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Set function - Wikipedia

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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 {\displaystyle \mathbb {R} \cup \{\pm \infty \},} which consists of the real numbers {\displaystyle \mathbb {R} } and {\displaystyle \pm \infty .}

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 {\displaystyle {\mathcal {F}}} is a family of sets over {\displaystyle \Omega } (meaning that {\displaystyle {\mathcal {F}}\subseteq \wp (\Omega )} where {\displaystyle \wp (\Omega )} denotes the powerset) then a set function on {\displaystyle {\mathcal {F}}} is a function {\displaystyle \mu } with domain {\displaystyle {\mathcal {F}}} and codomain {\displaystyle [-\infty ,\infty ]} 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 {\displaystyle {\mathcal {F}}} of sets over {\displaystyle \Omega }
Is necessarily true of {\displaystyle {\mathcal {F}}\colon }
or, is {\displaystyle {\mathcal {F}}} closed under: 		Directed
by {\displaystyle \,\supseteq } 		{\displaystyle A\cap B} {\displaystyle A\cup B} {\displaystyle B\setminus A} {\displaystyle \Omega \setminus A} {\displaystyle A_{1}\cap A_{2}\cap \cdots } {\displaystyle A_{1}\cup A_{2}\cup \cdots } {\displaystyle \Omega \in {\mathcal {F}}} {\displaystyle \varnothing \in {\mathcal {F}}} F.I.P.
π-system Yes Yes No No No No No No No No
Semiring Yes Yes No No No No No No Yes Never
Semialgebra (Semifield) Yes Yes No No No No No No Yes Never
Monotone class No No No No No only if {\displaystyle A_{i}\searrow } only if {\displaystyle A_{i}\nearrow } No No No
𝜆-system (Dynkin System) Yes No No only if
{\displaystyle A\subseteq B}		 Yes No only if {\displaystyle A_{i}\nearrow } or
they are disjoint		 Yes Yes Never
Ring (Order theory) Yes Yes Yes No No No No No No No
Ring (Measure theory) Yes Yes Yes Yes No No No No Yes Never
δ-Ring Yes Yes Yes Yes No Yes No No Yes Never
𝜎-Ring Yes Yes Yes Yes No Yes Yes No Yes Never
Algebra (Field) Yes Yes Yes Yes Yes No No Yes Yes Never
𝜎-Algebra (𝜎-Field) Yes Yes Yes Yes Yes Yes Yes Yes Yes Never
Dual ideal Yes Yes Yes No No No Yes Yes No No
Filter Yes Yes Yes Never Never No Yes Yes {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Prefilter (Filter base) Yes No No Never Never No No No {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Filter subbase No No No Never Never No No No {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Open Topology Yes Yes Yes No No No
(even arbitrary {\displaystyle \cup })		 Yes Yes Never
Closed Topology Yes Yes Yes No No
(even arbitrary {\displaystyle \cap })		 No Yes Yes Never
Is necessarily true of {\displaystyle {\mathcal {F}}\colon }
or, is {\displaystyle {\mathcal {F}}} closed under: 		directed
downward 		finite
intersections 		finite
unions 		relative
complements 		complements
in {\displaystyle \Omega } 		countable
intersections 		countable
unions 		contains {\displaystyle \Omega } contains {\displaystyle \varnothing } Finite
Intersection
Property

Additionally, a semiring is a π-system where every complement {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in {\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in {\displaystyle {\mathcal {F}}.}
{\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of {\displaystyle {\mathcal {F}}} and it is assumed that {\displaystyle {\mathcal {F}}\neq \varnothing .}

In general, it is typically assumed that {\displaystyle \mu (E)+\mu (F)} is always well-defined for all {\displaystyle E,F\in {\mathcal {F}},} or equivalently, that {\displaystyle \mu } does not take on both {\displaystyle -\infty } and {\displaystyle +\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 {\displaystyle \mu } is finitely additive:

Set difference formula: {\displaystyle \mu (F)-\mu (E)=\mu (F\setminus E){\text{ whenever }}\mu (F)-\mu (E)} is defined with {\displaystyle E,F\in {\mathcal {F}}} satisfying {\displaystyle E\subseteq F} and {\displaystyle F\setminus E\in {\mathcal {F}}.}

Null sets

A set {\displaystyle F\in {\mathcal {F}}} is called a null set (with respect to {\displaystyle \mu }) or simply null if {\displaystyle \mu (F)=0.} Whenever {\displaystyle \mu } is not identically equal to either {\displaystyle -\infty } or {\displaystyle +\infty } then it is typically also assumed that:

Variation and mass

The total variation of a set {\displaystyle S} is {\displaystyle |\mu |(S)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup\{|\mu (F)|:F\in {\mathcal {F}}{\text{ and }}F\subseteq S\}} where {\displaystyle |\,\cdot \,|} denotes the absolute value (or more generally, it denotes the norm or seminorm if {\displaystyle \mu } is vector-valued in a (semi)normed space). Assuming that {\displaystyle \cup {\mathcal {F}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\textstyle \bigcup \limits _{F\in {\mathcal {F}}}F\in {\mathcal {F}},} then {\displaystyle |\mu |\left(\cup {\mathcal {F}}\right)} is called the total variation of {\displaystyle \mu } and {\displaystyle \mu \left(\cup {\mathcal {F}}\right)} is called the mass of {\displaystyle \mu .}

A set function is called finite if for every {\displaystyle F\in {\mathcal {F}},} the value {\displaystyle \mu (F)} is finite (which by definition means that {\displaystyle \mu (F)\neq \infty } and {\displaystyle \mu (F)\neq -\infty }; an infinite value is one that is equal to {\displaystyle \infty } or {\displaystyle -\infty }). Every finite set function must have a finite mass.

Common properties of set functions

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A set function {\displaystyle \mu } on {\displaystyle {\mathcal {F}}} is said to be[1]

Arbitrary sums

As described in this article's section on generalized series, for any family {\displaystyle \left(r_{i}\right)_{i\in I}} of real numbers indexed by an arbitrary indexing set {\displaystyle I,} it is possible to define their sum {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} as the limit of the net of finite partial sums {\displaystyle F\in \operatorname {FiniteSubsets} (I)\mapsto \textstyle \sum \limits _{i\in F}r_{i}} where the domain {\displaystyle \operatorname {FiniteSubsets} (I)} is directed by {\displaystyle \,\subseteq .\,} Whenever this net converges then its limit is denoted by the symbols {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} while if this net instead diverges to {\displaystyle \pm \infty } then this may be indicated by writing {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}=\pm \infty .} Any sum over the empty set is defined to be zero; that is, if {\displaystyle I=\varnothing } then {\displaystyle \textstyle \sum \limits _{i\in \varnothing }r_{i}=0} by definition.

For example, if {\displaystyle z_{i}=0} for every {\displaystyle i\in I} then {\displaystyle \textstyle \sum \limits _{i\in I}z_{i}=0.} And it can be shown that {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}=\textstyle \sum \limits _{\stackrel {i\in I,}{r_{i}=0}}r_{i}+\textstyle \sum \limits _{\stackrel {i\in I,}{r_{i}\neq 0}}r_{i}=0+\textstyle \sum \limits _{\stackrel {i\in I,}{r_{i}\neq 0}}r_{i}=\textstyle \sum \limits _{\stackrel {i\in I,}{r_{i}\neq 0}}r_{i}.} If {\displaystyle I=\mathbb {N} } then the generalized series {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} converges in {\displaystyle \mathbb {R} } if and only if {\displaystyle \textstyle \sum \limits _{i=1}^{\infty }r_{i}} converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} converges in {\displaystyle \mathbb {R} } then both {\displaystyle \textstyle \sum \limits _{\stackrel {i\in I}{r_{i}>0}}r_{i}} and {\displaystyle \textstyle \sum \limits _{\stackrel {i\in I}{r_{i}<0}}r_{i}} also converge to elements of {\displaystyle \mathbb {R} } and the set {\displaystyle \left\{i\in I:r_{i}\neq 0\right\}} is necessarily countable (that is, either finite or countably infinite); this remains true if {\displaystyle \mathbb {R} } is replaced with any normed space.[proof 1] It follows that in order for a generalized series {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} to converge in {\displaystyle \mathbb {R} } or {\displaystyle \mathbb {C} ,} it is necessary that all but at most countably many {\displaystyle r_{i}} will be equal to {\displaystyle 0,} which means that {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}~=~\textstyle \sum \limits _{\stackrel {i\in I}{r_{i}\neq 0}}r_{i}} is a sum of at most countably many non-zero terms. Said differently, if {\displaystyle \left\{i\in I:r_{i}\neq 0\right\}} is uncountable then the generalized series {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} 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 {\displaystyle F_{1},F_{2},\ldots \,} in {\displaystyle {\mathcal {F}}} (and the usual countable series {\displaystyle \textstyle \sum \limits _{i=1}^{\infty }\mu \left(F_{i}\right)}) to arbitrarily many sets {\displaystyle \left(F_{i}\right)_{i\in I}} (and the generalized series {\displaystyle \textstyle \sum \limits _{i\in I}\mu \left(F_{i}\right)}).

Inner measures, outer measures, and other properties

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A set function {\displaystyle \mu } is said to be/satisfies[1]

If a binary operation {\displaystyle \,+\,} is defined, then a set function {\displaystyle \mu } is said to be

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

Relationships between set functions

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

Examples of set functions include:

The Jordan measure on {\displaystyle \mathbb {R} ^{n}} is a set function defined on the set of all Jordan measurable subsets of {\displaystyle \mathbb {R} ^{n};} it sends a Jordan measurable set to its Jordan measure.

The Lebesgue measure on {\displaystyle \mathbb {R} } is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue {\displaystyle \sigma }-algebra.[5]

Its definition begins with the set {\displaystyle \operatorname {Intervals} (\mathbb {R} )} of all intervals of real numbers, which is a semialgebra on {\displaystyle \mathbb {R} .} The function that assigns to every interval {\displaystyle I} its {\displaystyle \operatorname {length} (I)} is a finitely additive set function (explicitly, if {\displaystyle I} has endpoints {\displaystyle a\leq b} then {\displaystyle \operatorname {length} (I)=b-a}). This set function can be extended to the Lebesgue outer measure on {\displaystyle \mathbb {R} ,} which is the translation-invariant set function {\displaystyle \lambda ^{\!*\!}:\wp (\mathbb {R} )\to [0,\infty ]} that sends a subset {\displaystyle E\subseteq \mathbb {R} } to the infimum {\displaystyle \lambda ^{\!*\!}(E)=\inf \left\{\sum _{k=1}^{\infty }\operatorname {length} (I_{k}):{(I_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of open intervals with }}E\subseteq \bigcup _{k=1}^{\infty }I_{k}\right\}.} Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the 𝜎-algebra of all subsets {\displaystyle M\subseteq \mathbb {R} } that satisfy the Carathéodory criterion: {\displaystyle \lambda ^{\!*\!}(M)=\lambda ^{\!*\!}(M\cap E)+\lambda ^{\!*\!}(M\cap E^{c})\quad {\text{ for every }}S\subseteq \mathbb {R} } is a measure that called Lebesgue measure. Vitali sets are examples of non-measurable sets of real numbers.

Infinite-dimensional space

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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

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The only translation-invariant measure on {\displaystyle \Omega =\mathbb {R} } with domain {\displaystyle \wp (\mathbb {R} )} that is finite on every compact subset of {\displaystyle \mathbb {R} } is the trivial set function {\displaystyle \wp (\mathbb {R} )\to [0,\infty ]} that is identically equal to {\displaystyle 0} (that is, it sends every {\displaystyle S\subseteq \mathbb {R} } to {\displaystyle 0})[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 {\displaystyle [0,1].} In fact, such non-trivial set functions will exist even if {\displaystyle \mathbb {R} } is replaced by any other abelian group {\displaystyle G.}[7]

Theorem[8] — If {\displaystyle (G,+)} is any abelian group then there exists a finitely additive and translation-invariant[note 1] set function {\displaystyle \mu :\wp (G)\to [0,1]} of mass {\displaystyle \mu (G)=1.}

Extending set functions

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Extending from semialgebras to algebras

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Suppose that {\displaystyle \mu } is a set function on a semialgebra {\displaystyle {\mathcal {F}}} over {\displaystyle \Omega } and let {\displaystyle \operatorname {algebra} ({\mathcal {F}}):=\left\{F_{1}\sqcup \cdots \sqcup F_{n}:n\in \mathbb {N} {\text{ and }}F_{1},\ldots ,F_{n}\in {\mathcal {F}}{\text{ are pairwise disjoint }}\right\},} which is the algebra on {\displaystyle \Omega } generated by {\displaystyle {\mathcal {F}}.} The archetypal example of a semialgebra that is not also an algebra is the family {\displaystyle {\mathcal {S}}_{d}:=\{\varnothing \}\cup \left\{\left(a_{1},b_{1}\right]\times \cdots \times \left(a_{1},b_{1}\right]~:~-\infty \leq a_{i}<b_{i}\leq \infty {\text{ for all }}i=1,\ldots ,d\right\}} on {\displaystyle \Omega :=\mathbb {R} ^{d}} where {\displaystyle (a,b]:=\{x\in \mathbb {R} :a<x\leq b\}} for all {\displaystyle -\infty \leq a<b\leq \infty .}[9] Importantly, the two non-strict inequalities {\displaystyle \,\leq \,} in {\displaystyle -\infty \leq a_{i}<b_{i}\leq \infty } cannot be replaced with strict inequalities {\displaystyle \,<\,} since semialgebras must contain the whole underlying set {\displaystyle \mathbb {R} ^{d};} that is, {\displaystyle \mathbb {R} ^{d}\in {\mathcal {S}}_{d}} is a requirement of semialgebras (as is {\displaystyle \varnothing \in {\mathcal {S}}_{d}}).

If {\displaystyle \mu } is finitely additive then it has a unique extension to a set function {\displaystyle {\overline {\mu }}} on {\displaystyle \operatorname {algebra} ({\mathcal {F}})} defined by sending {\displaystyle F_{1}\sqcup \cdots \sqcup F_{n}\in \operatorname {algebra} ({\mathcal {F}})} (where {\displaystyle \,\sqcup \,} indicates that these {\displaystyle F_{i}\in {\mathcal {F}}} are pairwise disjoint) to:[9] {\displaystyle {\overline {\mu }}\left(F_{1}\sqcup \cdots \sqcup F_{n}\right):=\mu \left(F_{1}\right)+\cdots +\mu \left(F_{n}\right).} This extension {\displaystyle {\overline {\mu }}} will also be finitely additive: for any pairwise disjoint {\displaystyle A_{1},\ldots ,A_{n}\in \operatorname {algebra} ({\mathcal {F}}),} [9] {\displaystyle {\overline {\mu }}\left(A_{1}\cup \cdots \cup A_{n}\right)={\overline {\mu }}\left(A_{1}\right)+\cdots +{\overline {\mu }}\left(A_{n}\right).}

If in addition {\displaystyle \mu } is extended real-valued and monotone (which, in particular, will be the case if {\displaystyle \mu } is non-negative) then {\displaystyle {\overline {\mu }}} will be monotone and finitely subadditive: for any {\displaystyle A,A_{1},\ldots ,A_{n}\in \operatorname {algebra} ({\mathcal {F}})} such that {\displaystyle A\subseteq A_{1}\cup \cdots \cup A_{n},}[9] {\displaystyle {\overline {\mu }}\left(A\right)\leq {\overline {\mu }}\left(A_{1}\right)+\cdots +{\overline {\mu }}\left(A_{n}\right).}

Extending from rings to σ-algebras

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If {\displaystyle \mu :{\mathcal {F}}\to [0,\infty ]} is a pre-measure on a ring of sets (such as an algebra of sets) {\displaystyle {\mathcal {F}}} over {\displaystyle \Omega } then {\displaystyle \mu } has an extension to a measure {\displaystyle {\overline {\mu }}:\sigma ({\mathcal {F}})\to [0,\infty ]} on the σ-algebra {\displaystyle \sigma ({\mathcal {F}})} generated by {\displaystyle {\mathcal {F}}.} If {\displaystyle \mu } is σ-finite then this extension is unique.

To define this extension, first extend {\displaystyle \mu } to an outer measure {\displaystyle \mu ^{*}} on {\displaystyle 2^{\Omega }=\wp (\Omega )} by {\displaystyle \mu ^{*}(T)=\inf \left\{\sum _{n}\mu \left(S_{n}\right):T\subseteq \cup _{n}S_{n}{\text{ with }}S_{1},S_{2},\ldots \in {\mathcal {F}}\right\}} and then restrict it to the set {\displaystyle {\mathcal {F}}_{M}} of {\displaystyle \mu ^{*}}-measurable sets (that is, Carathéodory-measurable sets), which is the set of all {\displaystyle M\subseteq \Omega } such that {\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}(S\cap M^{\mathrm {c} })\quad {\text{ for every subset }}S\subseteq \Omega .} It is a {\displaystyle \sigma }-algebra and {\displaystyle \mu ^{*}} is sigma-additive on it, by Caratheodory lemma.

Restricting outer measures

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If {\displaystyle \mu ^{*}:\wp (\Omega )\to [0,\infty ]} is an outer measure on a set {\displaystyle \Omega ,} where (by definition) the domain is necessarily the power set {\displaystyle \wp (\Omega )} of {\displaystyle \Omega ,} then a subset {\displaystyle M\subseteq \Omega } is called {\displaystyle \mu ^{*}}–measurable or Carathéodory-measurable if it satisfies the following Carathéodory's criterion: {\displaystyle \mu ^{*}(S)=\mu ^{*}(S\cap M)+\mu ^{*}(S\cap M^{\mathrm {c} })\quad {\text{ for every subset }}S\subseteq \Omega ,} where {\displaystyle M^{\mathrm {c} }:=\Omega \setminus M} is the complement of {\displaystyle M.}

The family of all {\displaystyle \mu ^{*}}–measurable subsets is a σ-algebra and the restriction of the outer measure {\displaystyle \mu ^{*}} to this family is a measure.

  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. Probability Theory and Stochastic Modelling. Vol. 77. 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.

Proofs