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outer measure
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(Definition)
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Definition [1,2,3] Let $X$ be a set, and let $\mathcal{P}(X)$ be the power set of $X$ . An outer measure on $X$ is a function $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ satisfying the properties
- $\mu^\ast(\emptyset)=0$ .
- If $A\subset B$ are subsets in $X$ , then $\mu^\ast(A)\le \mu^\ast(B)$ .
- If $\{A_i\}$ is a countable collection of subsets of $X$ , then $$ \mu^\ast(\bigcup_i A_i) \le \sum_i \mu^\ast (A_i).$$
Here, we can make two remarks. First, from (1) and (2), it follows that $\mu^\ast$ is a positive function on $\mathcal{P}(X)$ . 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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Cross-references: empty sets, sequence, infinite, finite, positive, collection, countable, subsets, properties, function, power set
There are 5 references 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 4200 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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