# 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 $x\in X\ P(x)$“.