additive

Let $\varphi$ be some positive-valued set function defined on an algebra of sets $𝒜$. We say that $\varphi$ is additive if, whenever $A$ and $B$ are disjoint sets in $𝒜$, we have

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

Given any sequence $⟨A_i⟩$ 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 $B$, $\varphi (A\cup B)+\varphi (A\cap B)=\varphi (A)+\varphi (B)$.