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,
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 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,
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 Inte ...
,
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 su ...
,
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). Th ...
) 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
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
below.
Additive (or finitely additive) set functions
Let
be a
set function defined on an
algebra of sets
In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the r ...
with values in
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 o ...
). The function
is called or , if whenever
and
are
disjoint sets in
then
A consequence of this is that an additive function cannot take both
and
as values, for the expression
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
for any
disjoint sets in
σ-additive set functions
Suppose that
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 called ...
of pairwise disjoint sets in
holds then
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
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 ho ...
If for every
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
family of measurable
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
s
we say that
is
-additive. In particular, if
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'' that ...
(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
include the following.
Value of empty set
Either
or
assigns
to all sets in its domain, or
assigns
to all sets in its domain. ''Proof'': additivity implies that for every set
If
then this equality can be satisfied only by plus or minus infinity.
Monotonicity
If
is non-negative and
then
That is,
is a . Similarly, If
is non-positive and
then
Modularity
A
set function 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 ...
is called a and a
if whenever
and
are elements of
then
The above property is called and the argument below proves that modularity is equivalent to additivity.
Given
and
''Proof'': write
and
and
where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
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
and
is defined, then
Examples
An example of a -additive function is the function
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 ...
of the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, such that
If
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
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
to
(Cf.
ba space 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
, defined over the Lebesgue sets of the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s
by the formula
where
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
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
and if
then
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
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 measure
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his diss ...
s 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 ...
. Another example, also from quantum mechanics, is the
positive operator-valued measure.
See also
*
*
*
*
*
*
*
*
*
ba space – The set of bounded charges on a given sigma-algebra
References
{{reflist
Measure theory
Additive functions