Outer measure: Difference between revisions
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Let <math> X </math> be a nonempty set. An outer measure <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', Section 1.4 </ref> on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that | Let <math> X </math> be a nonempty set. An outer measure <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', Section 1.4 </ref> on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that | ||
* <math> \mu^* ( \emptyset) = 0 </math> | * <math> \mu^* ( \emptyset) = 0 </math>, | ||
* <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math> | * <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math>, | ||
* <math> \mu* (\cup_{j=1}^\infty A_j) \leq \sum_{j=1}^\infty \mu^*(A_j).</math> | * <math> \mu* (\cup_{j=1}^\infty A_j) \leq \sum_{j=1}^\infty \mu^*(A_j).</math> | ||
==References== | ==References== | ||
![{\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a514e94c721b4c864074ebea3cc863d66fa11ef2)
