Outer measure: Difference between revisions

Revision as of 14:20, 20 October 2020

Let $X$ be a nonempty set. An outer measure on the set $X$ is a function $\mu ^{*}:2^{X}\to [0,\infty ]$ such that

$\mu ^{*}(\emptyset )=0$
$\mu ^{*}(A)\leq \mu ^{*}(B)$ if $A\subseteq B$
$\mu *\left(\cup _{j=1}^{\infty }A_{j}\right)\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).$

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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 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^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math>
 * <math> \mu* \left(\cup_{j=1}^\infty A_j\right) \leq  \sum_{j=1}^\infty \mu^*(A_j).</math>