additive
Additive
2002-08-30 12:59:39
2008-11-23 06:19:25
Definition
countable additivity
countably additive
$\sigma$-additive
sigma-additive
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Let $\phi$ be some positive-valued set function defined on an algebra of sets $\mathcal{A}$. We say that $\phi$ is \emph{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 \emph{countably additive} or \emph{$\sigma$-additive}.
Useful properties of an additive set function $\phi$ include the following:
\begin{enumerate}
\item $\phi(\emptyset) = 0$.
\item If $A \subseteq B$, then $\phi(A) \leq \phi(B)$.
\item If $A \subseteq B$, then $\phi(B \setminus A) = \phi(B) - \phi(A)$.
\item Given $A$ and $B$, $\phi(A \cup B) + \phi(A \cap B) = \phi(A) + \phi(B)$.
\end{enumerate}