TheInfoList

In
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
(a branch of
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of
measure zero In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
, and is analogous to the notion of ''
almost surely In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
'' in
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, the
Lebesgue measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucli ...
is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent
French language French ( or ) is a Romance language The Romance languages, less commonly Latin or Neo-Latin languages, are the modern languages that evolved from Vulgar Latin Vulgar Latin, also known as Popular or Colloquial Latin is a range of inf ...

phrase ''presque partout''. A set with full measure is one whose complement is of measure zero. In probability theory, the terms ''almost surely'', ''almost certain'' and ''almost always'' refer to
event Event may refer to: Gatherings of people * Ceremony A ceremony (, ) is a unified ritual A ritual is a sequence of activities involving gestures, words, actions, or objects, performed in a sequestered place and according to a set sequence. Rit ...
s with
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

1 not necessarily including all of the outcomes. These are exactly the sets of full measure in a probability space. Occasionally, instead of saying that a property holds almost everywhere, it is said that the property holds for almost all elements (though the term
almost all In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
can also have other meanings).

# Definition

If $\left(X,\Sigma,\mu\right)$ is a
measure space A measure space is a basic object of measure theory Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after si ...
, a property $P$ is said to hold almost everywhere in $X$ if there exists a set $N \in \Sigma$ with $\mu\left(N\right) = 0$, and all $x\in X\setminus N$ have the property $P$. Another common way of expressing the same thing is to say that "almost every point satisfies $P\,$", or that "for almost every $x$, $P\left(x\right)$ holds". It is ''not'' required that the set $\$ has measure 0; it may not belong to $\Sigma$. By the above definition, it is sufficient that $\$ be contained in some set $N$ that is measurable and has measure 0.

# Properties

* If property $P$ holds almost everywhere and implies property ''$Q$'', then property ''$Q$'' holds almost everywhere. This follows from the
monotonicity Figure 3. A function that is not monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
of measures. * If $\left(P_n\right)$ is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction $\forall n P_n$ holds almost everywhere. This follows from the countable sub-additivity of measures. * By contrast, if $\left(P_x\right)_$ is an uncountable family of properties, each of which holds almost everywhere, then their conjunction $\forall x P_x$ does not necessarily hold almost everywhere. For example, if $\mu$ is Lebesgue measure on $X = \mathbf R$ and $P_x$ is the property of not being equal to $x$ (i.e. $P_x\left(y\right)$ is true if and only if $y \neq x$), then each $P_x$ holds almost everywhere, but the conjunction $\forall x P_x$ does not hold anywhere. As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction. This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point".

# 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 In the mathematical field of order theory, an ultrafilter on a given partially ordered set (poset) ''P'' is a certain subset of ''P,'' namely a maximal filter on ''P'', that is, a proper filter on ''P'' that cannot be enlarged to a bigger pr ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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.