Formal Definition
Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that
- The empty set is not an element of U
- 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.
- If A and B are elements of U, then so is the intersection of A and B.
- 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:
- There is no filter F finer than U, i.e., implies U = F.
- implies or .
- 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.
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