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additive (Definition)

Let $ \phi$ be some positive-valued real 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

$\displaystyle \phi(A \cup B) = \phi(A) + \phi(B) .$

Suppose $ \mathcal{A}$ is a $ \sigma$-algebra. Then, given any sequence $ \langle A_i \rangle$ of disjoint sets in $ \mathcal{A}$, if we have

$\displaystyle \phi\left( \bigcup A_i \right) = \sum \phi(A_i)$
we say that $ \phi$ is countably additive or $ \sigma$-additive.

Useful properties of an additive set function $ \phi$ include the following:

  1. $ \phi(\emptyset) = 0$.
  2. If $ A \subseteq B$, then $ \phi(A) \leq \phi(B)$.
  3. If $ A \subseteq B$, then $ \phi(B \setminus A) = \phi(B) - \phi(A)$.
  4. Given $ A$ and $ B$, $ \phi(A \cup B) + \phi(A \cap B) = \phi(A) + \phi(B)$.



"additive" is owned by Andrea Ambrosio. [ full author list (2) | owner history (1) ]
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Other names:  additivity
Also defines:  countable additivity, countably additive, $\sigma$-additive, sigma-additive
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Cross-references: set function, properties, sequence, disjoint, algebra of sets, real function
There are 62 references to this entry.

This is version 4 of additive, born on 2002-08-30, modified 2007-03-21.
Object id is 3400, canonical name is Additive.
Accessed 18338 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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