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.

— 1. Gelfand duality —

Given a compact Hausdorff space ${X}$, one can form the space ${C(X)}$ of continuous functions from ${X}$ to ${{\bf C}}$; such functions are necessarily bounded and compactly supported, and they form a commutative unital ${C^*}$-algebra. Conversely, given a commutative unital ${C^*}$-algebra ${{\mathcal A}}$, one can produce the Gelfand spectrum ${\mathtt{Spec}({\mathcal A})}$, which we define here as the collection of unital ${C^*}$-homomorphisms ${\lambda \colon {\mathcal A} \rightarrow {\bf C}}$ from ${{\mathcal A}}$ to ${{\bf C}}$. This spectrum can be viewed as a subset of ${{\bf C}^{\mathcal A}}$ and inherits a topology from the product topology on that latter space that turns the spectrum into a compact Hausdorff space. The classical Gelfand duality between compact Hausdorff spaces and unital commutative ${C^*}$-algebras asserts that these two operations ${X \mapsto C(X)}$ and ${{\mathcal A} \mapsto \mathtt{Spec}(A)}$ “essentially invert” each other: ${\mathtt{Spec}(C(X))}$ is homeomorphic to ${X}$, and ${C(\mathtt{Spec}({\mathcal A}))}$ is isomorphic as a commutative unital ${C^*}$-algebra to ${{\mathcal A}}$. In fact there is a more precise statement: the operations ${C}$, ${\mathtt{Spec}}$ are in fact contravariant functors that form a duality of categories between the category ${\mathbf{CH}}$ of compact Hausdorff spaces and the category ${\mathbf{CC^*Alg}_1}$ of commutative unital ${C^*}$-algebras. This duality of categories asserts, roughly speaking, that the operations ${C}$, ${\mathtt{Spec}}$ and the Gelfand duality isomorphisms interact in the “natural” fashion with respect to morphisms: for instance, any continuous map ${T: X \rightarrow Y}$ between compact Hausdorff spaces induces a commutative unital ${C^*}$-homomorphism ${C(T): C(Y) \rightarrow C(X)}$ defined by the pullback map ${C(T) f := f \circ T}$, and similarly any commutative unital ${C^*}$-homomorphism ${\Phi: {\mathcal A} \rightarrow {\mathcal B}}$ induces a continuous map ${\mathtt{Spec}(\Phi): \mathtt{Spec}({\mathcal B}) \rightarrow \mathtt{Spec}({\mathcal A})}$ defined by ${\mathtt{Spec}(\Phi) \lambda := \lambda \circ \Phi}$, and the homeomorphisms between ${X}$ and ${\mathtt{Spec}(C(X))}$ and between ${Y}$ and ${\mathtt{Spec}(C(Y))}$ commute with the continuous maps ${T: X \rightarrow Y}$ and ${\mathtt{Spec}(C(T)): \mathtt{Spec}(C(X)) \rightarrow \mathtt{Spec}(C(Y))}$ in the “natural” fashion. These sorts of properties are routine to verify and mostly consist of expanding out all the definitions.

It is natural to ask what the analogous Gelfand duality is for locally compact Hausdorff spaces. Somewhat confusingly, there appeared to be two such Gelfand dualities in the literature, which appeared at first glance to be incompatible with each other. Eventually we understood that there were two natural categories of locally compact Hausdorff spaces, and dually there were two natural categories of (non-unital) commutative ${C^*}$ algebras dual to them:

• The category ${\mathbf{LCH}}$ of locally compact Hausdorff spaces, whose morphisms consist of arbitrary continuous maps between those spaces.
• The subcategory ${\mathbf{LCH}_p}$ of locally compact Hausdorff spaces, whose morphisms consist of continuous proper maps between those spaces.
• The category ${\mathbf{CC^*Alg}_{nd}}$ of commutative ${C^*}$ algebras (not necessarily unital), whose morphisms were ${C^*}$ homomorphisms ${\Phi: \mathcal{A} \rightarrow \mathcal{B}}$ which were non-degenerate in the sense that ${\Phi({\mathcal A}){\mathcal B}}$ is dense in ${{\mathcal B}}$ (this was automatic in the unital case, but now needs to be imposed as an additional hypothesis).
• The larger category ${\mathbf{CC^*Alg}_{nd,\mathtt{Mult}}}$ of commutative ${C^*}$ algebras (not necessarily unital), whose morphisms were non-degenerate ${C^*}$ homomorphisms ${\Phi: \mathcal{A} \rightarrow \mathtt{Mult}(\mathcal{B})}$ into the multiplier algebra ${\mathtt{Mult}(\mathcal{B})}$ of ${\mathcal{B}}$. (It is not immediately obvious that one can compose two such morphisms together, but it can be shown that every such homomorphism has a unique extension to a homomorphism from ${\mathtt{Mult}(\mathcal{A})}$ to ${\mathtt{Mult}(\mathcal{B})}$, which can then be used to create a composition law.)

The map ${X \mapsto C_0(X)}$ that takes a locally compact space to the space of continuous functions that vanish at infinity, together with the previously mentioned Gelfand spectrum map ${{\mathcal A} \rightarrow \mathtt{Spec}({\mathcal A})}$, then forms a duality of categories between ${\mathbf{LCH}_p}$ and ${\mathbf{CC^*Alg}_{nd}}$, and between ${\mathbf{LCH}}$ and ${\mathbf{CC^*Alg}_{nd,\mathtt{Mult}}}$. Furthermore, these dualities of categories interact well with the two standard compactifications of locally compact spaces: the Stone-Cech compactification ${X \mapsto \beta X}$, and the Alexandroff compactification ${X \mapsto \mathtt{Alex}(X)}$ (also known as the one point compactification). From a category theoretic perspective, it is most natural to interpret ${\beta: \mathbf{LCH} \rightarrow \mathbf{CH}}$ as a functor from ${\mathbf{LCH}}$ to ${\mathbf{CH}}$, and to interpret ${\mathtt{Alex}: \mathbf{LCH}_p \rightarrow (\mathrm{pt} \downarrow \mathbf{CH})}$ as a functor from ${\mathbf{LCH}_p}$ to the category ${(\mathrm{pt} \downarrow \mathbf{CH})}$ of pointed compact Hausdorff spaces (the notation here is a special case of comma category notation). (Note in particular that a continuous map ${T: X \rightarrow Y}$ between locally compact Hausdorff spaces needs to be proper in order to guarantee a continuous extension ${\mathtt{Alex}(T): \mathtt{Alex}(X) \rightarrow \mathtt{Alex}(Y)}$ to the Alexandroff compactification, whereas no such properness condition is needed to obtain a continuous extension ${\beta T \colon \beta X \rightarrow \beta Y}$ to the Stone-Cech compactification.) In our paper, we summarized relationships between these functors (and some other related functors) in the following diagram, which commutes up to natural isomorphisms:

Thus for instance the space ${C_b(X)}$ of bounded continuous functions on a locally compact Hausdorff space ${X}$ is naturally isomorphic to the multiplier algebra of ${C_0(X)}$, and the Stone-Cech compactification ${\beta X}$ is naturally identified with the Gelfand spectrum of ${C_b(X)}$:

$\displaystyle \beta X \equiv \mathtt{Spec}(C_b(X)). \ \ \ \ \ (1)$

The coloring conventions in this paper are that (a) categories of “algebras” are displayed in red (and tend to be dual to categories of “spaces”, displayed in black; and (b) functors displayed in blue will be considered “casting functors” (analogous to type casting operators in various computing languages), and are used as the default way to convert an object or morphism in one category to an object or morphism in another. For instance, if ${X}$ is a compact Hausdorff space, the associated unital commutative ${C^*}$-algebra ${X_{\mathbf{CC^*Alg}_1}}$ is defined to be ${C(X)}$ by the casting convention.

Almost every component of the above diagram was already stated somewhere in the literature; our main contribution here is the synthesis of all of these components together into the single unified diagram.

— 2. Baire ${\sigma}$-algebras and Riesz duality —

Now we add measure theory to Gelfand duality, by introducing ${\sigma}$-algebras and probability measures on the (locally) compact Hausdorff side of the duality, and by introducing traces on the ${C^*}$-algebra side. Here we run into the issue that there are three natural choices of ${\sigma}$-algebra one can assign to a topological space ${X}$:

For compact Hausdorff spaces, the two types of Baire ${\sigma}$-algebras agree, but they can diverge for locally compact Hausdorff spaces. Similarly, the Borel and Baire algebras agree for compact metric spaces, but can diverge for more “uncountable” compact Hausdorff spaces. This is most dramatically exemplified by the Nedoma pathology, in which the Borel ${\sigma}$-algebra of the Cartesian square ${K \times K}$ of a locally compact Hausdorff space ${K}$ need not be equal to the product of the individual Borel ${\sigma}$-algebra, in contrast to the Baire ${\sigma}$-algebra which reacts well with the product space construction (even when there are uncountably many factors). In particular, the group operations on a locally compact Hausdorff group can fail to be Borel measurable, even though they are always Baire measurable. For these reasons we found it desirable to adopt a “Baire-centric” point of view, in which one prefers to use the Baire ${\sigma}$-algebras over the Borel ${\sigma}$-algebras in compact Hausdorff or locally compact Hausdorff settings. (However, in Polish spaces it seems Borel ${\sigma}$-algebras remain the more natural ${\sigma}$-algebra to use.) It turns out that the two Baire ${\sigma}$-algebras can be divided up naturally between the two categories ${\mathbf{LCH}, \mathbf{LCH}_p}$ of locally compact Hausdorff spaces, with the ${C_c}$-Baire ${\sigma}$-algebra naturally associated to ${\mathbf{LCH}_p}$ and the ${C_b}$ ${\sigma}$-algebra naturally associated to ${\mathbf{LCH}}$. The situation can be summarized by the following commuting diagram of functors between the various categories of (locally) compact Hausdorff spaces and the category ${\mathbf{ConcMes}}$ of concrete measurable spaces (sets equipped with a ${\sigma}$-algebra):

To each category of (locally) compact Hausdorff spaces, we can then define an associated category of (locally) compact Hausdorff spaces equipped with a Radon probability measure, where the underlying ${\sigma}$-algebra is as described above, and “Radon” means “compact ${G_\delta}$-inner regular”. For instance, ${\mathbf{CHPrb}}$ is the category of compact Hausdorff spaces equipped with a Baire probability measure ${\mu}$ with the inner regularity property

$\displaystyle \mu(E) = \sup \{ \mu(F): F \subset E, F \hbox{ is } G_\delta \},$

and one similarly defines ${\mathbf{LCHPrb}}$, ${\mathbf{LCH}_p \mathbf{Prb}}$, ${(\mathrm{pt} \downarrow \mathbf{CH})\mathbf{Prb}}$. (In the compact Hausdorff case the inner regularity property is in fact automatic, but in locally compact Hausdorff categories it needs to be imposed as an explicit hypothesis.) On the ${C^*}$-algebra side, categories such as ${\mathbf{CC^*Alg}_1}$ can be augmented to tracial categories ${\mathbf{CC^*Alg}_1^\tau}$ where the ${C^*}$ algebra ${{\mathcal A}}$ is now equipped with a trace ${\tau: {\mathcal A} \rightarrow {\bf C}}$, that is to say a non-negative linear functional of operator norm ${1}$. It then turns out that the Gelfand spectrum functors ${\mathtt{Spec}}$ from ${C^*}$-algebras to (locally) compact Hausdorff spaces can be augmented to “Riesz functors” ${\mathtt{Riesz}}$ from tracial ${C^*}$-algebras to (locally) compact Hausdorff spaces with Radon probability measures, where the probability measure in question is given by some form of the Riesz representation theorem; dually, the ${C_0}$ functors can similarly be augmented using the Lebesgue integral with respect to the given measure as the trace. This leads one to a complete analogue of the previous diagram of functors, but now in tracial and probabilistic categories, giving a new set of Gelfand-like dualities that we call “Riesz duality”:

Again, each component of this diagram was essentially already in the literature in either explicit or implicit form. In the paper we also review the Riesz representation theory for traces on ${C_b(X)}$ rather than ${C_c(X)}$, in which an additional “${\tau}$-smoothness” property is needed in order to recover a Radon probability measure on ${X}$. The distinction between Baire and Borel ${\sigma}$-algebras ends up being largely elided in the Riesz representation theory, as it turns out that every Baire-Radon probability measure (using either of the two Baire algebras) on a locally compact Hausdorff has a unique extension to a Borel-Radon probability measure (where for Borel measures, the Radon property is now interpreted as “compact inner regular” rather than “compact ${G_\delta}$ inner regular”).

— 3. The canonical model of opposite probability algebras —

Given a concrete probability space ${X = (X_{\mathbf{Set}}, X_{\mathbf{Bool}_\sigma}, \mu_X)}$ – a set ${X_{\mathbf{Set}}}$ equipped with a ${\sigma}$-algebra ${X_{\mathbf{Bool}_\sigma}}$ and a probability measure ${\mu_X \colon X_{\mathbf{Bool}_\sigma} \rightarrow [0,1]}$ – one can form the associated probability algebra ${(X_{\mathbf{Bool}_\sigma}/{\mathcal N}, \mu_X/{\sim})}$ by quotienting the ${\sigma}$-algebra ${X_{\mathbf{Bool}_\sigma}}$ by the null ideal ${{\mathcal N} := \{ E \in X_{\mathbf{Bool}_\sigma}: \mu_X(E)=0\}}$ and then similarly quotienting the measure ${\mu_X}$ to obtain an abstract countably additive probability measure ${\mu_X/{\sim}: X_{\mathbf{Bool}_\sigma}/{\mathcal N} \rightarrow [0,1]}$. More generally, one can consider abstract probability algebras ${({\mathcal B}, \mu)}$ where ${{\mathcal B}}$ is a ${\sigma}$-complete Boolean algebra and ${\mu: {\mathcal B}\rightarrow [0,1]}$ is a countably additive probability measure with the property that ${\mu(E)>0}$ whenever ${E \neq 0}$. This gives a category ${\mathbf{ProbAlg}}$. Actually to align things closer to the category of concrete probability spaces ${\mathbf{ConcPrb}}$, it is better to work with the opposite category ${\mathbf{ProbAlg}^{op}}$ of opposite probability algebras ${({\mathcal B}, \mu)^{op}}$ where the directions of the morphisms are reversed.

Many problems and concepts in ergodic theory are best phrased “up to null sets”, which can be interpreted category-theoretically by applying a suitable functor from a concrete category such as ${\mathbf{ConcPrb}}$ to the opposite probability algebra category ${\mathbf{ProbAlg}^{op}}$ to remove all the null sets. However, it is sometimes convenient to reverse this process and model an opposite probability algebra by a more concrete probability space, and to also model morphisms ${T: ({\mathcal A}, \mu) \rightarrow ({\mathcal B}, \nu)}$ between opposite probability algebras (${\sigma}$-complete Boolean homomorphisms from ${{\mathcal B}}$ to ${{\mathcal A}}$ that preserve measure) by concrete maps. Ideally the model spaces should also be compact Hausdorff spaces, and the model morphisms continuous maps, so that methods from topological dynamics may be applied. There are various ad hoc ways to create such models in “countable” settings, but it turns out that there is a canonical and “universal” model ${\mathtt{Conc}(X)}$ of any opposite probability algebra ${X}$ that one can construct in a completely functorial setting (so that any dynamics on the opposite probability algebra ${X}$ automatically carry over to concrete dynamics on ${\mathtt{Conc}(X)}$). The quickest way to define this model is by the formula

$\displaystyle \mathtt{Conc}(X) \equiv \mathtt{Riesz}(L^\infty(X))$

(compare with (1)), where ${L^\infty(X)}$ is the space of bounded abstractly measurable maps from ${X}$ to ${{\bf C}}$, which one can imbue with the structure of a tracial unital commutative ${C^*}$-algebra. This gives a (covariant full inclusion) functor from ${\mathbf{ProbAlg}^{op}}$ to ${\mathbf{CHPrb}}$ that has many pleasant properties, including:

• (Topological structure) ${\mathtt{Conc}(X)}$ is a compact Hausdorff space. In fact it is a Stone space with the additional property that every Baire-measurable set is equal to a clopen set modulo a Baire-meager set (such spaces we call ${\textbf{Stone}_\sigma}$-spaces in our paper). Furthermore a Baire set is null if and only if it is meager. Any opposite probability algebra morphism ${T: X \rightarrow Y}$ gives rise (in a functorial fashion) to a surjective continuous map ${\mathtt{Conc}(T): \mathtt{Conc}(X) \rightarrow \mathtt{Conc}(Y)}$ with the additional property that the inverse image of Baire-meager sets are Baire-meager. Furthermore, one has the (somewhat surprising) “strong Lusin property”: every element of ${L^\infty(\mathtt{Conc}(X))}$ has a unique representative in ${C(\mathtt{Conc}(X))}$ (thus, every bounded measurable function on ${\mathtt{Conc}(X)}$ is equal almost everywhere to a unique continuous function).
• (Concrete model) The opposite probability algebra of ${\mathtt{Conc}(X)}$ is (naturally) isomorphic to ${X}$.
• (Universality) There is a natural inclusion (in the category ${\mathbf{AbsPrb}}$ of abstract probability spaces) from ${\mathtt{Inc}(X)}$ (which is ${X}$ interpreted in ${\mathbf{AbsPrb}}$) into ${\mathtt{Conc}(X)}$, which is universal amongst all inclusions of ${\mathtt{Inc}(X)}$ into compact Hausdorff probability spaces.
• (Canonical extension) Every abstractly measurable map from ${\mathtt{Inc}(X)}$ to a compact Hausdorff space ${K}$ has a unique extension to a continuous map from ${\mathtt{Conc}(X)}$ to ${K}$.

This canonical model ${\mathtt{Conc}}$ seems quite analogous to the Stone-Cech compactification ${\beta}$. For instance the analogue of the canonical extension property for ${\beta}$ is that every continuous map from a locally compact space ${X}$ to a compact Hausdorff space ${K}$ has a unique extension to a continuous map from ${\beta X}$ to ${K}$. In both cases the model produced is usually too “large” to be separable, so this is a tool that is only available in “uncountable” frameworks.

There is an alternate description of the canonical model, which is basically due to Fremlin, and is based on Stone-type dualities rather than Riesz type dualities. Namely, one starts with the ${\sigma}$-complete Boolean algebra attached to the opposite probability space ${X}$, and constructs its Stone dual, which is a ${\mathbf{Stone}_\sigma}$ space. Every Baire set in this space is equal modulo Baire-meager sets to a clopen set, which by Stone duality is identifiable with an element of the probability algebra. The measure on ${X}$ then induces a measure on the ${\mathbf{Stone}_\sigma}$-space, and this compact Hausdorff probability space can serve as the canonical model ${\mathtt{Conc}(X)}$. The functoriality of this construction is closely tied to the functoriality of the Loomis-Sikorski theorem (discussed in this previous blog post). The precise connection in terms of functors and categories is a little complicated to describe, though, as the following diagram indicates:

One quick use of the canonical model is that it allows one to perform a variety of constructions on opposite probability algebras by passing to the canonical model and performing the analogous operation there. For instance, if one wants to construct a product ${\prod_{\alpha \in A} X_\alpha}$ of some number of opposite probability algebras, one can first take the product ${\prod_{\alpha \in A} \mathtt{Conc}(X)}$ of the concrete models (as a compact Hausdorff probability space), then extract the opposite probability algebra of that space. We will similarly use this model to construct group skew-products and homogeneous skew-products in a later paper.

— 4. Canonical disintegration —

Given a probability space ${(X,\mu_X)}$ and a probability-preserving factor map ${\pi: X \rightarrow Y}$ from ${X}$ to another probability space ${(Y,\mu_Y)}$, it is often convenient to look for a disintegration of ${\mu_X}$ into fibre measures ${\nu_y, y \in Y}$ with the property that the conditional expectation ${{\bf E}(f|Y)}$ of a function ${f \in L^\infty(X,\mu_X)}$ (defined as the orthogonal projection in ${L^2}$ to ${L^2(Y,\mu_Y)}$, viewed as a subspace in ${L^2(X,\mu_X)}$) is given pointwise almost everywhere by

$\displaystyle {\bf E}(f|Y) = \int_X f\ d\nu_y.$

The construction of such disintegrations originated in the work of Rohlin; the standard disintegration theorems require some sort of separability hypotheses on the spaces ${X,Y}$ (e.g., that ${X}$, ${Y}$ be standard Borel spaces) in order to avoid the issue of having to deal with an uncountable number of exceptional null sets. However, with the canonical model one can construct a canonical disintegration that does not require any of these hypotheses; the catch though is that the disintegration is now in the larger model space ${\mathtt{Conc}(X)}$ rather than in the original space ${X}$. Namely, because of the strong Lusin property ${L^\infty(\mathtt{Conc}(Y)) = C(\mathtt{Conc}(Y))}$, one can interpret the conditional expectation ${{\bf E}(f|Y)(y)}$ of a function ${f \in L^\infty(\mathtt{Conc}(X)) = C(\mathtt{Conc}(X))}$ evaluated at a point ${y \in \mathtt{Conc}(Y)}$ as a linear functional on ${C(\mathtt{Conc}(X))}$, and the Riesz representation theorem then produces the required measures ${\nu_y}$.

Among other things, the canonical disintegration makes it easy to construct relative products ${X_1 \times_Y X_2}$ of opposite probability algebras, and we believe it will also be of use in developing uncountable versions of Host-Kra structure theory.