Caratheodory's Theorem: Difference between revisions

Revision as of 22:25, 16 December 2020

Consider an out measure $\mu$ on $X$ . Define

${\mathcal {M}}=\{A\subseteq X:A\ {\text{is}}\ \mu -{\text{measurable}}\}$ .

Then ${\mathcal {M}}$ is a $\sigma$ -algebra and $\mu ^{*}$ is a measure on ${\mathcal {M}}$ .

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 Consider an out measure <math> \mu </math> on <math> X </math>. Define
-<math> \mathcal{M} = \{ A \subseteq X : A \text{is} \  \mu-\text{measurable} \} </math>.
+<math> \mathcal{M} = \{ A \subseteq X : A \ \text{is} \  \mu-\text{measurable} \} </math>.
 Then <math> \mathcal{M} </math> is a <math>\sigma</math>-algebra and <math> \mu^* </math> is a measure on <math> \mathcal{M} </math>.