Outer measure: Difference between revisions
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* <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^* (\ | * <math> \mu^* \left(\bigcup\limits_{j=1}^\infty A_j \right) \leq \sum_{j=1}^\infty \mu^*(A_j).</math> | ||
The second and third conditions in the definition of an outer measure is equivalent that <math> A \subseteq \ | The second and third conditions in the definition of an outer measure is equivalent that <math> A \subseteq \bigcup\limits_{i=1}^\infty B_i </math> implies <math>\mu^*(A) \leq \sum_{i=1}^\infty \mu^*(B_i)</math>. | ||
: '''Definition.''' A set <math> A \subset X </math> is called <math> \mu^* </math>-measurable if <math> \mu^*(E) = \mu^*(E \cap A) + \mu^* (E \cap A^c)</math> for all <math> E \subset X </math>. | : '''Definition.''' A set <math> A \subset X </math> is called <math> \mu^* </math>-measurable if <math> \mu^*(E) = \mu^*(E \cap A) + \mu^* (E \cap A^c)</math> for all <math> E \subset X </math>. | ||
Revision as of 18:20, 22 October 2020
- Definition. Let be a nonempty set. An outer measure [1] on the set is a function such that
![{\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a514e94c721b4c864074ebea3cc863d66fa11ef2)
- ,
- if ,
The second and third conditions in the definition of an outer measure is equivalent that implies .
- Definition. A set is called -measurable if for all .
Constructing a measure from an outer measure
References
- ↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, Section 1.4