Ultrafilter - Formal Definition

Formal Definition

Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that

1. The empty set is not an element of U
2. If A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U.
3. If A and B are elements of U, then so is the intersection of A and B.
4. If A is a subset of X, then either A or X \ A is an element of U. (Note: axioms 1 and 3 imply that A and X \ A cannot both be elements of U.)

A characterization is given by the following theorem. A filter U on a set X is an ultrafilter if any of the following conditions are true:

1. There is no filter F finer than U, i.e., implies U = F.
2. implies or .
3. or .

Another way of looking at ultrafilters on a set X is to define a function m on the power set of X by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Then m is a finitely additive measure on X, and every property of elements of X is either true almost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be countably additive.

For a filter F which is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.