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In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0). If A is a subset of X, then either A or X \ A is an element of the ultrafilter (here X \ A is the relative complement of A in X; that is, the set of all elements of X that are not in A). The concept can be generalized to Boolean algebras or even to general partial orders, and has many applications in set theory, model theory, and topology.


Formal definition

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

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

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.


The completeness of an ultrafilter U on a set is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition implies that the completeness of any ultrafilter is at least \aleph_0. An ultrafilter whose completeness is greater than \aleph_0—that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or σ-complete.

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