Sigma-additive set function
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, area, volume) 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 be a set function defined on an algebra of sets with values in (see the extended real number line). The function is called additive, or finitely additive, if, whenever and are disjoint sets in one has
One can prove by mathematical induction that an additive function satisfies
σ-additive set functions
τ-additive set functions
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.
If is non-negative and then That is, is a monotone set function. Similarly, If is non-positive and then
Given and Proof: write and and where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
The above property is called modularity, and we have just proved that modularity is equivalent to additivity. 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.
If and is defined, then
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
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 numbers by the formula
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
One may define additive functions with values in any additive monoid (for example any 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. Another example, also from quantum mechanics, is the positive operator-valued measure.
- Additive map – Z-module homomorphism
- Hahn–Kolmogorov theorem
- Measure (mathematics) – Generalization of mass, length, area and volume
- σ-finite measure
- Signed measure – Generalized notion of measure in mathematics
- Submodular set function – Set-to-real map with diminishing returns
- Subadditive set function
- ba space – The set of bounded charges on a given sigma-algebra Σ
- D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.
- Bhaskara Rao, K. P. S. (1983). Theory of charges: a study of finitely additive measures. M. Bhaskara Rao. London: Academic Press. p. 35. ISBN 0-12-095780-9. OCLC 21196971.