nLab
measurable space

Idea

Measurable spaces are a necessary prelude to the general theory of measure and integration. Basically, a measure is a recipe for computing the size — e.g. length, area, volume — of subsets of a given set $X$. The structure of a ‘measurable space’ picks out those subsets of $X$ for which the size is well-defined. These subsets are called ‘measurable’. A ‘measure’ on $X$ is then a recipe that assigns a number to each measurable subset saying how big it is.

In short: you get a measure space by placing a measure on a measurable space.

Ideally, all subsets would be measurable, but this contradicts the axiom of choice for the basic example of Lebesgue measure on the real line. Although it is possible to use nonstandard foundations of mathematics in which all subsets of the real line are Lebesgue measurable, any general theory that includes that example and is more general than those foundations requires some notion of measurable space.

In any case, measurable spaces are of some interest in their own right, even without a measure on them.

Definitions

We assume the law of excluded middle; see below for the constructive theory.

Given a set $X$, a $\sigma$-algebra is a collection of subsets of $X$ that is closed under complementation, countable unions, and countable intersections. A measurable space, by the usual modern definition, is a set $X$ equipped with a $\sigma$-algebra $\Sigma$. The elements of $\Sigma$ are called the measurable sets of $X$ (or more properly, the measurable subsets of $(X,\Sigma )$).

Historically, people have used more general notions that $\sigma$-algebras, such as $\delta$-rings and similar concepts whose names you can probably now guess; these are discussed at sigma-algebra.

Measurable functions

Given measurable spaces $X$ and $Y$, a measurable function from $X$ to $Y$ is a function $f:X\to Y$ such that the preimage $f^{*}(T)$ is measurable in $X$ whenever $T$ is measurable in $Y$. Measurable spaces and measurable functions form a category $\mathrm{Measble}$, which is topological over Set.

In classical measure theory, it is usually assumed that $Y$ is the real line (or a variation such as the extended real line or the complex plane) equipped with the Borel sets (see the examples below). Then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I\subseteq Y$ is an interval.

In some variations of measure theory based on $\delta$‑ or $\sigma$-rings instead of on $\sigma$-algebras, it is necessary to allow partial functions whose domain is a relatively measurable set. Classically (when $Y$ is the real line), one achieves (for purposes of integration) essentially the same result by requiring only that $f^{-1}(I)$ be measurable whenever $I\subseteq Y$ is an interval that does not contain $0$; in other words, one effectively assumes that $f$ is zero wherever it would otherwise be undefined.

Examples

• Of course, the power set of $X$ is closed under all operations, so it is a $\sigma$-algebra. Thus every set becomes a discrete measurable space.

• If $X$ is a topological space, then the $\sigma$-algebra generated by the open sets (or equivalently, by the closed sets) in $X$ is the Borel $\sigma$-algebra; its elements are called the Borel sets of $X$. In particular, the Borel sets of real numbers are the Borel sets in the real line with its usual topology.

• If a measurable space $(X,\Sigma )$ is equipped with a measure $\mu$, making it into a measure space, then the sets of measure zero form a $\sigma$-ideal of $\Sigma$ (that is an ideal that is also a sub-$\sigma$-ring). Let a null set be any set (measurable or not) contained in a set of measure zero; then the null sets form a $\sigma$-ideal in the power set of $X$. Call a set $\mu$-measurable if it is the union of a measurable set and a null set; then the $\mu$-measurable sets form a $\sigma$-algebra ${\Sigma }_{\mu }$ called the completion of $\Sigma$ under $\mu$, and the measurable space $(X,{\Sigma }_{\mu })$ is the completion of $(X,\Sigma )$.

• In particular, the Lebesgue-measurable sets in the real line are the elements of the completion of the Borel $\sigma$-algebra under Lebesgue measure.

Properties

Relation to von Neumann algebras

One version of the Gel’fand–Naimark theorem states that the category of commutative $W^{*}$-algebras is dual to the category of localisable measurable spaces. (As such, arbitrary $W^{*}$-algebras may be interpreted as ‘noncommutative’ measurable spaces in a sense analogous to noncommutative geometry.)

To make this work correctly, we cannot simply define localisability as a property of measurable spaces; instead, a localisable measurable space is a measurable space (a set $X$ with a $\sigma$-algebra $\Sigma$) with a $\sigma$-ideal $𝒩$ of $\Sigma$ and $𝒫X$ simultaneously (called the ideal of null sets) such that $\Sigma /𝒩$ is a complete lattice; and a morphism of localisable measurable spaces is a measurable function, with the property that the preimage of any null set is null, up to an equivalence relation where $f\cong g$ if $\{x\mid f(x)\ne g(x)\}$ is a null set.

The requirement that $\Sigma /𝒩$ be complete is the real localisability condition here; the trick of equipping a measurable space with a $\sigma$-ideal of null sets (or equivalently a $\sigma$-filter of full sets) and taking measurable functions only up to equivalence is a common one in other situations.

Localisable measurable spaces can also be studied via the lattice $\Sigma /𝒩$, which is a frame; the morphisms correspond to certain continuous maps between locales, and thus we are studying locales with extra structure, called measurable locales.

In alternative foundations

While Lebesgue measure on ${ℝ}^n$ can be done in very weak foundations, a general theory of measure and measurable spaces seems to require powerful set-theoretic machinery. Indeed, not much seems to be possible in predicative contexts, and the (nonpredicative) constructive theory is noticeably more complicated than the classical theory. On the other hand, the classical theory has its own complications, with nonmeasurable sets and functions that can be proved to exist but which seem to never arise in practice. Instead, there are classically false but apparently consistent foundations in which measure theory is extremely simple.

Predicative theory

The main problem for measure theory in predicative mathematics is getting your hands on a $\sigma$-algebra. Once you've got that, you've got a measurable space (obviously) and go on to measure space, where there are no new difficulties. However, what is (say) a Borel set in the real line? This is difficult, if not impossible, to explain predicatively. (In the case of Lebesgue measure, there are ways to describe the Lebesgue-measurable sets predicatively, but these do not seem to generalise to a broader theory.)

Note that there is no real problem in describing what, say, an open set is. Not only can this be done for the real line in the usual $ϵ$-$\delta$ way, but it is easy to take any collection of subsets of any set $X$, call that collection a subbase, and describe which sets are the open sets in the topology generated by that subbase. The reason is that there are only two steps in moving from a subbase to a topology, and while the latter step is too impredicative to allow one to speak of the set of all open sets, it's OK if you only want to talk about individual open sets. (To be explicit: given a collection $B$ to be used as subbase, a set $G$ is open if, for every point $x$, if $x\in G$, then there exist a natural number $n$ and elements $A_1,\dots ,A_n$ of $B$ such that $x\in A_i$ for each $i$ and, for every point $y$, if $y\in A_i$ for each $i$, then $y\in G$. Since we quantify only over points and natural numbers, not over sets or functions, this is a predicative definition, and it's easy to prove that the open sets satisfy the axioms of a topology.)

This cannot be done with $\sigma$-algebras, since we need uncountably many sets. To be sure, each individual step is predicative, and we can freely talk about $G_{\delta }$ sets and the like, but to define a Borel set we need to quantify over all countable ordinals. While it is possible to hypothesise the existence of an uncountable ordinal ${\omega }_1$ and be predicative ‘over’ ${\omega }_1$ (and after all, everything else in this section is only predicative over the first infinite ordinal ${\omega }_0$, which we only have if we accept an axiom of infinity), this cannot be constructed predicatively. (The immediate definition of ${\omega }_1$ as the Hartog's number of ${\omega }_0$ uses power sets; while the construction of an uncountable ordinal by applying the well-ordering theorem to the function set $N^N$ doesn't seem to use reasoning that requires the existence of power sets as long as you don't also throw in excluded middle, it does use reasoning that is not accepted by any predicative school that I know.)

So as far as I (Toby Bartels) can tell, there is no general predicative theory of measurable spaces, only an ad hoc theory of Lebsegue measurability. I would be delighted to learn otherwise!

Constructive theory

From a constructive perspective, there are a couple of related problems with the classical theory. One is that the notion of $\sigma$-algebra is highly suspicious, because it relies on an operation, complementation, that behaves very differently in the intuitionistic logic that constructive mathematics uses. The other is that, even you acept the definition of $\sigma$-algebra anyway (after all, the Lebesgue-measurable sets on the real line do still form one), there may be very few measurable functions.

Indeed, if we set aside the general theory of measurable spaces and simply do Lebesgue measure ad hoc in a constructive (even predicative) way, we find that instead of measurable functions we really want measurable partial functions whose domain of definition is a full set. This suggests that if we want to define the concept of measurable function, then we have to know what the full sets are.

There is a way out, due to Henry Cheng?, for both of these problems at once. Instead of dealing with individual sets, we will deal with pairs of disjoint sets. The intuition is that we use disjoint pairs $(A,B)$ such that $A\cup B$ is full —with $(A,\neg A)$ being the motivating example in the classical theory—, but we let the $\sigma$-algebra itself tell us which pairs those are. Once we fix a particular measure, we may find additional pairs whose union is full, somewhat like finding additional measurable sets when taking the completion in the classical theory (although taking the completion is a separate phenomenon here), but that's all right; the important thing is that each pair chosen really is full in any measure used (much as each set in a classical $\sigma$-algebra must actually be measurable by any measure used).

See details at Cheng space.

In dream mathematics

While measure theory only gets more complicated in constructive mathematics, it becomes much easier in dream mathematics.

… more coming …

Related concepts

• measurable locale

References

For Cheng's theory of measure spaces, see the 1985 edition of Bishop & Bridges, Constructive Analysis. (And the references therein, obviously, but I haven't read those.)

A discussion of the abstract properties of the category of localizable measurable spaces is in

• Alain Guichardet, Sur la catégorie des algèbres de von Neumann (French) Bull. Sci. Math. (2) 90 1966 41–64. (MSN)

A useful series of expositions along these lines is in

• Dmitri Pavlov, On measurable spaces

  • MO comment 1

  • MO comment 2

Revision plan

1. Describe several variations of definition:

   • with a filter of full sets,
   • localisable,
   • Cheng,
   • from a delta-ring etc,
   • localic,
   • finitary,
   • noncommutative.

2. Do measurable functions only afterwards, stressing almost-everywhere equality. Describe the relevant categories.

3. Mention measure spaces and anything else.