Outer measure: Difference between revisions
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Let <math> X </math> be a nonempty set. An outer measure on <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 on <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that | ||
(i) <math> \mu^* ( \emptyset) = 0 </math> | |||
(ii) <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math> | |||
(iii) <math> \mu* \left(\cup_{j=1}^\infty A_j\right) \leq \sum_{j=1}^\infty \mu^*(A_j).</math> | |||
Revision as of 14:18, 20 October 2020
Let be a nonempty set. An outer measure on is a function such that (i) (ii) if (iii)
![{\displaystyle \mu ^{*}:2^{X}\to [0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a514e94c721b4c864074ebea3cc863d66fa11ef2)
