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In measure theory (a branch of mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set
Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set ....
, i.e. is a set with measure zero, or in cases where the measure is not complete, contained within a set of measure zero. If used for properties of the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, the Lebesgue measure
Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
 is assumed unless otherwise stated. It is abbreviated a. e.; in older literature one can find p.






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In measure theory (a branch of mathematical analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set
Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set ....
, i.e. is a set with measure zero, or in cases where the measure is not complete, contained within a set of measure zero. If used for properties of the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, the Lebesgue measure
Lebesgue measure

In mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a length, area or volume to subsets of Euclidean space....
 is assumed unless otherwise stated. It is abbreviated a. e.; in older literature one can find p. p. instead, which stands for the equivalent French language
French language

French is a Romance language spoken around the world by around 80 million people as first language, by 190 million as second language, and by about another 200 million people as an acquired tongue, with significant speakers in 54 countries....
 phrase presque partout.

A set with full measure is one whose complement is of measure zero.

Occasionally, instead of saying that a property holds almost everywhere, one also says that the property holds for almost all elements, though the term almost all
Almost all

In mathematics, the phrase almost all has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finite setly many" or "all but a countable set" ; see almost....
 also has other meanings.

Here are some theorems that involve the term "almost everywhere":

  • If f : R ? R is a Lebesgue integrable function and f(x) = 0 almost everywhere, then




for all real numbers a < b.


  • If f : [a, b] ? R is a monotonic function, then f is differentiable
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
     almost everywhere.
  • If f : R ? R is Lebesgue measurable and




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




converges to f(x) as decreases to zero. The set E is called the Lebesgue set of f. Its compliment can be proved to have measure zero. In other words, the Lebesgue mean of f converges to f almost everywhere.


  • If f(x,y) is Borel measurable on R2 then for almost every x, the function y?f(x,y) is Borel measurable.


  • A bounded function
    Function (mathematics)

    The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
     f : [a, b] -> R is Riemann integrable
    Riemann integral

    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 ....
     if and only if it is continuous
    Continuous function

    In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
     almost everywhere.


Outside of the context of real analysis, the notion of a property true almost everywhere can be defined in terms of an ultrafilter
Ultrafilter

In the mathematics field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter , that cannot be enlarged ....
. For example, one construction of the hyperreal number
Hyperreal number

The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Isaac Newton and Gottfried Leibniz....
 system defines a hyperreal number as an equivalence class of sequences that are equal almost everywhere as defined by an ultrafilter.

In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
, the phrases become almost surely
Almost surely

In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory....
, almost certain or almost always, corresponding to a probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
 of 1.