| Version 3 |
Version 2 |
| Let $\phi$ be some real 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 |
Let $\phi$ be some real 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 |
| $$\phi(A \cup B) = \phi(A) + \phi(B) .$$ |
$$\phi(A \cup B) = \phi(A) + \phi(B) .$$ |
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| Suppose $\mathcal{A}$ is a $\sigma$-\PMlinkname{algebra}{SigmaAlgebra}. Then, given any sequence $\langle A_i \rangle$ of disjoint sets in $\mathcal{A}$, if we have |
Suppose $\mathcal{A}$ is a $\sigma$-\PMlinkname{algebra}{SigmaAlgebra}. Then, given any sequence $\langle A_i \rangle$ of disjoint sets in $\mathcal{A}$, if we have |
| $$\phi\left( \bigcup A_i \right) = \sum \phi(A_i)$$ |
$$\phi\left( \bigcup A_i \right) = \sum \phi(A_i)$$ |
| we say that $\phi$ is \emph{countably additive} or \emph{$\sigma$-additive}. |
we say that $\phi$ is \emph{countably additive}. |
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| Useful properties of an additive set function $\phi$ include the following: |
Useful properties of an additive set function $\phi$ include the following: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\phi(\emptyset) = 0$. |
\item $\phi(\emptyset) = 0$. |
| \item If $A \subseteq B$, then $\phi(A) \leq \phi(B)$. |
\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 If $A \subseteq B$, then $\phi(B \setminus A) = \phi(B) - \phi(A)$. |
| \item Given $A$ and $B$, $\phi(A \cup B) + \phi(A \cap B) = \phi(A) + \phi(B)$. |
\item Given $A$ and $B$, $\phi(A \cup B) + \phi(A \cap B) = \phi(A) + \phi(B)$. |
| \end{enumerate} |
\end{enumerate} |
