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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 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:
- $\phi(\emptyset) = 0$
- If $A \subseteq B$ then $\phi(A) \leq \phi(B)$
- If $A \subseteq B$ then $\phi(B \setminus A) = \phi(B) - \phi(A)$
- Given $A$ and $B$ $\phi(A \cup B) + \phi(A \cap B) = \phi(A) + \phi(B)$
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"additive" is owned by Andrea Ambrosio. [ full author list (2) | owner history (1) ]
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| Also defines: |
countable additivity, countably additive,  -additive, sigma-additive |
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Cross-references: properties, union, sequence, disjoint, algebra of sets, set function
There are 105 references to this entry.
This is version 7 of additive, born on 2002-08-30, modified 2008-11-23.
Object id is 3400, canonical name is Additive.
Accessed 22759 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) |
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Pending Errata and Addenda
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