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outer measure
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(Definition)
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Definition [1,2,3] Let  be a set, and let
 be the power set of  . An outer measure on  is a function
![$ \mu^\ast:\mathcal{P}(X)\to [0,\infty]$](http://images.planetmath.org:8080/cache/objects/4456/l2h/img5.png "$ \mu^\ast:\mathcal{P}(X)\to [0,\infty]$") satisfying the properties
-
 .
- If
 are subsets in  , then
 .
- If
 is a countable collection of subsets of  , then
Here, we can make two remarks. First, from (1) and (2), it follows that  is a positive function on
 . Second, property (3) also holds for any finite collection of subsets since we can always append an infinite sequence of empty sets to such a collection.
- 1
- A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
- 2
- A. Friedman, Foundations of Modern Analysis, Dover publications, 1982.
- 3
- G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
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"outer measure" is owned by mathcam. [ owner history (1) ]
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(view preamble)
Cross-references: empty sets, sequence, infinite, finite, positive, collection, countable, subsets, properties, function, power set
There is 1 reference to this entry.
This is version 3 of outer measure, born on 2003-07-15, modified 2005-05-17.
Object id is 4456, canonical name is OuterMeasure2.
Accessed 3509 times total.
Classification:
| AMS MSC: | 28A10 (Measure and integration :: Classical measure theory :: Real- or complex-valued set functions) | | | 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory) |
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Pending Errata and Addenda
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