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In measure theory (a branch of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
), 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, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
, and is analogous to the notion of ''
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
'' in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. 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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, the Lebesgue measure 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 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, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
s with
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
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 can also have other meanings).


Definition

If (X,\Sigma,\mu) is a measure space, a property P is said to hold almost everywhere in X if there exists a set N \in \Sigma with \mu(N) = 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(x) 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 In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
of measures. * If (P_n) 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 (P_x)_ 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(y) 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".


Examples

* If ''f'' : R → R is a
Lebesgue integrable In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
function and f(x) \ge 0 almost everywhere, then \int_a^b f(x) \, dx \geq 0 for all real numbers a < b with equality
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
f(x) = 0 almost everywhere. * If ''f'' : 'a'', ''b''→ R is a
monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
, then ''f'' is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
almost everywhere. * If ''f'' : R → R is
Lebesgue measurable In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
and \int_a^b , f(x), \, dx < \infty for all real numbers a < b , then there exists a set ''E'' (depending on ''f'') such that, if ''x'' is in ''E'', the Lebesgue mean \frac \int_^ f(t)\,dt converges to as \epsilon decreases to zero. The set ''E'' is called the Lebesgue set of ''f''. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of ''f'' converges to ''f'' almost everywhere. * A bounded
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is
Riemann integrable In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
if and only if it is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
almost everywhere. * As a curiosity, the decimal expansion of almost every real number in the interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
contains the complete text of
Shakespeare's plays Shakespeare's plays are a canon of approximately 39 dramatic works written by English poet, playwright, and actor William Shakespeare. The exact number of plays—as well as their classifications as tragedy, history, comedy, or otherwise—is a ...
, encoded in
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
; similarly for every other finite digit sequence, see
Normal number In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to b ...
.


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 (or "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 proper filter o ...
. 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, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
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.


References


Bibliography

* {{Measure theory Measure theory Mathematical terminology ja:ほとんど (数学)#ほとんど至るところで