This is a continuation of the previous post on the classification of complete probability spaces. Last time we set up the basic terminology and notation, and saw that for a complete probability space , the -algebra is countably generated mod 0 (which we denoted (CG0)) if and only if the pseudo-metric makes into a separable metric space. We also considered as a measured abstract -algebra with non non-trivial null sets; this is the point of view we will continue with in this post.
Our goal is to present the result from [HvN] and [Ro] (see references from last time) that classifies such measured abstract -algebras up to -algebra isomorphism. To this end, let be a separable measured abstract -algebra with no non-trivial null sets; let be the maximal element of , and suppose that . Note that is the minimal element of , which would correspond to if were a collection of subsets of some ambient space. In the abstract setting, we will write for this minimal element.
An element is an atom if it has no proper non-trivial subelement; that is, if implies or . By the assumption that has no non-trivial null sets, we have for every atom. Note that for any two atoms we have , and so .
Let be the set of atoms; then we have , so is (at most) countable. Let
then is non-atomic; it contains no atoms. Thus can be decomposed as a non-atomic part together with a countable collection of atoms. In particular, to classify we may assume without loss of generality that is non-atomic.
Consider the unit interval with Lebesgue measure; let denote the set of equivalence classes (mod 0) of measurable subsets of , and denote Lebesgue measure, so is a measured abstract -algebra. Moreover, is separable, non-atomic, has no non-trivial null sets, and has total weight 1.
The meaning of “isomorphism” here is that there is a bijection that preserves the Boolean algebra structure and carries to . That is: iff ; ; ; ; ; and finally, . We are most used to obtaining morphisms between -algebras as a byproduct of having a measurable map between the ambient sets; that is, given measurable spaces and , a measurable map gives a morphism . However, in the abstract setting we currently work in, there is no ambient space, so we cannot yet interpret this way. Eventually we will go from and ( back to the measure spaces that induced them, and then we will give conditions for to be induced by a function between those spaces, but for now we stick with the abstract viewpoint.
We sketch a proof of Theorem 1 that roughly follows [HvN, Theorem 1]; see also [Ro, §1.3]. Given a measurable set , we write for the equivalence class of ; in particular, we write for the equivalence class of the interval .
Observe that if is dense, then is generated by ; thus we can describe by finding such that
- whenever ,
- generates ,
and so we may reasonably set for .
To extend this further it is helpful to rephrase the last step. Consider
and observe that (1) can be rewritten as
Writing for the lexicographic order on , we observe that each of these three is of the form for some .
Each above is determined as follows: tells us whether to use or , and tells us whether to use or . This generalises very naturally to : given , let
Now put ; this gives elements (the last one is , and was omitted from our earlier bookkeeping). These have the property that whenever . Moreover, writing , the collection generates because does. Let ; now we have all the pieces of the construction that we asked for earlier.
Putting it all together, let . The following are now relatively straightforward exercises.
- is dense in since is non-atomic.
- For each there is with ; if such that this holds, then either or (or vice versa). For this step we actually need to choose a little more carefully to guarantee that each is non-trivial.
- The previous step guarantees that is a bijection between and .
- Since preserves the order on the generating collections, it preserves the -algebra structure as well.
- Since carries to on the generating collections, it carries to on the whole -algebra.
Thus we have produced a -algebra isomorphism , proving Theorem 1. Next time we will discuss conditions under which this can be extended to an isomorphism of the measure spaces themselves, and not just their abstract measured -algebras.