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 measure 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 set function on 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:

Set difference formula: is defined with satisfying and

Null sets

A set is called a null set (with respect to ) or simply null if Whenever is not identically equal to either or then it is typically also assumed that:

Variation and mass

The total variation of a set 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 total variation of and is called the mass of

A set function is called finite if for every the value is finite (which by definition means that and ; an infinite value is one that is equal to or ). Every finite set function must have a finite mass.

Common properties of set functions

A set function on is said to be[1]

Arbitrary sums

As described in this article's section on generalized series, for any family of real numbers indexed by an arbitrary indexing set it is possible to define their sum as the limit of the net of finite partial sums where the domain is directed by Whenever this net converges then its limit is denoted by the symbols while if this net instead diverges to 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 for every then And it can be shown that If then the generalized series converges in if and only if converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series converges in then both and also converge to elements of and the set is necessarily countable (that is, either finite or countably infinite); this remains true if is replaced with any normed space.[proof 1] It follows that in order for a generalized series to converge in or it is necessary that all but at most countably many will be equal to 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" is rarely extended from countably many sets in (and the usual countable series ) to arbitrarily many sets (and the generalized series ).

Inner measures, outer measures, and other properties

A set function is said to be/satisfies[1]

If a binary operation is defined, then a set function is said to be

Topology related definitions

If is a topology on then a set function is said to be:

Relationships between set functions

See also: Radon–Nikodym theorem and Lebesgue's decomposition theorem

If and are two set functions over then:


Examples of set functions include:

The Jordan measure on is a set function defined on the set of all Jordan measurable subsets of it sends a Jordan measurable set to its Jordan measure.

Lebesgue measure

The Lebesgue measure on is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue -algebra.[5]

Its definition begins with the set of all intervals of real numbers, which is a semialgebra on The function that assigns to every interval its is a finitely additive set function (explicitly, if has endpoints then ). This set function can be extended to the Lebesgue outer measure on 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

See also: Gaussian measure § Infinite-dimensional spaces, Abstract Wiener space, Feldman–Hájek theorem, and Radonifying function

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 that is finite on every compact subset of is the trivial set function that is identically equal to (that is, it sends every to )[6] 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 In fact, such non-trivial set functions will exist even if is replaced by any other abelian group [7]

Theorem[8] — If is any abelian group then there exists a finitely additive and translation-invariant[note 1] set function of mass

Extending set functions

See also: Carathéodory's extension theorem

Extending from semialgebras to algebras

Suppose that is a set function on a semialgebra over and let

which is the algebra on generated by The archetypal example of a semialgebra that is not also an algebra is the family
on where for all [9] Importantly, the two non-strict inequalities in cannot be replaced with strict inequalities since semialgebras must contain the whole underlying set that is, is a requirement of semialgebras (as is ).

If is finitely additive then it has a unique extension to a set function on defined by sending (where indicates that these are pairwise disjoint) to:[9]

This extension will also be finitely additive: for any pairwise disjoint [9]

If in addition is extended real-valued and monotone (which, in particular, will be the case if is non-negative) then will be monotone and finitely subadditive: for any such that [9]

Extending from rings to σ-algebras

See also: Pre-measure and Hahn–Kolmogorov theorem

If is a pre-measure on a ring of sets (such as an algebra of sets) over then has an extension to a measure on the σ-algebra generated by If is σ-finite then this extension is unique.

To define this extension, first extend to an outer measure on by

and then restrict it to the set of -measurable sets (that is, Carathéodory-measurable sets), which is the set of all such that
It is a -algebra and is sigma-additive on it, by Caratheodory lemma.

Restricting outer measures

See also: Outer measure § Measurability of sets relative to an outer measure

If is an outer measure on a set where (by definition) the domain is necessarily the power set of then a subset is called –measurable or Carathéodory-measurable if it satisfies the following Carathéodory's criterion:

where is the complement of

The family of all –measurable subsets is a σ-algebra and the restriction of the outer measure to this family is a measure.

See also


  1. ^ a b Durrett 2019, pp. 1–37, 455–470.
  2. ^ Durrett 2019, pp. 466–470.
  3. ^ Royden & Fitzpatrick 2010, p. 30.
  4. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
  5. ^ Kolmogorov and Fomin 1975
  6. ^ Rudin 1991, p. 139.
  7. ^ Rudin 1991, pp. 139–140.
  8. ^ Rudin 1991, pp. 141–142.
  9. ^ a b c d Durrett 2019, pp. 1–9.
  1. ^ The function being translation-invariant means that for every and every subset


  1. ^ Suppose the net converges to some point in a metrizable topological vector space (such as or a normed space), where recall that this net's domain is the directed set Like every convergent net, this convergent net of partial sums is a Cauchy net, which for this particular net means (by definition) that for every neighborhood of the origin in there exists a finite subset of such that for all finite supersets this implies that for every (by taking and ). Since is metrizable, it has a countable neighborhood basis at the origin, whose intersection is necessarily (since is a Hausdorff TVS). For every positive integer pick a finite subset such that for every If belongs to then belongs to Thus for every index that does not belong to the countable set


Further reading