Set function
Introduction 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  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  is a family of sets over  (meaning that  where  denotes the powerset) then a  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: ::  is defined with  satisfying  and  '''Null sets'''A set  is called a  (with respect to ) or simply  if  Whenever  is not identically equal to either  or  then it is typically also assumed that::  if '''Variation and mass'''The   iswhere  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  of  and  is called the  of  A set function is called  if for every  the value  is  (which by definition means that  and ; an  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 if it is valued in  if  for all pairwise disjoint finite sequences  such that * If  is closed under binary unions then  is finitely additive if and only if  for all disjoint pairs * If  is finitely additive and if  then taking  shows that  which is only possible if  or  where in the latter case,  for every  (so only the case  is useful). or  if in addition to being finitely additive, for all pairwise disjoint sequences  in  such that  all of the following hold:* The series on the left hand side is defined in the usual way as the limit * As a consequence, if  is any permutation/bijection then  this is because  and applying this condition (a) twice guarantees that both  and  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  to the new order  does not affect the sum of their measures. This is desirable since just as the union  does not depend on the order of these sets, the same should be true of the sums  and 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). * As with any convergent series of real numbers, by the Riemann series theorem, the series  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  is valued in 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  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 an  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 subsetsa  if it is countably additive, has a null empty set, and  does not take on both  and  as values. if every subset of every null set is null; explicitly, this means: whenever  and  is any subset of  then  and * Unlike many other properties, completeness places requirements on the set  (and not just on 's values). if there exists a sequence  in  such that  is finite for every index  and also  if there exists a subfamily  of pairwise disjoint sets such that  is finite for every  and also  (where ). * Every -finite set function is decomposable although not conversely. For example, the counting measure on  (whose domain is ) is decomposable but not -finite.a  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. * If  is valued in a normed space  then it is countably additive if and only if for any pairwise disjoint sequence  in   If  is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence  in  a  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  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. 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 if  whenever  satisfy  if it satisfies the following condition, known as :  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 . This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below. if  for all  such that  if  for all finite sequences  that satisfy  or  if  for all sequences  in  that satisfy * If  is closed under finite unions then this condition holds if and only if  for all  If  is non-negative then the absolute values may be removed. * If  is a measure then this condition holds if and only if  for all  in  If  is a probability measure then this inequality is Boole's inequality. * If  is countably subadditive and  with  then  is finitely subadditive. if  whenever  are disjoint with  if  for all  of sets  in  such that  with  and all  finite. * Lebesgue measure  is continuous from above but it would not be if the assumption that all  are eventually finite was omitted from the definition, as this example shows: For every integer  let  be the open interval  so that  where  if  for all  of sets  in  such that  if whenever  satisfies  then for every real  there exists some  such that  and an  if  is non-negative, countably subadditive, has a null empty set, and has the power set  as its domain.an  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. 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  for all  and  such that ===Topology related definitions===If  is a topology on  then a set function  is said to be:a  if it is a measure defined on the σ-algebra of all Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing ).a  if it is a measure defined on the σ-algebra of all Baire sets. if for every point  there exists some neighborhood  of this point such that  is finite. * If  is a finitely additive, monotone, and locally finite then  is necessarily finite for every compact measurable subset  if  whenever  is directed with respect to  and satisfies *  is directed with respect to  if and only if it is not empty and for all  there exists some  such that  and  or  if for every   if for every   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 ===Relationships between set functions===If  and  are two set functions over  then: is said to be  or , written  if for every set  that belongs to the domain of both  and  if  then * If  and  are -finite measures on the same measurable space and if  then the Radon–Nikodym derivative  exists and for every measurable  *  and  are called  if each one is absolutely continuous with respect to the other. is called a  of a measure  if  is -finite and they are equivalent. and  are , written  if there exist disjoint sets  and  in the domains of  and  such that   for all  in the domain of  and  for all  in the domain of
Examples Examples of set functions include:* The function  assigning densities to sufficiently well-behaved subsets  is a set function. * 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  with other sets given probabilities between  and * 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.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. 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 -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  that is finite on every compact subset of  is the trivial set function  that is identically equal to  (that is, it sends every  to )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
Extending set functions ===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 familyon  where  for all  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: This extension  will also be finitely additive: for any pairwise disjoint   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 ===Extending from rings to σ-algebras===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===If  is an outer measure on a set  where (by definition) the domain is necessarily the power set   of  then a subset  is called '''''' or '''''' if it satisfies the following : 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 * * * * * * * * * * * * * *
Notes '''Proofs'''
References *  *  *  A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover. *  *
Further reading * *  Regular set function at  Encyclopedia of Mathematics