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
A set function generally aims to measure subsets in some way.
Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
denotes the powerset) then a set function on
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.
Additionally, a semiring is a π-system where every complement
is equal to a finite disjoint union of sets in
is equal to a finite disjoint union of sets in
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".
is said to be[1] Arbitrary sums As described in this article's section on generalized series, for any family
as the limit of the net of finite partial sums
Whenever this net converges then its limit is denoted by the symbols
[proof 1] It follows that in order for a generalized series
is a sum of at most countably many non-zero terms.
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.
then: Examples of set functions include: The Jordan measure on
is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue
is a finitely additive set function (explicitly, if
This set function can be extended to the Lebesgue outer measure on
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.
)[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
In fact, such non-trivial set functions will exist even if
is any abelian group then there exists a finitely additive and translation-invariant[note 1] set function
The archetypal example of a semialgebra that is not also an algebra is the family
is finitely additive then it has a unique extension to a set function
where (by definition) the domain is necessarily the power set
–measurable subsets is a σ-algebra and the restriction of the outer measure