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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.