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, ${\textstyle \mu (A\cup B)=\mu (A)+\mu (B).}$ 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

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 $\scriptstyle {\mathcal {A}},$ one has $$\mu (A\cup B)=\mu (A)+\mu (B).$$ 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

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

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.

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 monotone set function. Similarly, If $\mu$ is non-positive and $A\subseteq B$ then $\mu (A)\geq \mu (B).$

Modularity

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 $\mu (A\setminus B)+\mu (B\setminus A)+2\mu (A\cap B).$

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.

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 𝜎-additive function is the function $\mu$ defined over the power set of the real numbers, such that $$\mu (A)={\begin{cases}1&{\mbox{ if }}0\in A\\0&{\mbox{ if }}0\notin A.\end{cases}}$$

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 $$\mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})$$ holds.

See measure and signed measure for more examples of 𝜎-additive functions.

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 numbers $\mathbb {R}$ by the formula $$\mu (A)=\lim _{k\to \infty }{\frac {1}{k}}\cdot \lambda (A\cap (0,k)),$$ where $\lambda$ denotes the Lebesgue measure and $\lim$ the Banach limit. 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 $$A_{n}=[n,n+1)$$ 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 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.

See also

• Additive map – Z-module homomorphism
• Hahn–Kolmogorov theorem – Theorem extending pre-measures to measuresPages displaying short descriptions of redirect targets
• Measure (mathematics) – Generalization of mass, length, area and volume
• σ-finite measure – Concept in measure theory
• Signed measure – Generalized notion of measure in mathematics
• Submodular set function – Set-to-real map with diminishing returns
• Subadditive set function
• τ-additivity

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