Measurable spaces are the traditional 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’. The measure on $X$ is then an operation 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 explicit notion of measurable space (or an alternative such as a measurable locale).
In any case, measurable spaces are of some interest in their own right, even without a measure on them.
We give first the usual notion, assuming the validity of excluded middle and power sets; see below for alternative versions, including the constructive and predicative theories.
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 $(X,\Sigma)$).
Given measurable spaces $X$ and $Y$, a measurable function from $X$ to $Y$ is a function $f\colon X \to Y$ such that the preimage $f^*(T)$ is measurable in $X$ whenever $T$ is measurable in $Y$. Measurable spaces and measurable functions form a category $Measble$, which is topological over Set.
In classical measure theory, it is usually assumed that $Y$ is the real line (or a variation) 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.
We will briefly examine variations of the notion of measurable space, from those most like the standard to those most unlike it. Most of these are discussed at articles dedicated to them.
Historically, people have used more general notions that $\sigma$-algebras, such as algebras, $\delta$-rings, and similar concepts whose names you can probably now guess; these are all discussed at sigma-algebra. These are all more general than $\sigma$-algebras, being possibly not closed under some operations. When using some of these more general rings of measurable sets, it is necessary to allow partial functions whose domain is a relatively measurable set as measurable functions; for details, see measurable function.
It is often convenient if a measurable space has, in addition to the $\sigma$-algebra of measurable sets, a $\sigma$-ideal of measurable null sets. That is, besides the set $X$ and the $\sigma$-alebra $\Sigma$, we have a collection $N \subseteq \Sigma$ that is closed under countable unions and taking subsets (within $\Sigma$). (The elements of $N$ are the measurable null sets; a null set is any subset of a measurable null set.) One can equivalently specify a $\delta$-filter of measurable full sets; the full sets are the complements of the null sets. Either way, this allows us to use almost measurable almost functions up to almost equality, as described at measurable function.
In constructive mathematics, because complementation doesn't behave nicely, the concept of $\sigma$-algebra is not so useful. It's also essential to use almost functions to avoid a paucity of measurable functions. One solution, due to Henry Cheng?, may be found at Cheng measurable space; briefly, we use disjoint pairs? $(A,B)$ of sets instead of individual measurable sets and use formal complements in the algebra, as well as a notion of full sets. Assuming excluded middle, a Cheng measurable space is actually equivalent to a measurable space equipped with null (or full) sets, as in the previous paragraph.
In order to have the most important theorems of measure theory, it is necessary and sufficient to restrict to localizable measures. Since localizability refers only to the null (or full) sets, we can actually speak of a localizable measurable space: a measurable space equipped with null (or full) sets as above, with the property that the boolean algebra of measurable sets modulo? the null sets is complete.
Another approach to measure theory, more abstract, is to ignore the set $X$ and use only the $\sigma$-algebra $\Sigma$, as an abstract boolean algebra equipped with countable suprema; this is called a measurable algebra? (or a measure algebra? when equipped with a measure). A measurable algebra might also can be equipped with a $\sigma$-ideal of null sets (or a $\delta$-filter of full sets), but really it is simpler to take the quotient algebra?, which is itself a perfectly good measurable algebra. Even if a measurable algebra is a complete lattice, it can still be pathological; but if it has enough normal measure?s, then we have a measurable locale; the category of measurable locales is equivalent to that of localizable measurable spaces (from the previous paragraph).
Yet another category equivalent to localizable measurable spaces and measurable locales is the opposite category of the category of commutative von Neumann algebras; this is really a version of the Gelfand–Neumark theorem. Then a noncommutative (localizable) measurable space is (the formal dual of) any von Neumann algebra. In this way, measure theory may be seen as a branch of operator algebra theory (at least if one assumes that only localizable measurable spaces are well enough behaved to be worthy of study).
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.
One version of the Gel’fand–Naimark theorem states that the category of commutative $W^*$-algebras is dual to the category of localizable measurable spaces. (As such, arbitrary $W^*$-algebras may be interpreted as ‘noncommutative’ measurable spaces in a sense analogous to noncommutative geometry.) See the references below.
To make this work correctly, we cannot simply define localizability as a property of measurable spaces; instead, a localizable measurable space is a measurable space (a set $X$ with a $\sigma$-algebra $\Sigma$) with a $\sigma$-ideal $\mathcal{N}$ of $\Sigma$ and $\mathcal{P}X$ simultaneously (called the ideal of null sets) such that $\Sigma/\mathcal{N}$ is a complete lattice; and a morphism of localizable 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 $\{ x \;|\; f(x) \neq g(x) \}$ is a null set.
The requirement that $\Sigma/\mathcal{N}$ be complete is the real localizability 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.
Localizable measurable spaces can also be studied via the lattice $\Sigma/\mathcal{N}$, 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 terms of topos theory, measured spaces are closely related to Boolean toposes (e.g. Jackson 06).
While Lebesgue measure on $\mathbb{R}^n$ 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.
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 $\epsilon$-$\delta$ 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 $A_1, \ldots, A_n$ 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 $\mathbf{N}^{\mathbf{N}}$ 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!
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.
While measure theory only gets more complicated in constructive mathematics, it becomes much easier in dream mathematics.
… more coming
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 and its relation to von Neumann algebras is in
A useful series of expositions along these lines is in
Dmitri Pavlov, On measurable spaces
See also
Describe several variations of definition:
Do measurable functions only afterwards, stressing almost-everywhere equality. Describe the relevant categories.
Mention measure spaces and anything else.