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 \R and \pm \infty. 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.

Definitions

If \mathcal{F} is a family of sets over \Omega (meaning that where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal{F} 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 \mu does not take on both - \infty and + \infty 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 \mu is finitely additive: Null sets A set is called a (with respect to \mu) or simply if \mu(F) = 0. Whenever \mu is not identically equal to either -\infty or +\infty then it is typically also assumed that:

<ul> <li>: if </li> </ul> **Variation and mass** The [Total variation (measure theory)](https://bliptext.com/articles/total-variation-measure-theory) S is where |\,\cdot\,| denotes the [absolute value](https://bliptext.com/articles/absolute-value) (or more generally, it denotes the [norm](https://bliptext.com/articles/norm-mathematics) or [semi](https://bliptext.com/articles/seminormed-space)[norm](https://bliptext.com/articles/norm-mathematics) if \mu is vector-valued in a ([semi](https://bliptext.com/articles/seminormed-space))[norm](https://bliptext.com/articles/norm-mathematics)ed space). Assuming that then is called the of \mu and is called the of \mu. A set function is called if for every the value \mu(F) is (which by definition means that and ; an is one that is equal to \infty or - \infty). Every finite set function must have a finite [mass](https://bliptext.com/articles/).

Common properties of set functions

A set function \mu on \mathcal{F} is said to be <ul>

<li> if it is valued in </li> <li>[Finitely additive set function](https://bliptext.com/articles/finitely-additive-set-function) if for all [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) finite sequences such that </li> <li>[Sigma-additive set function](https://bliptext.com/articles/sigma-additive-set-function) or [Sigma-additive set function](https://bliptext.com/articles/sigma-additive-set-function) if in addition to being finitely additive, for all [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sequences in \mathcal{F} such that all of the following hold: <li> <li>if is not infinite then this series must also [converge absolutely](https://bliptext.com/articles/absolute-convergence), which by definition means that must be finite. This is automatically true if \mu is [non-negative](https://bliptext.com/articles/) (or even just valued in the extended real numbers). <li>if is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if \mu is [non-negative](https://bliptext.com/articles/).</li> </ol> </li> <li>a [Pre-measure](https://bliptext.com/articles/pre-measure) if it is [non-negative](https://bliptext.com/articles/), [countably additive](https://bliptext.com/articles/sigma-additive-set-function) (including [finitely additive](https://bliptext.com/articles/)), and has a [null empty set](https://bliptext.com/articles/).</li> <li>a [Measure (mathematics)](https://bliptext.com/articles/measure-mathematics) if it is a [pre-measure](https://bliptext.com/articles/) whose domain is a [σ-algebra](https://bliptext.com/articles/algebra). That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a [null empty set](https://bliptext.com/articles/).</li> <li>a [Probability measure](https://bliptext.com/articles/probability-measure) if it is a measure that has a [mass](https://bliptext.com/articles/) of 1.</li> <li>an [Outer measure](https://bliptext.com/articles/outer-measure) if it is non-negative, [countably subadditive](https://bliptext.com/articles/), has a [null empty set](https://bliptext.com/articles/), and has the [power set](https://bliptext.com/articles/power-set) \wp(\Omega) as its domain. <li>a [Signed measure](https://bliptext.com/articles/signed-measure) if it is countably additive, has a [null empty set](https://bliptext.com/articles/), and \mu does not take on both - \infty and + \infty as values.</li> <li>[Complete measure](https://bliptext.com/articles/complete-measure) if every subset of every [null set](https://bliptext.com/articles/) is null; explicitly, this means: whenever and is any subset of F then and \mu(N) = 0. <li>[σ-finite measure](https://bliptext.com/articles/finite-measure) if there exists a sequence in \mathcal{F} such that is finite for every index i, and also </li> <li>[Decomposable measure](https://bliptext.com/articles/decomposable-measure) if there exists a subfamily of pairwise disjoint sets such that \mu(P) is finite for every and also (where ). <li>a [Vector measure](https://bliptext.com/articles/vector-measure) if it is a countably additive set function valued in a [topological vector space](https://bliptext.com/articles/topological-vector-space) X (such as a [normed space](https://bliptext.com/articles/normed-space)) whose domain is a [σ-algebra](https://bliptext.com/articles/algebra). <li>a [Complex measure](https://bliptext.com/articles/[complex](https://bliptext.com/articles/complex-number)-measure) if it is a countably additive [complex](https://bliptext.com/articles/complex-number)-valued set function whose domain is a [σ-algebra](https://bliptext.com/articles/algebra). <li>a [Random measure](https://bliptext.com/articles/random-measure) if it is a measure-valued [random element](https://bliptext.com/articles/random-element).</li> </ul> **Arbitrary sums** As described [in this article's section on generalized series](https://bliptext.com/articles/series-mathematics), for any family of [real numbers](https://bliptext.com/articles/real-number) indexed by an arbitrary [indexing set](https://bliptext.com/articles/indexing-set) I, it is possible to define their sum as the limit of the [net](https://bliptext.com/articles/net-mathematics) of finite partial sums where the domain is [directed](https://bliptext.com/articles/directed-set) by Whenever this [net](https://bliptext.com/articles/net-mathematics) converges then its limit is denoted by the symbols while if this [net](https://bliptext.com/articles/net-mathematics) instead diverges to \pm \infty then this may be indicated by writing Any sum over the empty set is defined to be zero; that is, if then by definition. For example, if z_i = 0 for every i \in I then And it can be shown that If I = \N then the generalized series converges in \R if and only if [converges unconditionally](https://bliptext.com/articles/unconditional-convergence) (or equivalently, [converges absolutely](https://bliptext.com/articles/absolute-convergence)) in the usual sense. If a generalized series converges in \R then both and also converge to elements of \R and the set is necessarily [countable](https://bliptext.com/articles/countable-set) (that is, either finite or [countably infinite](https://bliptext.com/articles/countably-infinite)); [this remains true](https://bliptext.com/articles/series-mathematics) if \R is replaced with any [normed space](https://bliptext.com/articles/normed-space). It follows that in order for a generalized series to converge in \R or \Complex, it is necessary that all but at most countably many r_i will be equal to 0, which means that is a sum of at most countably many non-zero terms. Said differently, if is uncountable then the generalized series does not converge. In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of "[countably additive](https://bliptext.com/articles/)" is rarely extended from countably many sets in \mathcal{F} (and the usual countable series ) to arbitrarily many sets (and the generalized series ).

Inner measures, outer measures, and other properties

A set function \mu is said to be/satisfies

<ul> <li> if whenever satisfy </li> <li>[Modular set function](https://bliptext.com/articles/modular-set-function) if it satisfies the following condition, known as : for all such that <li>[Submodular set function](https://bliptext.com/articles/submodular-set-function) if for all such that </li> <li> if for all finite sequences that satisfy </li> <li> or if for all sequences in \mathcal{F} that satisfy <li>[Superadditivity](https://bliptext.com/articles/superadditivity) if whenever are disjoint with </li> <li> if for all of sets in \mathcal{F} such that with and all finite. <li> if for all of sets in \mathcal{F} such that </li> <li> if whenever satisfies then for every real r > 0, there exists some such that and </li> [](https://bliptext.com/articles/)<[](https://bliptext.com/articles/)l[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)>[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/) 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set](https://bliptext.com/articles/power-set) \wp(\Omega) as its domain, and [+\infty is approached from below](https://bliptext.com/articles/).</li> <li>[Atomic measure](https://bliptext.com/articles/[atom](https://bliptext.com/articles/atom-measure-theory)ic-measure) if every measurable set of positive measure contains an [atom](https://bliptext.com/articles/atom-measure-theory).</li> </ul> If a [binary operation](https://bliptext.com/articles/binary-operation) \,+\, is defined, then a set function \mu is said to be <ul> <li> if for all and such that </li> </ul>

Topology related definitions

If \tau is a topology on \Omega then a set function \mu is said to be:

<ul> <li>a [Borel measure](https://bliptext.com/articles/borel-measure) if it is a measure defined on the σ-algebra of all [Borel sets](https://bliptext.com/articles/borel-set), which is the smallest σ-algebra containing all open subsets (that is, containing \tau).</li> <li>a [Baire measure](https://bliptext.com/articles/baire-measure) if it is a measure defined on the σ-algebra of all [Baire sets](https://bliptext.com/articles/baire-set).</li> <li>[Locally finite measure](https://bliptext.com/articles/locally-finite-measure) if for every point there exists some neighborhood of this point such that \mu(U) is finite. <li>[τ-additivity](https://bliptext.com/articles/additivity) if whenever is [directed](https://bliptext.com/articles/directed-set) with respect to and satisfies <li>[Inner regular measure](https://bliptext.com/articles/inner-regular-measure) or if for every </li> <li>[Outer regular measure](https://bliptext.com/articles/outer-regular-measure) if for every </li> <li>[Regular measure](https://bliptext.com/articles/regular-measure) if it is both inner regular and outer regular.</li> <li>a [Borel regular measure](https://bliptext.com/articles/borel-regular-measure) if it is a Borel measure that is also [Regular measure](https://bliptext.com/articles/regular-measure).</li> <li>a [Radon measure](https://bliptext.com/articles/radon-measure) if it is a regular and locally finite measure.</li> <li>[Strictly positive measure](https://bliptext.com/articles/strictly-positive-measure) if every non-empty open subset has (strictly) positive measure.</li> <li>a [Valuation (measure theory)](https://bliptext.com/articles/valuation-measure-theory) if it is non-negative, [monotone](https://bliptext.com/articles/), [modular](https://bliptext.com/articles/), has a [null empty set](https://bliptext.com/articles/), and has domain \tau.</li> </ul>

Relationships between set functions

If \mu and \nu are two set functions over \Omega, then:

<ul> <li>\mu is said to be [Absolute continuity (measure theory)](https://bliptext.com/articles/absolute-continuity-measure-theory) or [Domination (measure theory)](https://bliptext.com/articles/domination-measure-theory), written if for every set F that belongs to the domain of both \mu and \nu, if \nu(F) = 0 then \mu(F) = 0. <li>\mu and \nu are [Singular measure](https://bliptext.com/articles/singular-measure), written if there exist disjoint sets M and N in the domains of \mu and \nu such that \mu(F) = 0 for all in the domain of \mu, and \nu(F) = 0 for all in the domain of \nu.</li> </ul>

Examples

Examples of set functions include: The Jordan measure on \Reals^n is a set function defined on the set of all Jordan measurable subsets of \Reals^n; it sends a Jordan measurable set to its Jordan measure.

Lebesgue measure

The Lebesgue measure on \Reals is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue \sigma-algebra. Its definition begins with the set of all intervals of real numbers, which is a semialgebra on \Reals. The function that assigns to every interval I its is a finitely additive set function (explicitly, if I has endpoints a \leq b then ). This set function can be extended to the Lebesgue outer measure on \Reals, which is the translation-invariant set function that sends a subset to the infimum Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the 𝜎-algebra of all subsets that satisfy the Carathéodory criterion: is a measure that called Lebesgue measure. Vitali sets are examples of non-measurable sets of real numbers.

Infinite-dimensional space

As detailed in the article on infinite-dimensional Lebesgue measure, the only locally finite and translation-invariant Borel measure on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define Gaussian measures on infinite-dimensional topological vector spaces. The structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space.

Finitely additive translation-invariant set functions

The only translation-invariant measure on with domain \wp(\Reals) that is finite on every compact subset of \Reals is the trivial set function that is identically equal to 0 (that is, it sends every to 0) However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in [0, 1]. In fact, such non-trivial set functions will exist even if \Reals is replaced by any other abelian group G.

Extending set functions

Extending from semialgebras to algebras

Suppose that \mu is a set function on a semialgebra \mathcal{F} over \Omega and let which is the algebra on \Omega generated by The archetypal example of a semialgebra that is not also an algebra is the family on where for all Importantly, the two non-strict inequalities ,\leq, in cannot be replaced with strict inequalities ,<, since semialgebras must contain the whole underlying set \R^d; that is, is a requirement of semialgebras (as is ). If \mu is finitely additive then it has a unique extension to a set function on defined by sending (where ,\sqcup, indicates that these are pairwise disjoint) to: This extension will also be finitely additive: for any pairwise disjoint If in addition \mu is extended real-valued and monotone (which, in particular, will be the case if \mu is non-negative) then will be monotone and finitely subadditive: for any such that

Extending from rings to σ-algebras

If is a pre-measure on a ring of sets (such as an algebra of sets) \mathcal{F} over \Omega then \mu has an extension to a measure on the σ-algebra generated by If \mu is σ-finite then this extension is unique. To define this extension, first extend \mu to an outer measure \mu^* on by and then restrict it to the set of \mu^-measurable sets (that is, Carathéodory-measurable sets), which is the set of all such that It is a \sigma-algebra and \mu^ is sigma-additive on it, by Caratheodory lemma.

Restricting outer measures

If is an outer measure on a set \Omega, where (by definition) the domain is necessarily the power set \wp(\Omega) of \Omega, then a subset is called **' or **' if it satisfies the following : where is the complement of M. The family of all \mu^–measurable subsets is a σ-algebra and the restriction of the outer measure \mu^ to this family is a measure.