# 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. 1.

$\phi(\emptyset)=0$.

2. 2.

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

3. 3.

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

4. 4.

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

Title additive Additive 2013-03-22 13:00:58 2013-03-22 13:00:58 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 10 Andrea Ambrosio (7332) Definition msc 03E20 additivity countable additivity countably additive $\sigma$-additive sigma-additive