Function from sets to numbers
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 formula_1 which consists of the real numbers formula_2 and formula_3
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.
Definitions.
If formula_4 is a family of sets over formula_5 (meaning that formula_6 where formula_7 denotes the powerset) then a set function on formula_4 is a function formula_9 with domain formula_4 and codomain formula_11 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 formula_12 is always well-defined for all formula_13 or equivalently, that formula_9 does not take on both formula_15 and formula_16 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 formula_9 is finitely additive: 
 formula_18 is defined with formula_19 satisfying formula_20 and formula_21 
Null sets
A set formula_22 is called a (with respect to formula_9) or simply if formula_24 
Whenever formula_9 is not identically equal to either formula_26 or formula_27 then it is typically also assumed that:
Variation and mass
The formula_28 is
formula_29
where formula_30 denotes the absolute value (or more generally, it denotes the norm or seminorm if formula_9 is vector-valued in a (semi)normed space). 
Assuming that formula_32 then formula_33 is called the of formula_9 and formula_35 is called the of formula_36 
A set function is called if for every formula_37 the value formula_38 is (which by definition means that formula_39 and formula_40; an is one that is equal to formula_41 or formula_15). 
Every finite set function must have a finite mass. 
Common properties of set functions.
A set function formula_9 on formula_4 is said to be
Arbitrary sums
As described in this article's section on generalized series, for any family formula_45 of real numbers indexed by an arbitrary indexing set formula_46 it is possible to define their sum formula_47 as the limit of the net of finite partial sums formula_48 where the domain formula_49 is directed by formula_50 
Whenever this net converges then its limit is denoted by the symbols formula_47 while if this net instead diverges to formula_52 then this may be indicated by writing formula_53 
Any sum over the empty set is defined to be zero; that is, if formula_54 then formula_55 by definition.
For example, if formula_56 for every formula_57 then formula_58 
And it can be shown that formula_59 
If formula_60 then the generalized series formula_47 converges in formula_2 if and only if formula_63 converges unconditionally (or equivalently, converges absolutely) in the usual sense. 
If a generalized series formula_47 converges in formula_2 then both formula_66 and formula_67 also converge to elements of formula_2 and the set formula_69 is necessarily countable (that is, either finite or countably infinite); this remains true if formula_2 is replaced with any normed space. 
It follows that in order for a generalized series formula_47 to converge in formula_2 or formula_73 it is necessary that all but at most countably many formula_74 will be equal to formula_75 which means that formula_76 is a sum of at most countably many non-zero terms. 
Said differently, if formula_69 is uncountable then the generalized series formula_47 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 formula_79 in formula_4 (and the usual countable series formula_81) to arbitrarily many sets formula_82 (and the generalized series formula_83).
Inner measures, outer measures, and other properties.
A set function formula_9 is said to be/satisfies
If a binary operation formula_85 is defined, then a set function formula_9 is said to be
Topology related definitions.
If formula_87 is a topology on formula_5 then a set function formula_9 is said to be:
Relationships between set functions.
If formula_9 and formula_91 are two set functions over formula_92 then:
Examples.
Examples of set functions include:
The Jordan measure on formula_98 is a set function defined on the set of all Jordan measurable subsets of formula_99 it sends a Jordan measurable set to its Jordan measure.
Lebesgue measure.
The Lebesgue measure on formula_100 is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue formula_101-algebra. 
Its definition begins with the set formula_102 of all intervals of real numbers, which is a semialgebra on formula_103 
The function that assigns to every interval formula_104 its formula_105 is a finitely additive set function (explicitly, if formula_104 has endpoints formula_107 then formula_108). 
This set function can be extended to the Lebesgue outer measure on formula_109 which is the translation-invariant set function formula_110 that sends a subset formula_111 to the infimum
formula_112 
Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the 𝜎-algebra of all subsets formula_113 that satisfy the Carathéodory criterion:
formula_114
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 formula_115 with domain formula_116 that is finite on every compact subset of formula_100 is the trivial set function formula_118 that is identically equal to formula_96 (that is, it sends every formula_120 to formula_96)
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 formula_122 In fact, such non-trivial set functions will exist even if formula_100 is replaced by any other abelian group formula_124
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Theorem — 
If formula_125 is any abelian group then there exists a finitely additive and translation-invariant set function formula_126 of mass formula_127
Extending set functions.
Extending from semialgebras to algebras.
Suppose that formula_9 is a set function on a semialgebra formula_4 over formula_5 and let 
formula_131
which is the algebra on formula_5 generated by formula_133 
The archetypal example of a semialgebra that is not also an algebra is the family
formula_134
on formula_135 where formula_136 for all formula_137 Importantly, the two non-strict inequalities formula_138 in formula_139 cannot be replaced with strict inequalities formula_140 since semialgebras must contain the whole underlying set formula_141 that is, formula_142 is a requirement of semialgebras (as is formula_143).
If formula_9 is finitely additive then it has a unique extension to a set function formula_145 on formula_146 defined by sending formula_147 (where formula_148 indicates that these formula_149 are pairwise disjoint) to:
formula_150 
This extension formula_145 will also be finitely additive: for any pairwise disjoint formula_152 
formula_153 
If in addition formula_9 is extended real-valued and monotone (which, in particular, will be the case if formula_9 is non-negative) then formula_145 will be monotone and finitely subadditive: for any formula_157 such that formula_158
formula_159
Extending from rings to σ-algebras.
If formula_160 is a pre-measure on a ring of sets (such as an algebra of sets) formula_4 over formula_5 then formula_9 has an extension to a measure formula_164 on the σ-algebra formula_165 generated by formula_133 If formula_9 is σ-finite then this extension is unique.
To define this extension, first extend formula_9 to an outer measure formula_169 on formula_170 by 
formula_171 
and then restrict it to the set formula_172 of formula_169-measurable sets (that is, Carathéodory-measurable sets), which is the set of all formula_174 such that 
formula_175 It is a formula_101-algebra and formula_169 is sigma-additive on it, by Caratheodory lemma.
Restricting outer measures.
If formula_178 is an outer measure on a set formula_92 where (by definition) the domain is necessarily the power set formula_7 of formula_92 then a subset formula_174 is called formula_169–measurable or Carathéodory-measurable if it satisfies the following Carathéodory's criterion: 
formula_184
where formula_185 is the complement of formula_186 
The family of all formula_169–measurable subsets is a σ-algebra and the restriction of the outer measure formula_169 to this family is a measure.
Notes.
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Proofs
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Like every convergent net, this convergent net of partial sums formula_193 is a Cauchy net, which for this particular net means (by definition) that for every neighborhood formula_194 of the origin in formula_195 there exists a finite subset formula_196 of formula_104 such that 
formula_198 for all finite supersets formula_199
this implies that formula_200 for every formula_201 (by taking formula_202 and formula_203). 
Since formula_189 is metrizable, it has a countable neighborhood basis formula_205 at the origin, whose intersection is necessarily formula_206 (since formula_189 is a Hausdorff TVS).
For every positive integer formula_208 pick a finite subset formula_209 such that formula_210 for every formula_211 
If formula_212 belongs to formula_213 then formula_74 belongs to formula_215 
Thus formula_216 for every index formula_57 that does not belong to the countable set formula_218 formula_219 ;/ref ;