Sigma additivity

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 ${\displaystyle \scriptstyle {\mathcal {A}}}$.

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.

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]

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.

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.

• 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.