Functions on Boolean Rings and Measure Rings
We’re not just interested in Boolean rings as algebraic structures, but we’re also interested in real-valued functions on them. Given a function on a Boolean ring , we say that is additive, or a measure, -finite (on -rings), and so on analogously to the same concepts for set functions. We also say that a measure on a Boolean ring is “positive” if it is zero only for the zero element of the Boolean ring.
Now, if is the Boolean -ring that comes from a measurable space , then usually a measure on is not positive under this definition, since there exist sets of measure zero. However, remember that in measure theory we usually talk about things that happen almost everywhere. That is, we consider two sets — two elements of — to be “the same” if their difference is negligible — if the value of takes the value zero on this difference. If we let be the collection of -negligible sets, it turns out that is an ideal in the Boolean -ring . Indeed, if and are negligible, then so is , so is an Abelian subgroup. Further, if and , then , so is an ideal.
So we can form the quotient ring , which consists of the equivalence classes of elements which differ by an element of measure zero. This is equivalent to our old rhetorical trick of only considering properties up to “almost everywhere”. Using this new definition of “equals zero”, any measure on a Boolean -ring gives rise to a positive measure on the quotient -ring . In particular, given a measure space , we write for the Boolean -ring it gives rise to.
We say that a “measure ring” is a Boolean -ring together with a positive measure on . For instance, if is a measure space, then is a measure ring.. If is a Boolean -algebra we say that is a measure algebra. We say that measure rings and algebras are (totally) finite or -finite the same as for measure spaces. Measure rings, of course, form a category; a morphism from one measure algebra to another is a morphism of boolean -algebras so that for all .
I say that the mapping which sends a measure space to its associated measure algebra is a contravariant functor. Indeed, let be a morphism of measure spaces. That is, is a measurable function from to , so contains the pulled-back -algebra . This pull-back defines a map . Further, since is a morphism of measure spaces it must push forward to . That is, , or in other words . And so if then , thus the ideal is sent to the ideal , and so descends to a homomorphism between the quotient rings: . As we just said, , and thus we have a morphism of measure algebras . It’s straightforward to confirm that this assignment preserves identities and compositions.