Editing Ultrafilter (section)

==Ultrafilter on the power set of a set==
{{Main|Ultrafilter (set theory)}}

Given an arbitrary set <math>X,</math> its [[power set]] <math>{\mathcal P}(X),</math> ordered by [[set inclusion]], is always a Boolean algebra; hence the results of the above section apply. An (ultra)filter on <math>{\mathcal P}(X)</math> is often called just an "(ultra)filter on <math>X</math>".<ref name="notation warning" group=note/> Given an arbitrary set <math>X,</math> an ultrafilter on <math>{\mathcal P}(X)</math> is a set <math>\mathcal U</math> consisting of subsets of <math>X</math> such that:
#The empty set is not an element of <math>\mathcal U</math>.
#If <math>A</math> is an element of <math>\mathcal U</math> then so is every superset <math>B\supset A</math>.
#If <math>A</math> and <math>B</math> are elements of <math>\mathcal U</math> then so is the [[Intersection (set theory)|intersection]] <math>A\cap B</math>.
#If <math>A</math> is a subset of <math>X,</math> then either<ref name="exclusive or" group="note">Properties 1 and 3 imply that <math>A</math> and <math>X \setminus A</math> cannot {{em|both}} be elements of <math>U.</math></ref> <math>A</math> or its complement <math>X \setminus A</math> is an element of <math>\mathcal U</math>.

Equivalently, a family <math>\mathcal U</math> of subsets of <math>X</math> is an ultrafilter if and only if for any finite collection <math>\mathcal F</math> of subsets of <math>X</math>, there is some <math>x\in X</math> such that <math>\mathcal U\cap\mathcal F=F_x\cap\mathcal F</math> where <math>F_x=\{Y\subseteq X : x \in Y\}</math> is the principal ultrafilter seeded by <math>x</math>. In other words, an ultrafilter may be seen as a family of sets which "locally" resembles a principal ultrafilter.{{cn|date=December 2023}}

An equivalent form of a given <math>\mathcal U</math> is a [[2-valued morphism]], a function <math>m</math> on <math>{\mathcal P}(X)</math> defined as <math>m(A) = 1</math> if <math>A</math> is an element of <math>\mathcal U</math> and <math>m(A) = 0</math> otherwise. Then <math>m</math> is [[finitely additive]], and hence a {{em|[[Content (measure theory)|content]]}} on <math>{\mathcal P}(X),</math> and every property of elements of <math>X</math> is either true [[almost everywhere]] or false almost everywhere. However, <math>m</math> is usually not {{em|countably additive}}, and hence does not define a [[Measure (mathematics)|measure]] in the usual sense.

For a filter <math>\mathcal F</math> that is not an ultrafilter, one can define <math>m(A) = 1</math> if <math>A \in \mathcal F</math> and <math>m(A) = 0</math> if <math>X \setminus A \in \mathcal F,</math> leaving <math>m</math> undefined elsewhere.<ref name="Kruckman.2012">{{Cite web |title=Notes on Ultrafilters |url=https://math.berkeley.edu/~kruckman/ultrafilters.pdf|author=Alex Kruckman|date=November 7, 2012|publisher=Berkeley Math Toolbox Seminar}}</ref>