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In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive
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 ...
is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the In ...
,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The ...
) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term modular set function is equivalent to additive set function; see modularity below.


Additive (or finitely additive) set functions

Let \mu be a
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 ...
defined on an algebra of sets \scriptstyle\mathcal with values in \infty, \infty/math> (see the
extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
). The function \mu is called or , if whenever A and B are disjoint sets in \scriptstyle\mathcal, then \mu(A \cup B) = \mu(A) + \mu(B). A consequence of this is that an additive function cannot take both - \infty and + \infty as values, for the expression \infty - \infty is undefined. One can prove by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
that an additive function satisfies \mu\left(\bigcup_^N A_n\right)=\sum_^N \mu\left(A_n\right) for any A_1, A_2, \ldots, A_N disjoint sets in \mathcal.


σ-additive set functions

Suppose that \scriptstyle\mathcal is a σ-algebra. If for every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
A_1, A_2, \ldots, A_n, \ldots of pairwise disjoint sets in \scriptstyle\mathcal, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n), holds then \mu is said to be or . Every -additive function is additive but not vice versa, as shown below.


τ-additive set functions

Suppose that in addition to a sigma algebra \mathcal, we have a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
\tau. If for every directed family of measurable open sets \mathcal \subseteq \mathcal \cap \tau, \mu\left(\bigcup \mathcal \right) = \sup_ \mu(G), we say that \mu is \tau-additive. In particular, if \mu is
inner regular In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. Definition Let (''X'', ''T'') be a Hausdorff topological space and let Σ be a σ-algebra on ''X'' th ...
(with respect to compact sets) then it is τ-additive.D.H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.


Properties

Useful properties of an additive set function \mu include the following.


Value of empty set

Either \mu(\varnothing) = 0, or \mu assigns \infty to all sets in its domain, or \mu assigns - \infty to all sets in its domain. ''Proof'': additivity implies that for every set A, \mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing). If \mu(\varnothing) \neq 0, then this equality can be satisfied only by plus or minus infinity.


Monotonicity

If \mu is non-negative and A \subseteq B then \mu(A) \leq \mu(B). That is, \mu is a . Similarly, If \mu is non-positive and A \subseteq B then \mu(A) \geq \mu(B).


Modularity

A
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 ...
\mu on a
family of sets In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fa ...
\mathcal is called a and a if whenever A, B, A\cup B, and A\cap B are elements of \mathcal, then \phi(A\cup B)+ \phi(A\cap B) = \phi(A) + \phi(B) The above property is called and the argument below proves that modularity is equivalent to additivity. Given A and B, \mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B). ''Proof'': write A = (A \cap B) \cup (A \setminus B) and B = (A \cap B) \cup (B \setminus A) and A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A), where all sets in the union are disjoint. Additivity implies that both sides of the equality equal \mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B). However, the related properties of ''submodularity'' and ''subadditivity'' are not equivalent to each other. Note that modularity has a different and unrelated meaning in the context of complex functions; see
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
.


Set difference

If A \subseteq B and \mu(B) - \mu(A) is defined, then \mu(B \setminus A) = \mu(B) - \mu(A).


Examples

An example of a -additive function is the function \mu defined over the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, such that \mu (A)= \begin 1 & \mbox 0 \in A \\ 0 & \mbox 0 \notin A. \end If A_1, A_2, \ldots, A_n, \ldots is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n) holds. See measure and
signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not ...
for more examples of -additive functions. A ''charge'' is defined to be a finitely additive set function that maps \varnothing to 0. (Cf.
ba space In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X). If Σ i ...
for information about ''bounded'' charges, where we say a charge is ''bounded'' to mean its range its a bounded subset of ''R''.)


An additive function which is not σ-additive

An example of an additive function which is not σ-additive is obtained by considering \mu, defined over the Lebesgue sets of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s \R by the formula \mu(A) = \lim_ \frac \cdot \lambda(A \cap (0,k)), where \lambda denotes the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
and \lim the
Banach limit In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\inf ...
. It satisfies 0 \leq \mu(A) \leq 1 and if \sup A < \infty then \mu(A) = 0. One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets A_n = ,n + 1) for n = 0, 1, 2, \ldots The union of these sets is the positive reals, and \mu applied to the union is then one, while \mu applied to any of the individual sets is zero, so the sum of \mu(A_n)is also zero, which proves the counterexample.


Generalizations

One may define additive functions with values in any additive monoid (for example any Group (mathematics), group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
. Another example, also from quantum mechanics, is the positive operator-valued measure.


See also

* * * * * * * * *
ba space In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X). If Σ i ...
– The set of bounded charges on a given sigma-algebra


References

{{reflist Measure theory Additive functions