Sigma-additive set function

Let $\mu$ be some extended real-valued function defined on an algebra of sets ${\mathcal {A}}.$ We say that $\phi$ is additive if, whenever $A$ and $B$ are disjoint sets in ${\mathcal {A}}$, we have

$\mu (A\cup B)=\mu (A)+\mu (B).$

Suppose ${\mathcal {A}}$ is a $\sigma$-algebra. Then, given any sequence $\langle A_{i}\rangle$ of disjoint sets in ${\mathcal {A}}$, if we have

$\mu \left(\bigcup A_{i}\right)=\sum \mu (A_{i})$

we say that $\mu$ is countably additive or $\sigma$-additive.

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

1. $\mu (\varnothing )=0.$
2. If $\mu$ is non-negative and $A\subseteq B$, then $\mu (A)\leq \mu (B).$
3. If $A\subseteq B$, then $\mu (B\setminus A)=\mu (B)-\mu (A).$
4. Given $A$ and $B$, $\mu (A\cup B)+\mu (A\cap B)=\mu (A)+\mu (B).$

See also

• Measure (mathematics)
• additive function
• subadditive function

additive at PlanetMath.