additive

Let $\phi$ be some positive-valued set 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

\phi(A\cup B)=\phi(A)+\phi(B).

Given any sequence $\langle A_{i}\rangle$ of disjoint sets in A and whose union is also in A, if we have

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

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

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

1.

$\phi(\emptyset)=0$ .
2.

If $A\subseteq B$ , then $\phi(A)\leq\phi(B)$ .
3.

If $A\subseteq B$ , then $\phi(B\setminus A)=\phi(B)-\phi(A)$ .
4.

Given $A$ and $B$ , $\phi(A\cup B)+\phi(A\cap B)=\phi(A)+\phi(B)$ .

Title	additive
Canonical name	Additive
Date of creation	2013-03-22 13:00:58
Last modified on	2013-03-22 13:00:58
Owner	Andrea Ambrosio (7332)
Last modified by	Andrea Ambrosio (7332)
Numerical id	10
Author	Andrea Ambrosio (7332)
Entry type	Definition
Classification	msc 03E20
Synonym	additivity
Defines	countable additivity
Defines	countably additive
Defines	$\sigma$ -additive
Defines	sigma-additive