28 lines
1.4 KiB
Markdown
28 lines
1.4 KiB
Markdown
---
|
|
title: "Almost everywhere"
|
|
chunk: 2/2
|
|
source: "https://en.wikipedia.org/wiki/Almost_everywhere"
|
|
category: "reference"
|
|
tags: "science, encyclopedia"
|
|
date_saved: "2026-05-05T07:23:13.771862+00:00"
|
|
instance: "kb-cron"
|
|
---
|
|
|
|
== 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:
|
|
|
|
If U ∈ F and U ⊆ V then V ∈ F
|
|
The intersection of any two sets in F is in F
|
|
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.
|
|
|
|
== See also ==
|
|
Dirichlet's function, a function that is equal to 0 almost everywhere.
|
|
Cantor function
|
|
|
|
== References ==
|
|
|
|
== Bibliography ==
|
|
Billingsley, Patrick (1995). Probability and measure (3rd ed.). New York: John Wiley & Sons. ISBN 0-471-00710-2. |