additive
Let $\varphi $ be some positivevalued set function^{} defined on an algebra of sets^{} $\mathcal{A}$. We say that $\varphi $ is additive if, whenever $A$ and $B$ are disjoint sets in $\mathcal{A}$, we have
$$\varphi (A\cup B)=\varphi (A)+\varphi (B).$$ 
Given any sequence $\u27e8{A}_{i}\u27e9$ of disjoint sets in A and whose union is also in A, if we have
$$\varphi \left(\bigcup {A}_{i}\right)=\sum \varphi ({A}_{i})$$ 
we say that $\varphi $ is countably additive or $\sigma $additive.
Useful properties of an additive set function $\varphi $ include the following:

1.
$\varphi (\mathrm{\varnothing})=0$.

2.
If $A\subseteq B$, then $\varphi (A)\le \varphi (B)$.

3.
If $A\subseteq B$, then $\varphi (B\setminus A)=\varphi (B)\varphi (A)$.

4.
Given $A$ and $B$, $\varphi (A\cup B)+\varphi (A\cap B)=\varphi (A)+\varphi (B)$.
Title  additive 

Canonical name  Additive 
Date of creation  20130322 13:00:58 
Last modified on  20130322 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  sigmaadditive 