nLab categories of measure theory

Introduction

While areas of mathematics such as set theory, algebra, general topology, functional analysis, Lie theory have easily identifiable categories of main objects of study (e.g., categories of sets, groups, rings, modules, topological spaces, topological vector spaces, C*-algebras, Lie groups, Lie algebras), measure theory is not traditionally associated with a prominent, easily identifiable category.

For example, in Fremlin’s Measure Theory one reads (§234):

One of the striking features of measure theory, compared with other comparably abstract branches of pure mathematics, is the relative unimportance of any notion of ‘morphism’. The theory of groups, for instance, is dominated by the concept of ‘homomorphism’, and general topology gives a similar place to ‘continuous function’. In my view, the nearest equivalent in measure theory is the idea of ‘inverse-measure-preserving function’ (234A).

This article reviews existing categories in measure theory, explains why they do not match the existing practice of measure theory and real analysis, and proposes a very satisfying way to resolve these issues. The resulting category of compact strictly localizable enhanced measurable spaces enjoys excellent categorical properties and matches the existing practice of real analysis closely.

This article concentrates on measure theory as it is used in real analysis, probability theory, statistics, stochastic processes and other areas of analysis. In particular, given the existing practice in these fields, we take it for granted that we must identify functions that are equal almost everywhere. Other areas, such as descriptive set theory may need other criteria and other categories.

We also remark that the use of enhanced measurable spaces instead of measure spaces is for strictly esthetic reasons (like avoiding making noncanonical choices of measures), and everything works equally well with traditional measure spaces instead.

Elementary constructions

Measurable spaces

Recall that a measurable space is a pair $(X,M)$, where $X$ is a set and $M$ is a σ-algebra on $X$, i.e., a collection of subsets of $X$ closed under complements and countable unions. Elements of $M$ are known as measurable subsets of $X$.

Morphisms of measurable spaces $f\colon(X,M)\to(X',M')$ are known as measurable maps. These are maps of sets $f\colon X\to X'$ that reflect measurable sets, i.e., if $m'\in M'$, then $f^{-1}m'\in M$.

The resulting category of measurable spaces does not identify measurable maps that are equal almost everywhere, so cannot be used directly in real analysis and related fields. This is not merely an inconvenience: many important theorems in measure theory are false unless such an identification is made. For example, L^p-spaces would cease to be normed spaces, since plenty of nonzero elements would have norm 0. The Radon–Nikodym theorem and the Riesz representation theorem would fail too.

Measure spaces

Having recognized the need to identify measurable maps that are equal almost everywhere, we can see that the data of a measurable space is insufficient to perform such an identification, since there is no way of knowing which measurable sets have measure 0.

A measure space is a triple $(X,M,\mu)$, where $(X,M)$ is a measurable space and $\mu$ is a measure on $(X,M)$, i.e., a map of sets $\mu\colon M\to[0,\infty]$ that sends countable disjoint unions of sets to the sum of corresponding values.

Two measurable maps

$$f,g\colon(X,M,\mu)\to(X',M',\mu')$$

are equal almost everywhere if the set

$$[f\ne g]=\{x\in X\mid f(x)\ne g(x)\}$$

is contained in some set $m\in N_\mu$, where

$$N_\mu=\{m\in M\mid \mu(m)=0\}$$

is the σ-ideal of measurable sets with $\mu$-measure 0, also known as $\mu$-negligible sets.

Too much data required

Requiring a measurable space to be equipped with a specific choice of a measure forces us to make noncanonical choices that may be unnecessary.

Many constructions and theorems in measure theory do not require a specific choice of a measure but merely require us to know which measurable sets have measure 0. Examples include the Riesz representation theorem, the Radon–Nikodym theorem, or (less obviously) the construction of L^p-spaces. For the latter, observe that $L^1(X,M,\mu)$ is canonically isomorphic to the vector space of finite measures on $(X,M,N_\mu)$ via the Radon–Nikodym theorem; by taking $1/p$-th powers, we can give a similar definition for general L^p-spaces.

There are plenty of examples of situations where we do have a canonical choice of a σ-ideal of negligible sets even though we do not have a canonical measure.

For example, a subset of a smooth manifold is negligible if its preimage in any chart is a Lebesgue-negligible subset of $\mathbf{R}^n$.

Likewise, a subset of a locally compact group is negligible if it is negligible with respect to some (hence all) nonzero left-invariant or right-invariant measure.

Enhanced measurable spaces

The above considerations lead us to the notion of an enhanced measurable space.

Definition

An enhanced measurable space is a triple $(X,M,N)$, where $(X,M)$ is a measurable space and $N\subset M$ is a σ-ideal of $M$ whose elements are known as negligible sets.

We can keep the same definition of a measurable maps. Two measurable maps

$$f,g\colon(X,M,N)\to(X',M',N')$$

are equal almost everywhere if

$$[f\ne g]$$

is a subnegligible set, i.e., a subset of an element of $N$.

Any measure space $(X,M,\mu)$ yields an enhanced measurable space $(X,M,N_\mu)$ with $N_\mu$ as above.

A map of measure spaces is measurable if and only if it is measurable as a map of enhanced measurable spaces and two maps of measure spaces are equal almost everywhere if and only if they are equal almost everywhere as maps of enhanced measurable spaces.

An enhanced measurable space $(X,M,N)$ is complete if subnegligible sets are negligible, i.e., the σ-ideal $N$ is closed under passage to arbitrary subsets. This definition generalizes the usual notion of completeness of measure spaces.

Remark

Everything in this article could be done with conventional measure spaces $(X,M,\mu)$ instead of enhanced measurable spaces. Enhanced measurable spaces are only introduced to enhance the clarity of exposition and avoid making noncanonical choices of measures in some constructions. Also, separating measures from their underlying spaces makes the statements of some theorems cleaner, e.g., the Radon–Nikodym theorem now says that measures form a free module of rank one over measurable functions.

Measures on enhanced measurable spaces

A measure on an enhanced measurable space is a measure $\mu$ on the measurable space $(X,M)$ that vanishes on every element of $N$. A measure $\mu$ is faithful if for every $m\in M$ the relation $\mu(m)=0$ implies $m\in N$. A measure $\mu$ is semifinite if for every $m\in M$ such that $\mu(m)=\infty$ there is $m'\in M$ such that $m'\subset m$ and $0\lt\mu(m')\lt\infty$.

Typically, we are only interested in enhanced measurable spaces that admit a faithful semifinite measure, given that having a decent supply of measures is a prerequisite for developing measure theory. This property follows automatically from the strict localizability property introduced below.

An enhanced measurable space $(X,M,N)$ is σ-finite if it admits a faithful finite measure $\mu\colon M\to[0,1]$. An enhanced measurable space $(X,M,N)$ is locally determined if for every $a\subset X$ the following two properties hold.

• If for every σ-finite $m\in M$ we have $a\cap m\in M$, then also $a\in M$.

• If for every σ-finite $m\in M$ we have $a\cap m\in N$, then also $a\in N$.

By Fremlin (§213D), for every enhanced measurable space $(X,M,N)$ we can construct a complete locally determined enhanced measurable space $(X,M',N')$ such that $M\subset M'$, $N\subset N'$, so the identity map on $X$ is measurable, inducing a morphism $(X,M',N')\to(X,M,N)$.

One reasonable definition of a measurable map of enhanced measurable spaces $(X_1,M_1,N_1)\to(X_2,M_2,N_2)$ that need not be complete or locally determined is a map of sets $f\colon X_1\to X_2$ that reflects subnegligible sets as well as symmetric differences of subnegligible and σ-finite sets. Under such a definition, the above map $(X,M',N')\to(X,M,N)$ is an isomorphism, with its inverse again given by the identity map of sets $X\to X$.

It is for this reason that it is safe to assume enhanced measurable spaces to be complete from the start. Every strictly localizable enhanced measurable space is automatically locally determined, so there is no need for us to separately impose the property of local determinedness.

Precomposition does not respect equality almost everywhere

If we now attempt to construct a category of measure spaces or enhanced measurable spaces, we immediately run into a serious problem: precomposition does not respect equality almost everywhere.

To see this, suppose

$$f,g\colon(X,M,N)\to(X',M',N')$$

are measurable maps that are equal (or weakly equal) almost everywhere.

Suppose that

$$h\colon(X'',M'',N'')\to(X',M',N')$$

is a measurable map. If composition is to be compatible with equality almost everywhere, then $f\circ h$ would have to be equal (or weakly equal) to $g\circ h$ almost everywhere. However,

$$[f\circ h\ne g\circ h]=h^{-1}[f\ne g].$$

Although $[f\ne g]$ is subnegligible, there is absolutely no reason why its preimage $h^{-1}[f\ne g]$ should be subnegligible.

Indeed, it is quite easy to construct examples where this is false: the inclusion of the Cantor set into real numbers is a Borel measurable map. If we equip real numbers with the Lebesgue measure and the Cantor set with the product measure, then the inclusion map does not reflect subnegligible sets.

Thus, making the equivalence relation compatible with precomposition naturally leads us to an additional restriction on measurable maps.

Definition

A measurable map

$$f\colon(X,M,N)\to(X',M',N')$$

reflects negligible sets if for every $n'\in N'$ the preimage $f^{-1}n'$ is negligible. That is to say, the preimage map $f^{-1}$ preserves negligible sets.

By inspection, precomposition with a measurable map that reflects negligible sets preserves both equivalence relations: equality almost everywhere and weak equality almost everywhere.

A measurable map reflects subnegligible sets if and only if it reflects negligible sets.

If $N'=\{\emptyset\}$, then every measurable map $f$ reflects negligible sets. In particular, taking $(X',M',N')=(\mathbf{R},\mathbf{R}_{\mathsf{Borel}},\{\emptyset\})$, we see that real-valued measurable functions in the traditional sense of measure theory coincide with morphisms to real numbers in the new sense.

Negligibility reflection

Another way to look at the negligibility reflection condition introduced in the previous section is as follows.

Consider an enhanced measurable space $(X,M,N)$ such that $X\in N$, which implies $M=N$. If we are taking the ideology of measure theory seriously, measurable sets that differ by a negligible set should be indistinguishable. In particular, $X$ should be indistinguishable from the empty set. Thus, the inclusion map $(\emptyset,\{\emptyset\},\{\emptyset\})\to(X,M,N)$ should be an isomorphism in whatever category we are constructing.

Therefore, it is reasonable to expect that there are no morphisms $(X_1,M_1,N_1)\to(X,M,N)\cong(\emptyset,\{\emptyset\},\{\emptyset\})$ unless $X_1\in N_1$, i.e., the map is an isomorphism.

If $f\colon(X',M',N')\to(X'',M'',N'')$ is a measurable map and $n''\in N''$, consider the restriction of $f$ to a map $f^{-1}n''\to n''$. The restricted map is measurable in the induced structures. Since the codomain $n''$ is negligible, we expect $f^{-1}n''$ to be negligible. This leads us once again to the negligibility reflection condition.

Postcomposition and weak equality almost everywhere

Postcomposition respects equality almost everywhere.

To see this, suppose

$$f,g\colon(X,M,N)\to(X',M',N')$$

are measurable maps that are equal almost everywhere.

Suppose that

$$h\colon(X',M',N')\to(\{0,1\},2^{\{0,1\}},\{\emptyset\})$$

is a measurable map. Such measurable maps are in bijection with elements of $M'$: the bijection sends $h$ to $h^{-1}\{1\}$, and the inverse bijection sends a subset $m'\in M'$ to its characteristic function $\chi_{m'}$.

If composition is to be compatible with equality almost everywhere, then $h\circ f$ would have to be equal to $h\circ g$ almost everywhere. We have

$$[h\circ f\ne h\circ g]=f^{-1}m'\oplus g^{-1}m',$$

where $m'=h^{-1}\{1\}$ is as above. Since

$$f^{-1}m'\oplus g^{-1}m' \subset [f\ne g],$$

the set $f^{-1}m'\oplus g^{-1}m'$ is subnegligible.

The latter condition, i.e., the set $f^{-1}m'\oplus g^{-1}m'$ is subnegligible for all $m'\in M'$, is very much compatible with the ideology of measure theory: it says that the preimages of $f$ and $g$ are the same almost everywhere.

It is quite natural to ask whether the converse holds. For example, two continuous maps $f,g\colon X\to X'$ of $T_0$-topological spaces are equal if and only if for every open subset $U'\subset X'$ we have $f^{-1}U'=g^{-1}U'$. In measure theory, equality of sets should be replaced with equality almost everywhere, which amounts to requiring for every $m'\in M'$ the symmetric difference $f^{-1}m'\oplus g^{-1}m'$ to be negligible.

However, Fremlin (Example 343I) constructs an example of a measurable negligibility-reflecting map $f:(X,M,\mu)\to(X,M,\mu)$, where $(X,M,\mu)=\{0,1\}^{\mathfrak{c}}$ is a product of continuum many discrete measurable spaces $\{0,1\}$ such that for every $x\in X$ we have $f(x)\ne x$, but for every $m\in M$, the set $f^{-1}m\oplus m$ is negligible. Thus, the identity map $id_X$ is not equal to $f$ almost everywhere and yet, $id_X$ and $f$ have the same preimages, up to subnegligible sets.

Such a situation is pathological, and is it turns out, it is the condition of having the same preimages up to subnegligible sets that produces nonpathological theory when the two conditions differ (they do coincide for many spaces, as we will see below).

These considerations naturally lead us to a coarser equivalence relation.

Definition

Two measurable maps

$$f,g\colon(X,M,N)\to(X',M',N')$$

are weakly equal almost everywhere if for every $m'\in M'$ we have

$$f^{-1}m'\oplus g^{-1}m'\in N.$$

Equality almost everywhere implies weak equality almost everywhere. The converse is true if $(X',M',N')$ is countably separated (Fremlin, Proposition 343F), i.e., there is a countable subset $M''\subset M'$ that covers $X'$ and for any two points of $X'$ there is an element of $M''$ that contains exactly one of them.

An enhanced measurable space is countably separated if and only if it admits an injective measurable function into real numbers. Most measurable spaces (other than some counterexamples) in an introductory real analysis book are countably separated, which explains why the finer equivalence relation of equality almost everywhere is sufficient for many practical purposes.

The category of enhanced measurable spaces

We collect what we have learned so far.

• Measurable spaces $(X,M)$ do not have enough information: they lack the data of sets of measure 0.

• Measure spaces $(X,M,\mu)$ have too much information: they require us to make arbitrary unnecessary choices of measures $\mu$.

• Enhanced measurable spaces $(X,M,N)$ have just the right amount of information.

• The condition of measurability of maps, i.e., reflection of measurable sets, must be strengthened to also require reflection of negligible sets, to ensure its compatibility with precomposition.

• The equivalence relation of equality almost everywhere must be coarsened to weak equality almost everywhere to ensure morphisms are fully determined by their preimages up to subnegligible sets.

Assembling these properties together, we arrive at the following definition.

Definition

The category of enhanced measurable spaces is constructed as a quotient category. Objects are enhanced measurable spaces $(X,M,N)$, where $X$ is a set, $M$ is a σ-algebra, $N$ is a σ-ideal. Morphisms are maps of sets that reflect measurable sets and negligible sets. Two morphisms are equivalent if they are weakly equal almost everywhere.

Localizability

The category of enhanced measurable spaces captures the desired properties of measurable spaces and measurable maps as they are used in real analysis.

To complete the construction, we must eliminate pathological objects that invalidate the main theorems of measure theory. In traditional textbooks on measure theory, this is accomplished by restricting to σ-finite measure spaces and/or Radon measures.

We consider a simple generalization of σ-finiteness known as strict localizability, as well as an abstract formulation of Radon measures, known as compactness (introduced by Marczewski; not to be confused with compactness of topological spaces, to which it is closely related).

Localizability as a necessary prerequisite for measure theory

We review some classic results due to Irving E. Segal and John L. Kelley. As explained in the next section, the localizability property is still not quite enough to eliminate all the pathologies. However, it does explain the Boolean algebra side of the story.

Theorem

(Irving E. Segal, 1951; John L. Kelley, 1966.) The following properties are equivalent for an enhanced measurable space $(X,M,N)$ that admits a faithful semifinite measure.

• The Boolean algebra $M/N$ is complete.

• The Lebesgue decomposition for σ-ideals holds.

• The Hahn decomposition theorem holds for signed measures.

• The Radon–Nikodym theorem holds.

• The Riesz representation theorem holds: the canonical map $L^\infty\to(L^1)^*$ is an isomorphism.

• $L^\infty(X,M,N)$ is a commutative von Neumann algebra.

A localizable enhanced measurable space is an enhanced measurable space that admits a faithful semifinite measure and satisfies any of the above equivalent properties.

The takeaway from this result is that without the localizability property much of the measure theory as we know is false, so it is completely reasonable to require it from the start.

We can also establish an analogue of the above theorem for morphisms.

Theorem

(The relative Kelley–Segal theorem.) The following properties are equivalent for a morphism of localizable enhanced measurable spaces $f\colon(X,M,N)\to(X',M',N')$.

• The induced homomorphism of Boolean algebras $f^{-1}\colon M'/N'\to M/N$ is complete, i.e., preserves suprema.

• The map $f^{-1}$ preserves the generator of σ-ideals provided by the Lebesgue decomposition.

• The pushforward map $f_*$ on measures preserves the Hahn decomposition.

• The relative Radon–Nikodym theorem holds: $f_*(f^* h\cdot \mu)=h\cdot f_*\mu$.

• The induced map $f_*\colon L^1(X,M,N)\to L^1(X',M',N')$ is a bounded linear map of Banach spaces.

• The induced *-homomorphism $f^*$ of *-algebras of bounded real-valued functions on $X$ is a normal *-homomorphism of commutative von Neumann algebras.

A localizable map of enhanced measurable spaces is a morphism of enhanced measurable spaces that satisfies any of these equivalent properties.

Again, the takeaway here is that we really want to only use localizable maps if we want to avoid a complete breakdown of measure theory as we know it.

Surprisingly, once we impose conditions on objects (namely, strict localizability and compactness) that eliminate pathological objects, it turns out that all maps between such objects are localizable. This is a deep theorem essentially due to Fremlin (implicitly contained in §451Q).

Enhanced measurable spaces and complete Boolean algebras

In the previous section we constructed (essentially by definition) a functor

$$LEMS \to CBAlg^{op}$$

from the category of localizable enhanced measurable spaces and localizable maps to the opposite category of complete Boolean algebras and complete homomorphisms of Boolean algebras.

This functor lands inside complete Boolean algebras $A$ that admit a faithful semifinite measure, i.e., a map of sets $\mu\colon A\to[0,\infty]$ that is completely additive, vanishes only on 0, and $1=\sup \mu^{-1}[0,\infty)$. We refer to such Boolean algebras as localizable Boolean algebras.

Thus, the above functor descends to a functor

$$LEMS \to LBAlg^{op}.$$

The opposite category of $LBAlg$ is known as the category of measurable locales, since every complete morphism of complete Boolean algebras is a morphism of frames, so localizable Boolean algebras form a full subcategory of frames.

Stone and Gelfand dualities for localizable Boolean algebras

The Stone duality is an equivalence of categories from the opposite category of Boolean algebras to the category of Stone spaces (or Stone locales). It restricts to an equivalence of categories from the opposite category of complete Boolean algebras and complete homomorphisms of Boolean algebras to the category of Stonean spaces (or Stonean locales) and open maps. Restricting further to the full category of localizable Boolean algebras, the essential image of the Stonean duality functor is precisely the full subcategory of hyperstonean spaces (or hyperstonean locales) and open maps (Pavlov, Propositions 2.72, 2.73).

The Gelfand duality is an equivalence of categories from the opposite category of (unital) commutative C*-algebras and (unital) *-homomorphisms and the category of compact Hausdorff spaces (or compact regular locales). It restricts to an equivalence of categories from the opposite category of commutative von Neumann algebras and (unital) normal *-homomorphisms to the category of hyperstonean spaces (or hyperstonean locales) and open maps. Furthermore, this equivalence factors through the opposite category of localizable Boolean algebras: the relevant functor sends a commutative von Neumann algebra to the localizable Boolean algebra of its projections and a (unital) normal *-homomorphism of commutative von Neumann algebras to the induced complete homomorphism of Boolean algebras of projections (Pavlov, Theorem 3.20).

Theorem

(Pavlov, Propositions 2.72, 2.73; Theorem 3.20.) The following four categories are equivalent.

• The opposite category of commutative von Neumann algebras and unital normal *-homomorphisms.

• The opposite category of localizable Boolean algebras and complete homomorphisms.

• The category of hyperstonean locales and open maps.

• The category of hyperstonean spaces and open maps.

We take this theorem as a strong evidence that whatever good category of enhanced measurable spaces we end up constructing, it better be equivalent to the above four categories.

We have already constructed a functor

$$LEMS \to LBAlg^{op}.$$

In the next section, we construct a functor

$$LBAlg^{op}\to LEMS$$

that exhibits $LBAlg^{op}$ as a retract (in particular, a full subcategory) of $LEMS$.

The Loomis–Sikorski construction

The functor

$$LBAlg^{op}\to LEMS$$

is constructed as the composition of functors

$$LBAlg^{op}\to HStonean\to LEMS,$$

where the first functor is the hyperstonean duality functor explained in the previous section.

The second functor is known as the Loomis–Sikorski construction (Loomis, 1947; Sikorski, 1948; see also Pavlov, Definition 5.2) and is defined as follows. Send a hyperstonean topological space $(X,U)$ to the enhanced measurable space $(X,M,N)$, where $N$ is the σ-ideal of nowhere dense sets (equivalently in the case of hyperstonean spaces: meager sets) and $M$ is the σ-algebra of subsets with the property of Baire, i.e., symmetric differences of clopen subsets and nowhere dense sets.

The quotient Boolean algebra $M/N$ is easily seen to be precisely the complete Boolean algebra of clopen subsets of $X$ and the latter is precisely the inverse of the Stone spectrum functor evaluated on $(X,U)$. Thus, the Loomis–Sikorski construction indeed exhibits $LBAlg^{op}$ as a retract of $LEMS$.

Summary

We have seen that in order to have a theory resembing conventional measure theory, we have to work with localizable enhanced measurable spaces and localizable morphisms.

As it turns out, the essential image of the Loomis–Sikorski functor is the full subcategory of enhanced measurable spaces comprising precisely compact and strictly localizable enhanced measurable spaces, defined in the next section.

In particular (and rather surprisingly), all morphisms of compact strictly localizable enhanced measurable spaces are automatically localizable, even though there are examples (one of which is given below) of nonlocalizable morphisms from a σ-finite enhanced measurable space to a completely atomic (i.e., discrete) enhanced measurable space, both of which are localizable.

Thus, if one finds the necessity of extending the above equivalence of four categories to include some form of enhanced measurable spaces convincing enough, one can skip the next section and proceed directly to the summary. Below, we provide additional independent motivation for strict localizability and compactness that does not rely on the above reasoning.

The subtle points: strict localizability and compactness

Strict localizability

We already established in the previous section that it is highly desirable (e.g., for Gelfand duality) to be able to lift complete homomorphisms of Boolean algebras $M'/N'\to M/N$ to morphisms of enhanced measurable spaces $(X,M,N)\to(X',M',N')$.

Consider the simplest case when $(X',M',N')=(X',2^{X'},\{\emptyset\})$ is a discrete enhanced measurable space.

Then complete homomorphisms $2^{X'}\to M/N$ can be identified with $X'$-indexed families of elements $p_i\in M/N$ ($p\in X'$) such that $i\ne j$ implies $p_i p_j=0$ and $\sup_i p_i=1$. That is to say, the elements $p_i$ form a partition of 1 in the complete Boolean algebra $M/N$.

Likewise, measurable maps $(X,M,N)\to(X',2^{X'},\{\emptyset\})$ can be identified with $X'$-indexed families of elements $P_i\in M$ such that $i\ne j$ implies $P_i\cap P_j=\emptyset$ and $\bigcup_i P_i=X$. That is to say, the sets $P_i$ form a (disjoint) partition of $X$.

Lifting complete homomorphisms $2^{X'}\to M/N$ to morphisms of enhanced measurable spaces $(X,M,N)\to(X',2^{X'},\{\emptyset\})$ amounts to constructing $P_i$ such that the equivalence class of $P_i$ in $M/N$ equals $p_i$.

Pick some arbitrary representative $Q_i$ for every $p_i$. For every $i\ne j$ we have $p_i p_j = 0$, hence $Q_i \cap Q_j \in N$. Furthermore, $X$ is the essential supremum of $Q_i$. The question is then whether it is possible to find $P_i$ such that $P_i\oplus Q_i\in N$ and $P_i$ are disjoint.

Surprisingly, the answer is negative in general.

An enhanced measurable space $(X,M,N)$ is strictly localizable if it is isomorphic to the coproduct of a set-indexed family of σ-finite enhanced measurable spaces.

Coproducts of enhanced measurable spaces are computed in the expected manner:

$$\coprod_i (X_i,M_i,N_i)=\left(\coprod_i X_i,\prod_i M_i,\prod_i N_i\right),$$

i.e., a subset of $\coprod_i X_i$ is measurable or negligible if and only if its intersection with every $X_i$ is measurable or negligible, respectively.

A rather delicate example of a complete locally determined localizable measure space that is not strictly localizable can be found in Fremlin (§216E).

It is not difficult to see that a localizable enhanced measurable space $(X,M,N)$ admits liftings of complete homomorphisms $2^{X'}\to M/N$ if and only if it is strictly localizable. Indeed, if $(X,M,N)$ is localizable and admits liftings, then the lifting of a σ-finite partition $p_i\in M/N$ of $1\in M/N$ is a strictly localizable partition of $X$. Conversely, if $(X,M,N)$ is strictly localizable, then any lifting problem for $p_i$ can be solved separately for every σ-finite $(X_i,M_i,N_i)$, with individual solutions combined using disjoint unions. In the σ-finite case, only countably many $Q_i$ can be nonnegligible, which allows us to take $P_i=\emptyset$ for all $i$ such that $Q_i$ is negligible, whereas for the remaining $Q_i$ we take $P_i=Q_i\setminus \bigcup_{j\lt i}Q_j$, for some fixed well-ordering of indices $i$.

Surprisingly, the lifting property for complete homomorphisms out of complete atomic Boolean algebras implies the lifting property for complete homomorphisms out of arbirary localizable Boolean algebras. Thus, the latter property holds if and only if the enhanced measurable space is strictly localizable.

This is the rather deep von Neumann–Maharam theorem, an exposition of which can be found in Fremlin (§341M).

Thus, if we want to be able to lift complete homomorphisms of Boolean algebras (or, equivalently, normal unital *-homomorphisms of commutative von Neumann algebras) to measurable maps of measurable spaces, we have to assume strict localizability.

Compactness

Suppose

$$f\colon(X,M,N)\to(X',M',N')$$

is a morphism of enhanced measurable spaces.

What subset of $X'$ should we take as the image of $f$?

The ordinary image of $f$ as a map of sets does not result in a useful notion. Indeed, suppose $M=N$, i.e., $X\in N$. Then every map of sets $X\to X'$ defines a morphism of enhanced measurable spaces, since every subset of $X$ is a subnegligible set. If the cardinality of $X$ is at least the cardinality of $X'$, then every subset of $X'$ is the image of some measurable map $f\colon X\to X'$. In particular, such a subset need not be measurable.

On the other hand, we have already seen that the enhanced measurable space $(X,M,N)$ is isomorphic to the empty space $(\emptyset,\{\emptyset\},\{\emptyset\})$. It is reasonable to assume that the notion of the image of a morphism should be invariant under isomorphisms of enhanced measurable spaces. In particular, we expect the image of $f$ to be empty, or equivalent to the empty set, i.e., negligible.

To resolve the contradiction, we modify the definition to allow for images-up-to-a-subnegligible subset.

Definition

Suppose

$$f\colon(X,M,N)\to(X',M',N')$$

is a morphism of enhanced measurable spaces. The essential image of $f$ (if it exists) is an element $A\in M'$ such that $f^{-1}(X'\setminus A)\in N$ and for every $B\in M'$ such that $f^{-1}(X'\setminus B)\in N$ we have $A\setminus B\in N'$.

All essential images of $f$ taken together form a (unique) equivalence class in $M'/N'$.

For example, in the above example, the essential image of $f$ is always negligible, even if the ordinary image coincides with $X$.

However, the essential image of $f$ does not always exist. Consider a set $X$ together with a probability measure $\mu\colon 2^X\to[0,1]$ such that for every $x\in X$ we have $\mu(\{x\})=0$. Such measures exist if and only if $X$ has the cardinality of a real-valued-measurable cardinal, a type of a large cardinal. The identity map of sets $X\to X$ defines a morphism of enhanced measurable spaces

$$f\colon(X,2^X,N_\mu)\to(X,2^X,\{\emptyset\}).$$

Indeed, given any $A\in 2^X$ such that $X\setminus A\in N$, for every $a\in A$ we have $X\setminus B\in N$, where $B=A\setminus\{a\}$. Thus, if $A$ is the essential image of $f$, then we must have $A=\emptyset$, a contradiction since $X\setminus A\notin N$.

To eliminate pathological examples such as the one given above, we need to require $(X,M,N)$ to be a compact enhanced measurable space. Compactness is the “abstract essence” of the notion of Radon measure that is not affixed to any particular topology on the underlying set. It was first introduced by Marczewski.

Definition

An enhanced measurable space $(X,M,N)$ is compact if there is a compact class $K\subset M$ such that for any $m\in M\setminus N$ there is $k\in K\setminus N$ such that $k\subset m$. A collection $K\subset 2^X$ of subsets of a set $X$ is a compact class if for any $K'\subset K$ the following finite intersection property holds: if for any finite $K''\subset K'$ we have $\bigcap K''\ne\emptyset$, then also $\bigcap K'\ne\emptyset$.

To connect to the notion of a compact topological space, observe (Fremlin, Lemma 342D(a)) that $K\subset 2^X$ is a compact class if and only if there is a compact topology on the set $X$ such that every element of $K$ is a closed (and hence compact) subset of $X$. Thus, an enhanced measurable space is compact if and only if it admits a compact topology such that every nonnegligible measurable set contains a nonnegligible subset that is closed (Fremlin, Corollary 342F).

A Radon enhanced measurable space is an enhanced measurable space equipped with a structure of a Hausdorff topological space such that the σ-algebra of measurable sets contains open sets and there is a faithful measure that is locally finite (every point has a neighborhood of finite measure) and inner regular with respect to compact subsets (the measure of any measurable subset is the supremum of measures of its compact subsets). (See Fremlin Definition~411H(b).) Every Radon enhanced measurable space is compact and strictly localizable by Fremlin (Proposition 416W(a), Theorem 414J).

The following theorem answers the question of existence of the essential image in the affirmative in the case when the domain of $f$ is a compact enhanced measurable space. The crucial ingredient is a deep result of Fremlin (§451Q) that shows that a compact enhanced measurable space whose σ-ideal of negligible sets is induced by a faithful semifinite measure cannot be partitioned into an uncountable family of negligible sets whose arbitrary unions are measurable subsets.

Theorem

(Pavlov, Proposition 4.65.) Suppose $(X, M, N)$ is a compact enhanced measurable space that admits a faithful semifinite measure, $(X',M',N')$ is a strictly localizable enhanced measurable space, and $f\colon (X, M, N) \to (X', M', N')$ is a morphism of enhanced measurable spaces. Then $f$ admits an essential image.

It is easy to see that the essential image of $f$ (if it exists) must be the complement of the essential supremum $p$ of all $m'\in M'$ such that $f^{-1}m'\in N$. The difficult part is to show that $f^{-1}p\in N$, and this is exactly what Fremlin’s theorem achieves.

As an added bonus, we obtain the following counterpart to the von Neumann–Maharam co-lifting theorem for strictly localizable spaces.

Theorem

(von Neumann 1932; C.~Ionescu Tulcea 1965; Vesterstrøm–Wils 1969; Edgar 1976; Graf 1980; see also Fremlin, Theorem~343B(iv).) Suppose $(X,M,N)$ is an enhanced measurable space that admits a faithful semifinite measure. Assume $X\notin N$. The following two conditions are equivalent.

• For every $m\in M\setminus N$ there is $a\in M\setminus N$ such that $a\subset m$ and $a$ is compact. (If $(X,M,N)$ is strictly localizable, this condition implies compactness of $X$.)

• For every strictly localizable enhanced measurable space $(X',M',N')$ and complete homomorphism of localizable Boolean algebras $\phi\colon M/N\to M'/N'$ there is a morphism of enhanced measurable spaces $(X',M',N')\to(X,M,N)$ that induces $\phi$.

Equivalence to the four other categories

Previously, we constructed a functor

$$LBAlg \to CSLEMS$$

from localizable Boolean algebras and complete homomorphisms to enhanced measurable spaces. The Loomis–Sikorski construction applied to hyperstonean spaces produces compact and strictly localizable spaces, thus landing in the category $CSLEMS$ of compact strictly localizable enhanced measurable spaces.

We also constructed a functor

$$LEMS \to LBAlg, \qquad (X,M,N)\mapsto M/N,$$

which we can now restrict to a functor

$$CSLEMS \to LBAlg.$$

Previously, we saw that the compustion

$$LBAlg \to CSLEMS \to LBAlg$$

is isomorphic to the identity functor.

Showing that the composition

$$CSLEMS \to LBAlg \to CSLEMS$$

is isomorphic to the identity functor is much more difficult. Indeed, the construction of the isomorphism and its inverse requires the properties of compactness and strict localizability and uses deep theorems due to von Neumann–Maharam and Ionescu Tulcea.

Theorem

(Pavlov, Theorem 5.19.) The category $CSLEMS$ is equivalent to the following four categories:

• opposite category of commutative von Neumann algebras

• opposite category of localizable Boolean algebras (i.e., measurable locales)

• hyperstonean locales

• hyperstonean spaces

Summary

The following category is suitable for measure theory in the sense that it matches the existing practice of real analysis, probability theory, stochastic processes, etc.

• Objects are (complete) compact strictly localizable enhanced measurable spaces $(X,M,N)$.

• Morphisms $(X,M,N)\to(X',M',N')$ are equivalence classes of maps of sets $X\to X'$ that reflect measurable and negligible sets.

• Two morphisms $f$ and $g$ are equivalent if they are weakly equivalent almost everywhere for every $m'\in M'$ we have $f^{-1}m'\oplus g^{-1}m'\in N$.

The following remarks are in order.

• Enhanced measurable spaces $(X,M,N)$ have just the right amount of information: enough to define (weak) equality almost everywhere, not too much to require constructions of noncanonical measures.

• However, one could use conventional measure spaces $(X,M,\mu)$ instead, using $N_\mu$ as the σ-ideal of negligible sets.

• To ensure that composition descends to the quotient category, measurable maps must reflect negligible sets.

• The equivalence relation of equality almost everywhere must be coarsened to weak equality almost everywhere to ensure morphisms are fully determined by their preimages up to subnegligible sets.

• Completeness is optional, but simplifies some aspects of the presentation. Local determinacy is implied by strict localizability.

• The property of strict localizability is a straightforward generalization of the notion of a σ-finite space.

• The property of compactness is an abstraction of the definition of a Radon measure.

These properties eliminate pathologies and ensure the following desirable properties of objects and morphisms.

• The resulting category is equivalent to four other categories: opposite of commutative von Neumann algebras, opposite of localizable Boolean algebras (i.e., measurable locales), hyperstonean locales, hyperstonean spaces.

• Compact strictly localizable spaces are localizable and hence satisfy the traditional theorems of elementary measure theory: the Radon–Nikodym theorem, the Hahn and Jordan decomposition theorems, the Riesz representation theorem, bounded measurable functions form a von Neumann algebra, the Boolean algebra $M/N$ of equivalence classes of measurable sets is complete.

• Morphisms of compact strictly localizable enhanced measurable spaces induce complete homomorphisms of localizable Boolean algebras.

• Morphisms of enhanced measurable spaces admit essential images.

Categorical properties

The category $CSLEMS$ has excellent categorical properties: it is complete and cocomplete, admits a closed monoidal structure whose product is the measure-theoretic product, is comonadic over sets and over compact Hausdorff spaces. It also admits a commutative Giry-type probability monad (Furber).

Related concepts

• Stone duality, Gelfand duality

• commutative von Neumann algebra

• hyperstonean locale, hyperstonean space

• enhanced measurable spaces, measurable space, measure space, measure theory

References

• Irving E. Segal, Equivalences of Measure Spaces. American Journal of Mathematics 73:2 (1951), 275. doi.

• Edward Marczewski, On compact measures, Fundamenta Mathematicae 40 (1) (1953) 113–124. doi.

• John L. Kelley, Decomposition and representation theorems in measure theory. Mathematische Annalen 163:2 (1966), 89-94. doi.

• David H. Fremlin, Measure Theory.

• Dmitri Pavlov, Gelfand-type duality for commutative von Neumann algebras, Journal of Pure and Applied Algebra 226:4 (2022), 106884, 1–53. doi.

• Robert Furber?, A Probability Monad on Measure Spaces, PDF.

• Robert Furber?, Commutative W*-algebras as a Markov Category (Extended Abstract), PDF.