An ultrafilter on a set is a collection of subsets of with the following properties:
- and ;
- is closed under finite intersection;
- if and , then
- fr every , either or .
A trivial example of an ultrafilter is the collection of all sets containing some fixed element of . Such ultrafilters are called principal. It is not trivial that there are any non-principal ultrafilters, but this can be proved using Zorn’s lemma.
The following facts about filters, which follow easily from the basic definition, will be used in this post. Let be an ultrafilter on a set .
- If is partitioned into finitely many sets , the precisely one belongs to .
- If and do not belong to then neither does .
- If any finite set belongs to then is a principal filter.
Examples 1: generalized limits
We can think of the process of taking limits of sequence as a linear functional defined on the convergent sequences.
Can we generalize by finding a linear functional that is defined on all bounded sequences and not just all convergent on ? In order for it to count as a generalization, we would like to be linear, and we would like to equal whenever is convergent sequence.
If is a non-principal ultrafilter, and is a sequence that takes values in , then we can define a limit along as follows. Let be the collection of all subintervals of such that belongs . Then the ultrafilter properties of imply that has all ultrafilter properties but restricted to intervals.
From this it follows that is something like a “principal interval-ultrafilter”. More precisely, it contains all open intervals that contain some particular point . To see this, for each partition into finitely many subintervals of length at most . Then one of these subintervals belongs to . So for every we have an interval of length that belongs to . Now let . Since is closed under intersection, belongs to . Let be the intersection of the closures of the (which are non-empety and nested). If is any open interval containing , then contains some , so belongs to .
Thus ,we have found a number a with the following property: for every , the set belongs to . Moreover, it is easy to see that this is unique. We write it as . It is easy to cheak that is linear.
To see ever more clearly how this ties in with the usual notion of a limit, note that converges to if and only if for every , the set belongs to the cofinite filter.
Set systems as quantifiers
It is often better to think of a set system as a quantifiers. In particular, if is an ultrafilter then one often finds oneself writing sentences of the form , as we have already seen. But it can be much easier to deal with these sentences if one instead writes . One can read this as “For -almost every “.