Almost Everywhere - Definition Using Ultrafilters

Definition Using Ultrafilters

Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that:

1. If U ∈ F and U ⊆ V then V ∈ F
2. The intersection of any two sets in F is in F
3. The empty set is not in F

A property P of points in X holds almost everywhere, relative to an ultrafilter F, if the set of points for which P holds is in F.

For example, one construction of the hyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.

The definition of almost everywhere in terms of ultrafilters is closely related to the definition in terms of measures, because each ultrafilter defines a finitely-additive measure taking only the values 0 and 1, where a set has measure 1 if and only if it is included in the ultrafilter.