PlanetMath: atom (measure theory)

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atom (measure theory)

(Definition)

Let $(X, \mathcal{B},\mu)$ be a measure space. A set $A \in \mathcal{B}$ is called an atom if has positive measure and contains no measurable subsets $B \subset A$ such that $0 < \mu (B)< \mu(A)$ .

An equivalent definition can be: has positive measure and for every measurable subset $B \subset A$ , either $\mu(B)=0$ or $\mu(A-B)=0$ .

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Cross-references: equivalent, subsets, measurable, contains, measure, positive, measure space
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This is version 1 of atom (measure theory), born on 2007-11-26.
Object id is 10064, canonical name is AtomMeasureTheory.
Accessed 648 times total.

Classification:

28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)

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