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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.

Families of sets over v t e
Is necessarily true of or, is closed under:	Directed by									F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if	only if
𝜆-system (Dynkin System)				only if			only if or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter (Filter base)
Filter subbase
Open Topology							(even arbitrary )			Never
Closed Topology						(even arbitrary )				Never
Is necessarily true of or, is closed under:	directed downward	finite intersections	finite unions	relative complements	complements in	countable intersections	countable unions	contains	contains	Finite Intersection Property
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in are arbitrary elements of and it is assumed that

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.

A set function on is said to be^[1]

• non-negative if it is valued in
• finitely additive if for all pairwise disjoint finite sequences such that
• countably additive or σ-additive^[2] if in addition to being finitely additive, for all pairwise disjoint sequences in such that all of the following hold:
  1. 
  2. if is not infinite then this series must also converge absolutely, which by definition means that must be finite. This is automatically true if is non-negative (or even just valued in the extended real numbers).
  3. 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 is non-negative.
• a pre-measure if it is non-negative, countably additive (including finitely additive), and has a null empty set.
• a measure if it is a pre-measure whose domain is a σ-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.
• a probability measure if it is a measure that has a mass of
• an outer measure if it is non-negative, countably subadditive, has a null empty set, and has the power set as its domain.
  • Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets
• a signed measure if it is countably additive, has a null empty set, and does not take on both and as values.
• complete if every subset of every null set is null; explicitly, this means: whenever and is any subset of then and
• 𝜎-finite if there exists a sequence in such that is finite for every index and also
• decomposable if there exists a subfamily of pairwise disjoint sets such that is finite for every and also (where ).
• a vector measure if it is a countably additive set function valued in a topological vector space (such as a normed space) whose domain is a σ-algebra.
• a complex measure if it is a countably additive complex-valued set function whose domain is a σ-algebra.
  • By definition, a complex measure never takes as a value and so has a null empty set.
• a random measure if it is a measure-valued random element.

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 ).

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

• monotone if whenever satisfy
• modular if it satisfies the following condition, known as modularity: for all such that
  • Every finitely additive function on a field of sets is modular.
  • In geometry, a set function valued in some abelian semigroup that possess this property is known as a valuation. This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.
• submodular if for all such that
• finitely subadditive if for all finite sequences that satisfy
• countably subadditive or σ-subadditive if for all sequences in that satisfy
• superadditive if whenever are disjoint with
• continuous from above if for all non-increasing sequences of sets in such that with and all finite.
• continuous from below if for all non-decreasing sequences of sets in such that
• infinity is approached from below if whenever satisfies then for every real there exists some such that and
• an outer measure if is non-negative, countably subadditive, has a null empty set, and has the power set as its domain.
• an inner measure if is non-negative, superadditive, continuous from above, has a null empty set, has the power set as its domain, and is approached from below.
• atomic if every measurable set of positive measure contains an atom.

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

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

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.

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.

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.

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

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]

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.

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.

• Absolute continuity (measure theory) – Form of continuity for functions
• Boolean ring – Algebraic structure in mathematics
• Cylinder set measure
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Hadwiger's theorem – Theorem in integral geometry
• Hahn decomposition theorem – Measurability theorem
• Invariant measure – Concept in mathematics
• Lebesgue's decomposition theorem – Theorem in mathematical measure theory
• Positive and negative sets
• Radon–Nikodym theorem – Expressing a measure as an integral of another
• Riesz–Markov–Kakutani representation theorem – Statement about linear functionals and measures
• Ring of sets – Family closed under unions and relative complements
• σ-algebra – Algebraic structure of set algebra
• Vitali–Hahn–Saks theorem

Proofs

• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
• Kolmogorov, Andrey; Fomin, Sergei V. (2012) [1957]. Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
• A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0
• Royden, Halsey; Fitzpatrick, Patrick (15 January 2010). Real Analysis (4 ed.). Boston: Prentice Hall. ISBN 978-0-13-143747-0. OCLC 456836719.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

• Sobolev, V.I. (2001) [1994], "Set function", Encyclopedia of Mathematics, EMS Press
• Regular set function at Encyclopedia of Mathematics