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counting measure (Definition)

Let $(X,\borel)$ be a measurable space. The measure $\mu$ on $X$ defined by

$\displaystyle \mu(A) = \left\{ \begin{array}{ll} n & \text{if}\, A\, \text{ has exactly }\, n\, \text{ elements} \\ \infty & \text{otherwise.} \end{array}\right.$    

for all $A\in\borel$ is called the counting measure on $X$ . Usually this is applied when $X$ is countable, e.g. $\naturals$ or $\integers$ .



"counting measure" is owned by mathwizard. [ full author list (2) | owner history (1) ]
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See Also: measure


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support of integrable function with respect to counting measure is countable (Result) by Wkbj79
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Cross-references: countable, measure, measurable space
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This is version 4 of counting measure, born on 2002-02-18, modified 2007-08-13.
Object id is 2096, canonical name is CountingMeasure.
Accessed 7664 times total.

Classification:
AMS MSC: 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
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