In various areas of mathematics, classification theorems give a more or less complete understanding of what kinds of behaviour are possible. For example, in linear algebra we learn that up to isomorphism, is the only real vector space with dimension , and every linear operator on a finite-dimensional vector space can be put into Jordan normal form via a change of coordinates; this means that many questions in linear algebra can be answered by understanding properties of Jordan normal form. A similar classification result is available in measure theory, but the preliminaries are a little more involved. In this and the next post I will describe the classification result for complete probability spaces, which gives conditions under which such a space is equivalent to the unit interval with Lebesgue measure.
The main original references for these results are a 1942 paper by Halmos and von Neumann [“Operator methods in classical mechanics. II”, Ann. of Math. (2), 43 (1942), 332–350, and a 1949 paper by Rokhlin [“On the fundamental ideas of measure theory”, Mat. Sbornik N.S. 25(67) (1949). 107–150, English translation in Amer. Math. Soc. Translation 1952 (1952). no. 71, 55 pp.]. I will refer to these as [HvN] and [Ro], respectively.
1. Equivalence of measure spaces
First we must establish exactly which class of measure spaces we work with, and under what conditions two measure spaces will be thought of as equivalent. Let be the unit interval and the Borel and Lebesgue -algebras, respectively; let be Lebesgue measure (on either of these). To avoid having to distinguish between and , let us agree to only work with complete measure spaces; this is no great loss, since given an arbitrary metric space we can pass to its completion .
The most obvious notion of isomorphism is that two complete measure spaces and are isomorphic if there is a bijection such that are measurable and ; that is, given we have if and only if , and in this case .
In the end we want to loosen this definition a little bit. For example, consider the space of all infinite binary sequences, equipped with the Borel -algebra associated to the product topology (or if you prefer, the metric ). Let be the -Bernoulli measure on ; that is, for each the cylinder gets weight . Then there is a natural correspondence between the completion and given by
By looking at dyadic intervals one can readily verify that ; however, is not a bijection because for every we have .
The points at which is non-injective form a -null set (since there are only countably many of them), so from the point of view of measure theory, it is natural to disregard them. This motivates the following definition.
Definition 1 Two measure spaces and are isomorphic mod 0 if there are measurable sets and such that , together with a bijection such that are measurable and .
From now on we will be interested in the question of classifying complete measure spaces up to isomorphism mod 0. The example above suggests that is a reasonable candidate for a `canonical’ complete measure space that many others are equivalent to, and we will see that this is indeed the case.
Notice that the total measure is clearly an invariant of isomorphism mod 0, and hence we restrict our attention to probability spaces, for which .
2. Separability, etc.
Let be a probability space. We describe several related conditions that all give senses in which can be understood via countable objects.
The -algebra carries a natural pseudo-metric given by . Write if ; this is an equivalence relation on , and we write for the space of equivalence classes. The function induces a metric on in the natural way, and we say that is separable if the metric space is separable; that is, if it has a countable dense subset.
Another countability condition is this: call “countably generated” if there is a countable subset such that is the smallest -algebra containing . We write (CG) for this property; for example, the Borel -algebra in satisfies (CG) because we can take to be the set of all intervals with rational endpoints. (In [HvN], such an is called “strictly separable”, but we avoid the word “separable” as we have already used it in connection with the metric space .)
In and of itself, (CG) is not quite the right sort of property for our current discussion, because it does not hold when we pass to the completion; the Lebesgue -algebra is not countably generated (one can prove this using cardinality estimates). Let us say that satisfies property (CG0) (for “countably generated mod 0”) if there is a countably generated -algebra with the property that for every , there is with . In other words, we have . Note that is countably generated mod 0 by taking . (In [HvN], such an is called “separable”; the same property is used in §2.1 of [Ro] with the label , rendered in a font that I will not attempt to duplicate here.)
In fact, the approximation of by satisfies an extra condition. Let us write (CG0+) for the following condition on : there is a countably generated such that for every , there is with and . This is satisfied for and . (In [HvN], such an is called “properly separable”; the same property is used in §2.1 of [Ro] with the label .)
The four properties introduced above are related as follows.
The first two implications are immediate, and their converses fail in general:
- The Lebesgue -algebra satisfies (CG0+) but not (CG).
- Let . Then satisfies (CG0) but not (CG0+).
Now we prove that (CG0) and separability are equivalent. First note that if is a countable subset, then the algebra generated by is also countable; in particular, is separable if and only if there is a countable algebra that is dense with respect to , and similarly in the definition of (CG0) the generating set can be taken to be an algebra. To show equivalence of (CG0) and separability it suffices to show that given an algebra and , we have
First we prove by proving that is a -algebra, and hence contains ; this will show that (CG0) implies separability.
- Closure under : if then there are such that . Since and (since it is an algebra), this gives .
- Closure under : given , let . To show that , note that given any , there are such that . Let and ; note that
Moreover by continuity from below we have , so , and thus for sufficiently large we have . This holds for all , so .
Now we prove , thus proving that is “large enough” that separability implies (CG0). Given any , there are such that . Let We get
and similarly, , which gives
Then , which completes the proof of .
The first half of the argument above (the direction) appears in this MathOverflow answer to a question discussing the relationship between different notions of separability, which ultimately inspired this post. That answer (by Joel David Hamkins) also suggests one further notion of “countably generated”, distinct from all of the above; say that satisfies (CCG) (for “completion of countably generated”) if there is a countably generated -algebra such that , where is the completion of with respect to the measure . One quickly sees that
Both reverse implications fail; the Lebesgue -algebra satisfies (CCG) but not , and an example satisfying separability (and hence (CG0)) but not (CCG) was given in that same MathOverflow answer (the example involves ordinal numbers and something called the “club filter”, which I will not go into here).
3. Abstract -algebras
It is worth looking at some of the previous arguments through a different lens, that will also appear next time when we discuss the classification problem.
Recall the space of equivalence classes from earlier, where means that . Although elements of are not subsets of , we can still speak of the “union” of two such elements by choosing representatives from the respective equivalence classes; that is, given , we choose representatives and (so ), and consider the “union” of and to be the equivalence class of ; write this as . One can easily check that this is well-defined; if and , then .
This shows that induces a binary operation on the space ; similarly, induces a binary operation , complementation induces an operation , and set inclusion induces a partial order . These give the structure of a Boolean algebra; say that is an abstract Boolean algebra if it has a partial order , binary operations , , and a unary operation , satisfying the same rules as inclusion, union, intersection, and complementation:
- is the join of and (the minimal element such that ), and is the meet of and (the maximal element such that );
- the distributive laws and hold;
- there is a maximal element whose complement is the minimal element;
- and .
For the form of this list I have followed this blog post by Terry Tao, which gives a good in-depth discussion of some other issues relating to concrete and abstract Boolean algebras and -algebras.
Exercise 1 Using the four axioms above, prove the following properties:
- is the unique element satisfying (4) — that is, if and , then ;
- de Morgan’s laws: and .
If you get stuck, see Chapter IV, Lemma 1.2 in A Course in Universal Algebra by Burris and Sankappanavar.
In fact inherits just a little bit more, since (and hence ) can be iterated countably many times. We add this as a fifth axiom, and say that an abstract Boolean algebra is an abstract -algebra if in addition to (1)–(4) it satisfies
- any countable family has a least upper bound and a greatest lower bound .
A measured abstract -algebra is a pair , where is an abstract -algebra and is a function satisfying the usual properties: and whenever for all . (Note that is playing the role of , but we avoid the latter notation to remind ourselves that elements of do not need to be represented as subsets of some ambient space.)
The operations induce a binary operator on by
which is the abstract analogue of set difference, and so a measured abstract -algebra carries a pseudo-metric defined by
If has the property that for all , then this becomes a genuine metric.
In particular, if is a measure space and is the space of equivalence classes modulo (equivalence mod 0), then induces a function , which we continue to denote by , such that is a measured abstract -algebra; this has the property that for all non-trivial , and so it defines a metric as above.
Given an abstract -algebra and a subset , the algebra (-algebra) generated by is the smallest algebra (-algebra) in that contains . Now we can interpret the equivalence (1) from the previous section (which drove the correspondence between (CG0) and separability) in terms of the measured abstract -algebra .
Next time we will see how separability (or equivalently, (CG0)) can be used to give a classification result for abstract measured -algebras, which at first requires us to take the abstract point of view introduced in this section. Finally, we will see what is needed to go from there to a similar result for probability spaces.