Sigma-additive set function

In mathematics, additivity (specifically finite additivity) and sigma additivity (also called countable additivity) of a function (often a measure) defined on subsets of a given set 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.

Additive (or finitely additive) set functions

Let $\mu$ be a 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 $$\mu \left(\bigcup _{n=1}^{N}A_{n}\right)=\sum _{n=1}^{N}\mu (A_{n})$$ for any $A_{1},A_{2},\dots ,A_{N}$ disjoint sets in $\scriptstyle {\mathcal {A}}.$

σ-additive set functions

Suppose that $\scriptstyle {\mathcal {A}}$ is a σ-algebra. If for any 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. Any 𝜎-additive function is additive but not vice versa, as shown below.

τ-additive set functions

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

Properties

Basic properties

Useful properties of an additive function $\mu$ include the following:

1. Either $\mu (\varnothing )=0,$ or $\mu$ assigns $\infty$ to all sets in its domain, or $\mu$ assigns $-\infty$ to all sets in its domain.
2. If $\mu$ is non-negative and $A\subseteq B$ then $\mu (A)\leq \mu (B).$
3. If $A\subseteq B$ and $\mu (B)-\mu (A).$ is defined, then $\mu (B\setminus A)=\mu (B)-\mu (A).$
4. Given $A$ and $B,$ $\mu (A\cup B)-\mu (A\cap B)=\mu (A)+\mu (B).$

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 by the formula $$\mu (A)=\lim _{k\to \infty }{\frac {1}{k}}\cdot \lambda \left(A\cap \left(0,k\right)\right),$$ where $\lambda$ denotes the Lebesgue measure and lim the Banach limit.

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 $$A_{n}=\left[n,n+1\right)$$ 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 $\mu \left(A_{n}\right)$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

• signed measure – Generalized notion of measure in mathematics
• measure (mathematics) – Generalization of mass, length, area and volume
• additive map – Z-module homomorphism
• subadditive function – Property of some mathematical functionsPages displaying short descriptions of redirect targets
• σ-finite measure – Concept in measure theory
• Hahn–Kolmogorov theorem – Theorem extending pre-measures to measuresPages displaying short descriptions of redirect targets
• τ-additivity

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