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 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 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:
Null sets
A set is called a or simply if
Whenever is not identically equal to either or then it is typically also assumed that:

: if

Variation and [|mass]
The [Total variation (measure theory)|] is
where denotes the absolute value.
Assuming that then is called the of and is called the of
A set function is called if for every the value is .
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
[Finitely additive set function|] 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 .
[Sigma-additive set function|] or [Sigma-additive set function|] if in addition to being finitely additive, for all pairwise disjoint sequences in such that all of the following hold:
1. - 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 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
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.
  - 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. Since unconditional convergence is guaranteed by above, this condition is automatically true if is valued in
3. if is infinite then it is also required that the value of at least one of the series be finite. This is automatically true if is non-negative.
a [Pre-measure|] if it is non-negative, countably additive, and has a null empty set.
a [Measure (mathematics)|] 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 measure|] 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 .
[σ-finite measure|] if there exists a sequence in such that is finite for every index and also
[Decomposable measure|] if there exists a subfamily of pairwise disjoint sets such that is finite for every and also .
- Every -finite set function is decomposable although not conversely. For example, the counting measure on is decomposable but not -finite.
a [Vector measure|] if it is a countably additive set function valued in a topological vector 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 [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 in the usual sense.
If a generalized series converges in then both and also converge to elements of and the set is necessarily countable ; 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 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 to arbitrarily many sets .

Inner measures, outer measures, and other properties

A set function is said to be/satisfies

if whenever satisfy
[Modular set function|] 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 [Valuation (geometry)|]. This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.
[Submodular set function|] 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.
[Superadditivity|] 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 [#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 measure|] 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 [Borel measure|] if it is a measure defined on the σ-algebra of all Borel sets, which is the smallest σ-algebra containing all open subsets.
a [Baire measure|] if it is a measure defined on the σ-algebra of all Baire sets.
[Locally finite measure|] 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
[τ-additivity|] 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
[Inner regular measure|] or if for every
[Outer regular measure|] if for every
[Regular measure|] if it is both inner regular and outer regular.
a [Borel regular measure|] if it is a Borel measure that is also [Regular measure|].
a [Radon measure|] if it is a regular and locally finite measure.
[Strictly positive measure|] if every non-empty open subset has positive measure.
a [Valuation (measure theory)|] 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 [Absolute continuity (measure theory)|] or [Domination (measure theory)|], 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 [Equivalence (measure theory)|] if each one is absolutely continuous with respect to the other. is called a [Equivalence (measure theory)#Supporting measure|] of a measure if is -finite and they are equivalent.
and are [Singular measure|], 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.
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 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
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 family
on 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.
If is finitely additive then it has a unique extension to a set function on defined by sending to:
This extension will also be finitely additive: for any pairwise disjoint
If in addition is extended real-valued and monotone 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 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, 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 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.