Application of ultrafilters

An ultrafilter on a set X is a collection \mathcal{U} of subsets of X with the following properties:

  1. \emptyset \notin\mathcal{U} and X\in\mathcal{U};
  2. \mathcal{U} is closed under finite intersection;
  3. if A\in\mathcal{U} and A\subset B, then B\in\mathcal{U}
  4. fr every A\subset X, either A\in\mathcal{U} or X\setminus A\in\mathcal{U}.

A trivial example of an ultrafilter is the collection of all sets containing some fixed element x of X. 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 \mathcal{U} be an ultrafilter on a set X.

  1. If X is partitioned into finitely many sets A_1,\dots,A_n, the precisely one A_i belongs to \mathcal{U}.
  2. If F and G do not belong to \mathcal{U} then neither does F\cup G.
  3. If any finite set belongs to \mathcal{U} then \mathcal{U} 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 L by finding a linear functional \phi 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 \phi to be linear, and we would like \phi(a) to equal L(a) whenever a is convergent sequence.

If \mathcal{U} is a non-principal ultrafilter, and (a_1,a_2,\dots) is a sequence that takes values in [-1,1], then we can define a limit along \mathcal{U} as follows. Let \mathcal{J} be the collection of all subintervals J of [-1,1] such that \{n:a_n\in J\} belongs \mathcal{U}. Then the ultrafilter properties of \mathcal{U} imply that \mathcal{J} has all ultrafilter properties but restricted to intervals.

From this it follows that \mathcal{J} is something like a “principal interval-ultrafilter”. More precisely, it contains all open intervals that contain some particular point a. To see this, for each n\in\mathbf{N} partition [-1,1] into finitely many subintervals of length at most 1/n. Then one of these subintervals belongs to \mathcal{J}. So for every n we have an interval \mathcal{J}_n of length 1/n that belongs to \mathcal{J}. Now let I_n=J_1\cap \dots\cap J_n. Since \mathcal{J} is closed under intersection, I_n belongs to \mathcal{J}. Let \{a\} be the intersection of the closures of the  I_n (which are non-empety and nested). If U is any open interval containing a, then U contains some I_n, so belongs to \mathcal{U}.

Thus ,we have found a number a with the following property: for every \varepsilon>0, the set \{n:|a_n-a|<\varepsilon\} belongs to \mathcal{U}. Moreover, it is easy to see that this a is unique. We write it as \lim_{\mathcal{U}}a_n. It is easy to cheak that \lim_{\mathcal{U}} is linear.

To see ever more clearly how this ties in with the usual notion of a limit, note that a_n converges to a if and only if for every \varepsilon>0, the set \{n:|a_n-a|<\varepsilon\} 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 \mathcal{U} is an ultrafilter then one often finds oneself writing sentences of the form \{x\in X:P(x)\}\in\mathcal{U}, as we have already seen. But it can be much easier to deal with these sentences if one instead writes \mathcal{U}x\in X\ P(x). One can read this as  “For \mathcal{U}-almost every x\in X\ P(x)“.

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