Sigma-additive set function

In mathematics, an additive set function is a function ${\textstyle \mu }$ 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, ${\textstyle \mu (A\cup B)=\mu (A)+\mu (B).}$ 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 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, ${\textstyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}).}$

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[edit]

Let $\mu$ be a set function defined on an algebra of sets $\scriptstyle {\mathcal {A}}$ with values in $[-\infty ,\infty ]$ (see the extended real number line). The function $\mu$ is called additive or finitely additive, if whenever $A$ and $B$ are disjoint sets in $\scriptstyle {\mathcal {A}},$ 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 that an additive function satisfies $\mu \left(\bigcup _{n=1}^{N}A_{n}\right)=\sum _{n=1}^{N}\mu \left(A_{n}\right)$ for any $A_{1},A_{2},\ldots ,A_{N}$ disjoint sets in ${\textstyle {\mathcal {A}}.}$

σ-additive set functions[edit]

Suppose that $\scriptstyle {\mathcal {A}}$ is a σ-algebra. If for every sequence $A_{1},A_{2},\ldots ,A_{n},\ldots$ of pairwise disjoint sets in $\scriptstyle {\mathcal {A}},$ $\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}),$ holds then $\mu$ is said to be countably additive or 𝜎-additive. Every 𝜎-additive function is additive but not vice versa, as shown below.

τ-additive set functions[edit]

Suppose that in addition to a sigma algebra ${\textstyle {\mathcal {A}},}$ we have a topology $\tau .$ If for every directed family of measurable open sets ${\textstyle {\mathcal {G}}\subseteq {\mathcal {A}}\cap \tau ,}$ $\mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G),$ we say that $\mu$ is $\tau$ -additive. In particular, if $\mu$ is inner regular (with respect to compact sets) then it is τ-additive.^[1]

Generalizations[edit]

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.

– Z-module homomorphism

Additive map

– Theorem extending pre-measures to measures

Hahn–Kolmogorov theorem

– Generalization of mass, length, area and volume

Measure (mathematics)

– Concept in measure theory

σ-finite measure

– Generalized notion of measure in mathematics

Signed measure

– Set-to-real map with diminishing returns

Submodular set function

Subadditive set function

τ-additivity

– The set of bounded charges on a given sigma-algebra

ba space

This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.