}$! Thus for instance the sum of two measurable functions ${F: X \rightarrow K}$ need not remain measurable, which makes even the very definition of a measurable cocycle or measurable coboundary problematic (or at least unnatural). This phenomenon is known as the Nedoma pathology. A standard example arises when ${K}$ is the uncountable torus ${{\mathbf T}^{{\bf R}}}$, endowed with the product topology. Crucially, the Borel ${\sigma}$-algebra ${{\mathcal B}(K)}$ generated by this uncountable product is not the product ${{\mathcal B}(\mathbf{T})^{\otimes {\bf R}}}$ of the factor Borel ${\sigma}$-algebras (the discrepancy ultimately arises from the fact that topologies permit uncountable unions, but ${\sigma}$-algebras do not); relating to this, the product ${\sigma}$-algebra ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$ is not the same as the Borel ${\sigma}$-algebra ${{\mathcal B}(K \times K)}$, but is instead a strict sub-algebra. If the group operations on ${K}$ were measurable, then the diagonal set

$\displaystyle K^\Delta := \{ (k,k') \in K \times K: k = k' \} = \{ (k,k') \in K \times K: k - k' = 0 \}$

would be measurable in ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$. But it is an easy exercise in manipulation of ${\sigma}$-algebras to show that if ${(X, {\mathcal X}), (Y, {\mathcal Y})}$ are any two measurable spaces and ${E \subset X \times Y}$ is measurable in ${{\mathcal X} \otimes {\mathcal Y}}$, then the fibres ${E_x := \{ y \in Y: (x,y) \in E \}}$ of ${E}$ are contained in some countably generated subalgebra of ${{\mathcal Y}}$. Thus if ${K^\Delta}$ were ${{\mathcal B}(K) \otimes {\mathcal B}(K)}$-measurable, then all the points of ${K}$ would lie in a single countably generated ${\sigma}$-algebra. But the cardinality of such an algebra is at most ${2^{\alpha_0}}$ while the cardinality of ${K}$ is ${2^{2^{\alpha_0}}}$, and Cantor’s theorem then gives a contradiction.

To resolve this problem, we give ${K}$ a coarser ${\sigma}$-algebra than the Borel ${\sigma}$-algebra, namely the Baire ${\sigma}$-algebra ${{\mathcal B}^\otimes(K)}$, thus coarsening the measurable space structure on ${K = (K,{\mathcal B}(K))}$ to a new measurable space ${K_\otimes := (K, {\mathcal B}^\otimes(K))}$. In the case of compact Hausdorff abelian groups, ${{\mathcal B}^{\otimes}(K)}$ can be defined as the ${\sigma}$-algebra generated by the characters ${\hat k: K \rightarrow {\mathbf T}}$; for more general compact abelian groups, one can define ${{\mathcal B}^{\otimes}(K)}$ as the ${\sigma}$-algebra generated by all continuous maps into metric spaces. This ${\sigma}$-algebra is equal to ${{\mathcal B}(K)}$ when ${K}$ is metrisable but can be smaller for other ${K}$. With this measurable structure, ${K_\otimes}$ becomes a measurable group; it seems that once one leaves the metrisable world that ${K_\otimes}$ is a superior (or at least equally good) space to work with than ${K}$ for analysis, as it avoids the Nedoma pathology. (For instance, from Plancherel’s theorem, we see that if ${m_K}$ is the Haar probability measure on ${K}$, then ${L^2(K,m_K) = L^2(K_\otimes,m_K)}$ (thus, every ${K}$-measurable set is equivalent modulo ${m_K}$-null sets to a ${K_\otimes}$-measurable set), so there is no damage to Plancherel caused by passing to the Baire ${\sigma}$-algebra.

Passing to the Baire ${\sigma}$-algebra ${K_\otimes}$ fixes the most severe problems with an uncountable Moore-Schmidt theorem, but one is still faced with an issue of having to potentially take an uncountable union of null sets. To avoid this sort of problem, we pass to the framework of abstract measure theory, in which we remove explicit mention of “points” and can easily delete all null sets at a very early stage of the formalism. In this setup, the category of concrete measurable spaces is replaced with the larger category of abstract measurable spaces, which we formally define as the opposite category of the category of ${\sigma}$-algebras (with Boolean algebra homomorphisms). Thus, we define an abstract measurable space to be an object of the form ${{\mathcal X}^{\mathrm{op}}}$, where ${{\mathcal X}}$ is an (abstract) ${\sigma}$-algebra and ${\mathrm{op}}$ is a formal placeholder symbol that signifies use of the opposite category, and an abstract measurable map ${T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Y}^{\mathrm{op}}}$ is an object of the form ${(T^*)^{\mathrm{op}}}$, where ${T^*: {\mathcal Y} \rightarrow {\mathcal X}}$ is a Boolean algebra homomorphism and ${\mathrm{op}}$ is again used as a formal placeholder; we call ${T^*}$ the pullback map associated to ${T}$.  [UPDATE: It turns out that this definition of a measurable map led to technical issues.  In a forthcoming revision of the paper we also impose the requirement that the abstract measurable map be $\sigma$-complete (i.e., it respects countable joins).] The composition ${S \circ T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Z}^{\mathrm{op}}}$ of two abstract measurable maps ${T: {\mathcal X}^{\mathrm{op}} \rightarrow {\mathcal Y}^{\mathrm{op}}}$, ${S: {\mathcal Y}^{\mathrm{op}} \rightarrow {\mathcal Z}^{\mathrm{op}}}$ is defined by the formula ${S \circ T := (T^* \circ S^*)^{\mathrm{op}}}$, or equivalently ${(S \circ T)^* = T^* \circ S^*}$.

Every concrete measurable space ${X = (X,{\mathcal X})}$ can be identified with an abstract counterpart ${{\mathcal X}^{op}}$, and similarly every concrete measurable map ${T: X \rightarrow Y}$ can be identified with an abstract counterpart ${(T^*)^{op}}$, where ${T^*: {\mathcal Y} \rightarrow {\mathcal X}}$ is the pullback map ${T^* E := T^{-1}(E)}$. Thus the category of concrete measurable spaces can be viewed as a subcategory of the category of abstract measurable spaces. The advantage of working in the abstract setting is that it gives us access to more spaces that could not be directly defined in the concrete setting. Most importantly for us, we have a new abstract space, the opposite measure algebra ${X_\mu}$ of ${X}$, defined as ${({\bf X}/{\bf N})^*}$ where ${{\bf N}}$ is the ideal of null sets in ${{\bf X}}$. Informally, ${X_\mu}$ is the space ${X}$ with all the null sets removed; there is a canonical abstract embedding map ${\iota: X_\mu \rightarrow X}$, which allows one to convert any concrete measurable map ${f: X \rightarrow Y}$ into an abstract one ${[f]: X_\mu \rightarrow Y}$. One can then define the notion of an abstract action, abstract cocycle, and abstract coboundary by replacing every occurrence of the category of concrete measurable spaces with their abstract counterparts, and replacing ${X}$ with the opposite measure algebra ${X_\mu}$; see the paper for details. Our main theorem is then

Theorem 2 (Uncountable Moore-Schmidt theorem) Let ${\Gamma}$ be a discrete group acting abstractly on a ${\sigma}$-finite measure space ${X}$. Let ${K}$ be a compact Hausdorff abelian group. Then a ${K_\otimes}$-valued abstract measurable cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is an abstract coboundary if and only if for each character ${\hat k \in \hat K}$, the ${\mathbf{T}}$-valued cocycles ${\langle \hat k, \rho \rangle = ( \langle \hat k, \rho_\gamma \rangle )_{\gamma \in \Gamma}}$ are abstract coboundaries.

With the abstract formalism, the proof of the uncountable Moore-Schmidt theorem is almost identical to the countable one (in fact we were able to make some simplifications, such as avoiding the use of the ergodic decomposition). A key tool is what we call a “conditional Pontryagin duality” theorem, which asserts that if one has an abstract measurable map ${\alpha_{\hat k}: X_\mu \rightarrow {\bf T}}$ for each ${\hat k \in K}$ obeying the identity ${ \alpha_{\hat k_1 + \hat k_2} - \alpha_{\hat k_1} - \alpha_{\hat k_2} = 0}$ for all ${\hat k_1,\hat k_2 \in \hat K}$, then there is an abstract measurable map ${F: X_\mu \rightarrow K_\otimes}$ such that ${\alpha_{\hat k} = \langle \hat k, F \rangle}$ for all ${\hat k \in \hat K}$. This is derived from the usual Pontryagin duality and some other tools, most notably the completeness of the ${\sigma}$-algebra of ${X_\mu}$, and the Sikorski extension theorem.

We feel that it is natural to stay within the abstract measure theory formalism whenever dealing with uncountable situations. However, it is still an interesting question as to when one can guarantee that the abstract objects constructed in this formalism are representable by concrete analogues. The basic questions in this regard are:

• (i) Suppose one has an abstract measurable map ${f: X_\mu \rightarrow Y}$ into a concrete measurable space. Does there exist a representation of ${f}$ by a concrete measurable map ${\tilde f: X \rightarrow Y}$? Is it unique up to almost everywhere equivalence?
• (ii) Suppose one has a concrete cocycle that is an abstract coboundary. When can it be represented by a concrete coboundary?

For (i) the answer is somewhat interesting (as I learned after posing this MathOverflow question):

• If ${Y}$ does not separate points, or is not compact metrisable or Polish, there can be counterexamples to uniqueness. If ${Y}$ is not compact or Polish, there can be counterexamples to existence.
• If ${Y}$ is a compact metric space or a Polish space, then one always has existence and uniqueness.
• If ${Y}$ is a compact Hausdorff abelian group, one always has existence.
• If ${X}$ is a complete measure space, then one always has existence (from a theorem of Maharam).
• If ${X}$ is the unit interval with the Borel ${\sigma}$-algebra and Lebesgue measure, then one has existence for all compact Hausdorff ${Y}$ assuming the continuum hypothesis (from a theorem of von Neumann) but existence can fail under other extensions of ZFC (from a theorem of Shelah, using the method of forcing).
• For more general ${X}$, existence for all compact Hausdorff ${Y}$ is equivalent to the existence of a lifting from the ${\sigma}$-algebra ${\mathcal{X}/\mathcal{N}}$ to ${\mathcal{X}}$ (or, in the language of abstract measurable spaces, the existence of an abstract retraction from ${X}$ to ${X_\mu}$).
• It is a long-standing open question (posed for instance by Fremlin) whether it is relatively consistent with ZFC that existence holds whenever ${Y}$ is compact Hausdorff.

Our understanding of (ii) is much less complete:

• If ${K}$ is metrisable, the answer is “always” (which among other things establishes the countable Moore-Schmidt theorem as a corollary of the uncountable one).
• If ${\Gamma}$ is at most countable and ${X}$ is a complete measure space, then the answer is again “always”.

In view of the answers to (i), I would not be surprised if the full answer to (ii) was also sensitive to axioms of set theory. However, such set theoretic issues seem to be almost completely avoided if one sticks with the abstract formalism throughout; they only arise when trying to pass back and forth between the abstract and concrete categories.