Sigma additivity

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, and σ-additivity implies additivity.

Additive (or finitely additive) set functions

Let ${\displaystyle \mu }$ be a function defined on an algebra of sets ${\displaystyle \scriptstyle {\mathcal {A}}}$ with values in [−∞, +∞] (see the extended real number line). The function ${\displaystyle \mu }$ is called additive, or finitely additive, if, whenever A and B are disjoint sets in ${\displaystyle \scriptstyle {\mathcal {A}}}$, one has

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

(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 that an additive function satisfies

${\displaystyle \mu \left(\bigcup _{n=1}^{N}A_{n}\right)=\sum _{n=1}^{N}\mu (A_{n})}$

for any ${\displaystyle A_{1},A_{2},\dots ,A_{N}}$ disjoint sets in ${\displaystyle \scriptstyle {\mathcal {A}}}$.

σ-additive set functions

Suppose that ${\displaystyle \scriptstyle {\mathcal {A}}}$ is a σ-algebra. If for any sequence ${\displaystyle A_{1},A_{2},\dots ,A_{n},\dots }$ of pairwise disjoint sets in ${\displaystyle \scriptstyle {\mathcal {A}}}$, one has

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

we say that μ is 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 ${\displaystyle \scriptstyle {\mathcal {A}}}$, we have a topology τ. If for any directed family of measurable open sets ${\displaystyle \scriptstyle {\mathcal {G}}}$⊆${\displaystyle \scriptstyle {\mathcal {A}}}$∩τ,

${\displaystyle \mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G)}$,

we say that μ is τ-additive. In particular, if μ is inner regular (with respect to compact sets) then it is τ-additive.[1]

Properties

Basic properties

Useful properties of an additive function μ include the following:

1. Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain.
2. If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).
3. If A ⊆ B and μ(B) − μ(A) is defined, then μ(B \ A) = μ(B) − μ(A).
4. Given A and B, μ(A ∪ B) + μ(A ∩ B) = μ(A) + μ(B).

Examples

An example of a σ-additive function is the function μ defined over the power set of the real numbers, such that

${\displaystyle \mu (A)={\begin{cases}1&{\mbox{ if }}0\in A\\0&{\mbox{ if }}0\notin A.\end{cases}}}$

If ${\displaystyle A_{1},A_{2},\dots ,A_{n},\dots }$ 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

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

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 μ, defined over the Lebesgue sets of the real numbers by the formula

${\displaystyle \mu (A)=\lim _{k\to \infty }{\frac {1}{k}}\cdot \lambda \left(A\cap \left(0,k\right)\right),}$

where λ 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

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

for n=0, 1, 2, ... The union of these sets is the positive reals, and μ applied to the union is then one, while μ applied to any of the individual sets is zero, so the sum of μ(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

• signed measure
• measure (mathematics)
• additive map
• subadditive function
• σ-finite measure
• Hahn–Kolmogorov theorem
• τ-additivity

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

References

1. D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.