Caratheodory's Theorem: Difference between revisions
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Consider an out measure <math> \mu </math> on <math> X </math>. Define | Consider an out measure <math> \mu </math> on <math> X </math>. Define | ||
<math> \mathcal{M} = \{ A \subseteq X : A is \mu-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>. | Then <math> \mathcal{M} </math> is a <math>\sigma</math>-algebra and <math> \mu^* </math> is a measure on <math> \mathcal{M} </math>. | ||
Revision as of 22:25, 16 December 2020
Statement
Consider an out measure on . Define
.
Then is a -algebra and is a measure on .
