Stone’s Representation Theorem III
We conclude our coverage of Stone’s representation theorem with a version for Boolean -algebra. Each such algebra is isomorphic to a -algebra of subsets of some space, modulo a -ideal.
We start as we did for any Boolean algebra, by using the map sending to the Boolean algebra of clopen subsets of the Stone space . This algebra is, of course, our identified base. Let be the -ring generated by this base.
When we first dealt with measure rings, we quotiented out by the ideal of negligible sets. We don’t have a measure on , so we don’t have measurable sets, but we do have something almost as good. Just as when we discussed Baire spaces, we can use nowhere-denseness as a topological stand-in for negligibility. In fact, we’ll define a “meager” set to be any countable union of nowhere-dense sets, and we’ll let be the collection of all meager sets in . It is straightforward to verify that is a -ideal in .
A quick side note: classically, meager sets were said to be “of the first category”. Any other sets were said to be “of the second category”. This is the root of the term “category” in the Baire category theorem.
Now if is a sequence of clopen sets, then we can find a unique preimage . Since is a -algebra, we can take the countable union of these elements, and then apply to the result. That is, we can define the set
Now, of course, we can also take the countable union of the sets themselves. If were an isomorphism of -algebras, then this union would be exactly itself; but we aren’t usually so lucky. Instead, I say that the difference
is nowhere-dense, and thus countable unions of clopen sets are clopen modulo meager sets.
Indeed, the countable union of the (open) sets is open, and so its complement is closed. The difference above is the intersection of the (closed) set with the (closed) complement of the union, and is thus closed. So if it were dense on some nonempty open set it would have to actually contain . But since is open it must be the union of some collection of basic clopen sets, and we can take to be one of these sets. That is, and for each . Since these relations only involve clopen sets, we find that and . But there’s no way this can happen if is the union of the !
So the map takes elements of to clopen sets in , and then on to their equivalence classes in , and this map commutes with countable unions. All that remains to show that this is an isomorphism is to show that no two clopen sets in represent the same equivalence class. That is, if and are distinct clopen sets, then cannot be meager. Equivalently, no nonempty clopen set is meager.
But this is just a consequence of the Baire category theorem, here in its formulation for compact Hausdorff spaces. Indeed, if a clopen set in any Baire space — and a compact Hausdorff space is Baire — were the countable union of closed nowhere-dense sets, then its interior would be empty. But since it’s open its interior is itself, and thus is would have to be empty. Thus if and are clopen sets representing the same equivalence class modulo then their (clopen) symmetric difference is meager, and thus empty. That is, .
No comments yet.