In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and
A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
If is a family of sets over (meaning that where denotes the powerset) then a is a function with domain and codomain or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures.
The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.
In general, it is typically assumed that is always well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is finitely additive:
is defined with satisfying and
Null sets
A set is called a (with respect to ) or simply if
Whenever is not identically equal to either or then it is typically also assumed that:
if
Variation and mass
The is
where denotes the absolute value (or more generally, it denotes the norm or seminorm if is vector-valued in a (semi)normed space).
Assuming that then is called the of and is called the of
A set function is called if for every the value is (which by definition means that and ; an is one that is equal to or ).
Every finite set function must have a finite mass.
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Dans ce cours on définira et étudiera la notion de mesure et d'intégrale contre une mesure dans un cadre général, généralisant ce qui a été fait en Analyse IV dans le cas réel.
On verra aussi quelques
The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
This is an introductory course to the concentration of measure phenomenon - random functions that depend on many random variables tend to be often close to constant functions.
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent).
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. Formally, a complex measure on a measurable space is a complex-valued function that is sigma-additive. In other words, for any sequence of disjoint sets belonging to , one has As for any permutation (bijection) , it follows that converges unconditionally (hence absolutely).
In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
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