Set function

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 $\R \cup \,$ 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 $\mathcal$ is a family of sets over $\Omega$ (meaning that $\mathcal \subseteq \wp(\Omega)$ where $\wp(\Omega)$ denotes the powerset) then a is a function $\mu$ with domain $\mathcal$ and codomain \infty, \infty/math> or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures.
The domain is 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 $\mu(E) + \mu(F)$ is always well-defined for all $E, F \in \mathcal,$ 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:
:: $\mu(F) - \mu(E) = \mu(F \setminus E) \text \mu(F) - \mu(E)$ is defined with $E, F \in \mathcal \text E \subseteq F \text F \setminus E \in \mathcal.$

Null sets

A set $F \in \mathcal$ 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:

• : $\mu(\varnothing) = 0$ if $\varnothing \in \mathcal.$

Variation and mass

The $S$ is
$, \mu, (S) := \sup \$
where $, \,\cdot\,,$ denotes the absolute value (or more generally, it denotes the norm or seminorm if $\mu$ is vector-valued in a (semi)normed space).
Assuming that $\cup \mathcal := \bigcup_ F \in \mathcal,$ then $, \mu, \left(\cup \mathcal\right)$ is called the of $\mu$ and $\mu\left(\cup \mathcal\right)$ is called the of $\mu.$

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

Common properties of set functions

A set function $\mu$ on $\mathcal$ is said to be

• if it is valued in , \infty


• if $\textstyle\sum\limits_^n \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_^n F_i\right)$ for all pairwise disjoint finite sequences $F_1, \ldots, F_n \in \mathcal$ such that $\textstyle\bigcup\limits_^n F_i \in \mathcal.$
  * If $\mathcal$ is closed under binary unions then $\mu$ is finitely additive if and only if $\mu(E \cup F) = \mu(E) + \mu(F)$ for all disjoint pairs $E, F \in \mathcal.$
  * If $\mu$ is finitely additive and if $\varnothing \in \mathcal$ then taking $E := F := \varnothing$ shows that $\mu(\varnothing) = \mu(\varnothing) + \mu(\varnothing)$ which is only possible if $\mu(\varnothing) = 0$ or $\mu(\varnothing) = \pm \infty,$ where in the latter case, $\mu(E) = \mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing) = \mu(E) + (\pm \infty) = \pm \infty$ for every $E \in \mathcal$ (so only the case $\mu(\varnothing) = 0$ is useful).
  


• or if in addition to being finitely additive, for all pairwise disjoint sequences $F_1, F_2, \ldots\,$ in $\mathcal$ such that $\textstyle\bigcup\limits_^\infty F_i \in \mathcal,$ all of the following hold:
  
  
  
  1. $\textstyle\sum\limits_^\infty \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_^\infty F_i\right)$
     * The series on the left hand side is defined in the usual way as the limit $\textstyle\sum\limits_^\infty \mu\left(F_i\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).$
     * As a consequence, if $\rho : \N \to \N$ is any permutation/bijection then $\textstyle\sum\limits_^ \mu\left(F_i\right) = \textstyle\sum\limits_^ \mu\left(F_\right);$ this is because $\textstyle\bigcup\limits_^ F_i = \textstyle\bigcup\limits_^ F_$ and applying this condition (a) twice guarantees that both $\textstyle\sum\limits_^ \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_^ F_i\right)$ and $\mu\left(\textstyle\bigcup\limits_^ F_\right) = \textstyle\sum\limits_^ \mu\left(F_\right)$ hold. By definition, a convergent series with this property is said to be unconditionally convergent. Stated in plain English, this means that rearranging/relabeling the sets $F_1, F_2, \ldots$ to the new order $F_, F_, \ldots$ does not affect the sum of their measures. This is desirable since just as the union $F := \textstyle\bigcup\limits_ F_i$ does not depend on the order of these sets, the same should be true of the sums $\mu(F) = \mu\left(F_1\right) + \mu\left(F_2\right) + \cdots$ and $\mu(F) = \mu\left(F_\right) + \mu\left(F_\right) + \cdots\,.$
  
  
  2. if $\mu\left(\textstyle\bigcup\limits_^ F_i\right)$ is not infinite then this series $\textstyle\sum\limits_^\infty \mu\left(F_i\right)$ must also converge absolutely, which by definition means that $\textstyle\sum\limits_^\infty \left, \mu\left(F_i\right)\$ must be finite. This is automatically true if $\mu$ is non-negative (or even just valued in the extended real numbers).
     * As with any convergent series of real numbers, by the Riemann series theorem, the series $\textstyle\sum\limits_^\infty \mu\left(F_i\right) = \mu\left(F_1\right) + \mu\left(F_2\right) + \cdots + \mu\left(F_N\right)$ converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if $\mu$ is valued in \infty, \infty
  
  
  3. if $\mu\left(\textstyle\bigcup\limits_^ F_i\right) = \textstyle\sum\limits_^ \mu\left(F_i\right)$ is infinite then it is also required that the value of at least one of the series $\textstyle\sum\limits_ \mu\left(F_i\right) \; \text \; \textstyle\sum\limits_ \mu\left(F_i\right) \;$ be finite (so that the sum of their values is well-defined). This is automatically true if $\mu$ is non-negative.
  
  
  
  


• a if it is non-negative, countably additive (including finitely additive), and has a null empty set.


• a 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 if it is a measure that has a mass of $1.$


• an if it is non-negative, countably subadditive, has a null empty set, and has the power set $\wp(\Omega)$ as its domain.
  * Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets


• a if it is countably additive, has a null empty set, and $\mu$ does not take on both $- \infty$ and $+ \infty$ as values.


• if every subset of every null set is null; explicitly, this means: whenever $F \in \mathcal \text \mu(F) = 0$ and $N \subseteq F$ is any subset of $F$ then $N \in \mathcal$ and $\mu(N) = 0.$
  * Unlike many other properties, completeness places requirements on the set $\operatorname \mu = \mathcal$ (and not just on $\mu$'s values).


• if there exists a sequence $F_1, F_2, F_3, \ldots\,$ in $\mathcal$ such that $\mu\left(F_i\right)$ is finite for every index $i,$ and also $\textstyle\bigcup\limits_^\infty F_n = \textstyle\bigcup\limits_ F.$


• if there exists a subfamily $\mathcal \subseteq \mathcal$ of pairwise disjoint sets such that $\mu(P)$ is finite for every $P \in \mathcal$ and also $\textstyle\bigcup\limits_ = \textstyle\bigcup\limits_ F$ (where $\mathcal = \operatorname \mu$).
  * Every -finite set function is decomposable although not conversely. For example, the counting measure on $\R$ (whose domain is $\wp(\R)$) is decomposable but not -finite.


• a if it is a countably additive set function $\mu : \mathcal \to X$ valued in a topological vector space $X$ (such as a normed space) whose domain is a σ-algebra.
  * If $\mu$ is valued in a normed space $(X, \, \cdot\, )$ then it is countably additive if and only if for any pairwise disjoint sequence $F_1, F_2, \ldots\,$ in $\mathcal,$ $\lim_ \left\, \mu\left(F_1\right) + \cdots + \mu\left(F_n\right) - \mu\left(\textstyle\bigcup\limits_^\infty F_i\right)\right\, = 0.$ If $\mu$ is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence $F_1, F_2, \ldots\,$ in $\mathcal,$ $\lim_ \left\, \mu\left(F_n \cup F_ \cup F_ \cup \cdots\right)\right\, = 0.$


• a if it is a countably additive complex-valued set function $\mu : \mathcal \to \Complex$ whose domain is a σ-algebra.
  * By definition, a complex measure never takes $\pm \infty$ as a value and so has a null empty set.


• a if it is a measure-valued random element.

Arbitrary sums

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

For example, if $z_i = 0$ for every $i \in I$ then $\textstyle\sum\limits_ z_i = 0.$
And it can be shown that $\textstyle\sum\limits_ r_i = \textstyle\sum\limits_ r_i + \textstyle\sum\limits_ r_i = \textstyle\sum\limits_ r_i.$
If $I = \N$ then the generalized series $\textstyle\sum\limits_ r_i$ converges in $\R$ if and only if $\textstyle\sum\limits_^\infty r_i$ converges unconditionally (or equivalently, converges absolutely) in the usual sense.
If a generalized series $\textstyle\sum\limits_ r_i$ converges in $\R$ then both $\textstyle\sum\limits_ r_i$ and $\textstyle\sum\limits_ r_i$ also converge to elements of $\R$ and the set $\left\$ is necessarily countable (that is, either finite or countably infinite); this remains true if $\R$ is replaced with any normed space.
It follows that in order for a generalized series $\textstyle\sum\limits_ r_i$ to converge in $\R$ or $\Complex,$ it is necessary that all but at most countably many $r_i$ be equal to $0,$ which means that $\textstyle\sum\limits_ r_i ~=~ \textstyle\sum\limits_ r_i$ is a sum of at most countably many non-zero terms.
Said differently, if $\left\$ is uncountable then the generalized series $\textstyle\sum\limits_ r_i$ 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 $F_1, F_2, \ldots\,$ in $\mathcal$ (and the usual countable series $\textstyle\sum\limits_^\infty \mu\left(F_i\right)$) to arbitrarily many sets $\left(F_i\right)_$ (and the generalized series $\textstyle\sum\limits_ \mu\left(F_i\right)$).

Inner measures, outer measures, and other properties

A set function $\mu$ is said to be/satisfies

• if $\mu(E) \leq \mu(F)$ whenever $E, F \in \mathcal$ satisfy $E \subseteq F.$


• if it satisfies the following condition, known as : $\mu(E \cup F) + \mu(E \cap F) = \mu(E) + \mu(F)$ for all $E, F \in \mathcal$ such that $E \cup F, E \cap F \in \mathcal.$
  * 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 . This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.


• if $\mu(E \cup F) + \mu(E \cap F) \leq \mu(E) + \mu(F)$ for all $E, F \in \mathcal$ such that $E \cup F, E \cap F \in \mathcal.$


• if $, \mu(F), \leq \sum_^n \left, \mu\left(F_i\right)\$ for all finite sequences $F, F_1, \ldots, F_n \in \mathcal$ that satisfy $F \;\subseteq\; \bigcup_^n F_i.$


• or if $, \mu(F), \leq \sum_^ \left, \mu\left(F_i\right)\$ for all sequences $F, F_1, F_2, F_3, \ldots\,$ in $\mathcal$ that satisfy $F \;\subseteq\; \bigcup_^ F_i.$
  * If $\mathcal$ is closed under finite unions then this condition holds if and only if $, \mu(F \cup G), \leq, \mu(F), + , \mu(G),$ for all $F, G \in \mathcal.$ If $\mu$ is non-negative then the absolute values may be removed.
  * If $\mu$ is a measure then this condition holds if and only if $\mu\left(\bigcup_^ F_i\right) \leq \sum_^ \mu\left(F_i\right)$ for all $F_1, F_2, F_3, \ldots\,$ in $\mathcal.$ If $\mu$ is a probability measure then this inequality is Boole's inequality.
  * If $\mu$ is countably subadditive and $\varnothing \in \mathcal$ with $\mu(\varnothing) = 0$ then $\mu$ is finitely subadditive.


• if $\mu(E) + \mu(F) \leq \mu(E \cup F)$ whenever $E, F \in \mathcal$ are disjoint with $E \cup F \in \mathcal.$


• if $\lim_ \mu\left(F_i\right) = \mu\left(\bigcap_^ F_i\right)$ for all of sets $F_1 \supseteq F_2 \supseteq F_3 \cdots\,$ in $\mathcal$ such that $\bigcap_^ F_i \in \mathcal$ with $\mu\left(\bigcap_^ F_i\right)$ and all $\mu\left(F_i\right)$ finite.
  * Lebesgue measure $\lambda$ is continuous from above but it would not be if the assumption that all $\mu\left(F_i\right)$ are eventually finite was omitted from the definition, as this example shows: For every integer $i,$ let $F_i$ be the open interval $(i, \infty)$ so that $\lim_ \lambda\left(F_i\right) = \lim_ \infty = \infty \neq 0 = \lambda(\varnothing) = \lambda\left(\bigcap_^ F_i\right)$ where $\bigcap_^ F_i = \varnothing.$


• if $\lim_ \mu\left(F_i\right) = \mu\left(\bigcup_^ F_i\right)$ for all of sets $F_1 \subseteq F_2 \subseteq F_3 \cdots\,$ in $\mathcal$ such that $\bigcup_^ F_i \in \mathcal.$


• if whenever $F \in \mathcal$ satisfies $\mu(F) = \infty$ then for every real $r > 0,$ there exists some $F_r \in \mathcal$ such that $F_r \subseteq F$ and $r \leq \mu\left(F_r\right) < \infty.$


• an if $\mu$ is non-negative, countably subadditive, has a null empty set, and has the power set $\wp(\Omega)$ as its domain.


• an if $\mu$ is non-negative, superadditive, continuous from above, has a null empty set, has the power set $\wp(\Omega)$ as its domain, and $+\infty$ is approached from below.

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

• if $\mu(\omega + F) = \mu(F)$ for all $\omega \in \Omega$ and $F \in \mathcal$ such that $\omega + F \in \mathcal.$

Topology related definitions

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

• a if it is defined on the σ-algebra of all Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing $\tau$).


• a if it is defined on the σ-algebra of all Baire sets.


• if for every point $\omega \in \Omega$ there exists some neighborhood $U \in \mathcal \cap \tau$ of this point such that $\mu(U)$ is finite.
  * If $\mu$ is a finitely additive, monotone, and locally finite then $\mu(K)$ is necessarily finite for every compact measurable subset $K.$


• if $\mu\left(\bigcup \mathcal\right) = \sup_ \mu(D)$ whenever $\mathcal \subseteq \tau \cap \mathcal$ is directed with respect to $\,\subseteq\,$ and satisfies $\bigcup \mathcal := \bigcup_ D \in \mathcal.$
  * $\mathcal$ is directed with respect to $\,\subseteq\,$ if and only if it is not empty and for all $A, B \in \mathcal$ there exists some $C \in \mathcal$ such that $A \subseteq C$ and $B \subseteq C.$


• or if for every $F \in \mathcal,$ $\mu(F) = \sup \.$


• if for every $F \in \mathcal,$ $\mu(F) = \inf \.$


• if it is both inner regular and outer regular.


• a if it is a Borel measure that is also .


• a if it is a regular and locally finite measure.


• if every non-empty open subset has (strictly) positive measure.


• a if it is non-negative, monotone, modular, has a null empty set, and has domain $\tau.$

Relationships between set functions

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

• $\mu$ is said to be or , written $\mu \ll \nu,$ if for every set $F$ that belongs to the domain of both $\mu \text \nu,$ if $\nu(F) = 0$ then $\mu(F) = 0.$
  * If $\mu$ and $\nu$ are $\sigma$-finite measures on the same measurable space and if $\mu \ll \nu,$ then the Radon–Nikodym derivative $\frac$ exists and for every measurable $F,$ $\mu(F) = \int_F \frac d \nu.$


• $\mu \text \nu$ are , written $\mu \perp \nu,$ if there exist disjoint sets $M$ and $N$ in the domains of $\mu \text \nu$ such that $M \cup N = \Omega,$ $\mu(F) = 0$ for all $F \subseteq M$ in the domain of $\mu,$ and $\nu(F) = 0$ for all $F \subseteq N$ in the domain of $\nu.$

Examples

Examples of set functions include:
* The function $d(A) = \lim_ \frac,$ assigning densities to sufficiently well-behaved subsets $A \subseteq \,$ is a set function.
* The Lebesgue measure is a set function that assigns a non-negative real number to any set of real numbers, that is in Lebesgue $\sigma$-algebra.Kolmogorov and Fomin 1975
* A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is $1,$ with other sets given probabilities between $0$ and $1.$
* A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
* A is a set-valued random variable. See the article random compact set.

Extending set functions

Extending from semialgebras to algebras

Suppose that $\mu$ is a set function on a semialgebra $\mathcal$ over $\Omega$ and let
$\operatorname(\mathcal) := \left\,$
which is the algebra on $\Omega$ generated by $\mathcal.$
The archetypal example of a semialgebra that is not also an algebra is the family
$\mathcal_d := \ \cup \left\$
on $\Omega := \R^d$ where $(a, b] := \$ for all $-\infty \leq a < b \leq \infty.$ Importantly, the two non-strict inequalities $\,\leq\,$ in $-\infty \leq a_i < b_i \leq \infty$ cannot be replaced with strict inequalities $\,<\,$ since semialgebras must contain the whole underlying set $\R^d;$ that is, $\R^d \in \mathcal_d$ is a requirement of semialgebras (as is $\varnothing \in \mathcal_d$).

If $\mu$ is finitely additive then it has a unique extension to a set function $\overline$ on $\operatorname(\mathcal)$ defined by sending $F_1 \sqcup \cdots \sqcup F_n \in \operatorname(\mathcal)$ (where $\,\sqcup\,$ indicates that these $F_i \in \mathcal$ are pairwise disjoint) to:
$\overline\left(F_1 \sqcup \cdots \sqcup F_n\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).$
This extension $\overline$ will also be finitely additive: for any pairwise disjoint $A_1, \ldots, A_n \in \operatorname(\mathcal),$
$\overline\left(A_1 \cup \cdots \cup A_n\right) = \overline\left(A_1\right) + \cdots + \overline\left(A_n\right).$

If in addition $\mu$ is extended real-valued and monotone (which, in particular, will be the case if $\mu$ is non-negative) then $\overline$ will be monotone and finitely subadditive: for any $A, A_1, \ldots, A_n \in \operatorname(\mathcal)$ such that $A \subseteq A_1 \cup \cdots \cup A_n,$
$\overline\left(A\right) \leq \overline\left(A_1\right) + \cdots + \overline\left(A_n\right).$

Extending from rings to σ-algebras

If \mu : \mathcal \to , \infty/math> is a pre-measure on a ring of sets (such as an algebra of sets) $\mathcal$ over $\Omega$ then $\mu$ has an extension to a measure \overline : \sigma(\mathcal) \to , \infty/math> on the σ-algebra $\sigma(\mathcal)$ generated by $\mathcal.$ If $\mu$ is σ-finite then this extension is unique.

To define this extension, first extend $\mu$ to an outer measure $\mu^*$ on $2^\Omega = \wp(\Omega)$ by
$\mu^*(T) = \inf \left\$
and then restrict it to the set $\mathcal_M$ of $\mu^*$-measurable sets (that is, Carathéodory-measurable sets), which is the set of all $M \subseteq \Omega$ such that
$\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega.$ It is a $\sigma$-algebra and $\mu^*$ is sigma-additive on it, by Caratheodory lemma.

Restricting outer measures

If \mu^* : \wp(\Omega) \to , \infty/math> 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 $M \subseteq \Omega$ is called or if it satisfies the following :
$\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega,$
where $M^\mathrm := \Omega \setminus M$ 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.

See also

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Notes

References

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* A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover.
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Further reading

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Regular set function
a
Encyclopedia of Mathematics

{{Measure theory

Basic concepts in set theory
Functions and mappings
Measures (measure theory)