Outer measure: Difference between revisions
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Let <math> X </math> be a nonempty set. An outer measure on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that | |||
* <math> \mu^* ( \emptyset) = 0 </math> | : '''Definition.''' 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'', §1.4 </ref> on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that | ||
* <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math> | * <math> \mu^* ( \emptyset) = 0 </math>, | ||
* <math> \mu* \left(\ | * <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math>, | ||
* <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 are equivalent to the condition 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>. | |||
==Examples of Outer Measures== | |||
The standard example of an outer measure is the Lebesgue outer measure, defined on subsets of <math>\mathbb{R}</math>. | |||
:<math> \mu^*(A) = \inf \left\{ \sum_{i=1}^\infty |b_i - a_i| : A \subseteq \bigcup_{i=1}^\infty (a_i, b_i) \right\}. </math> | |||
A near-generalization of the Lebesgue outer measure is given by | |||
:<math> \mu^*_F(A) = \inf \left\{ \sum_{i=1}^\infty |F(b_i) - F(a_i)| : A \subseteq \bigcup_{i=1}^\infty (a_i, b_i] \right\}, </math> | |||
where <math>F</math> is any [[right-continuous]] function <ref name="Folland2">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §1.5</ref>. | |||
Given a measure space <math>(X, \mathcal{M}, \mu)</math>, one can always define an outer measure <math>\mu^*</math><ref name="Craig">Craig, Katy. ''MATH 201A HW 3''. UC Santa Barbara, Fall 2020.</ref> by | |||
:<math> \mu^*(A) = \inf \left\{ \mu(B) : A \subseteq B, B \in \mathcal{M} \right\}. </math> | |||
==References== | |||
Latest revision as of 06:58, 2 December 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 are equivalent to the condition that implies .
- Definition. A set is called -measurable if for all .
Examples of Outer Measures
The standard example of an outer measure is the Lebesgue outer measure, defined on subsets of .
A near-generalization of the Lebesgue outer measure is given by
![{\displaystyle \mu _{F}^{*}(A)=\inf \left\{\sum _{i=1}^{\infty }|F(b_{i})-F(a_{i})|:A\subseteq \bigcup _{i=1}^{\infty }(a_{i},b_{i}]\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f663363eb887ffffd6bb2181fdde85032c8c8687)
where is any right-continuous function [2].
Given a measure space , one can always define an outer measure [3] by
