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measure [2014/03/21 11:11] 127.0.0.1 external edit |
measure [2015/04/01 10:30] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $\Sigma \in \mathrm{SigmaAlgebra}(X) $ | | | @#55CCEE: context | @#55CCEE: $\Sigma \in \mathrm{SigmaAlgebra}(X) $ | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $ \mu\in \mathrm{measure}(\Sigma)$ | | | @#FFBB00: definiendum | @#FFBB00: $ \mu\in \mathrm{measure}(\Sigma)$ | | ||
| - | |||
| | @#55CCEE: context | @#55CCEE: $E\in \Sigma$ | | | @#55CCEE: context | @#55CCEE: $E\in \Sigma$ | | ||
| | @#55CCEE: context | @#55CCEE: $S\in \mathrm{Sequence}(\Sigma)$ | | | @#55CCEE: context | @#55CCEE: $S\in \mathrm{Sequence}(\Sigma)$ | | ||
| - | + | | @#AAFFAA: inclusion | @#AAFFAA: $\mu:\Sigma\to \overline{\mathbb R} $ | | |
| - | | @#55EE55: postulate | @#55EE55: $ \mu:\Sigma\to \overline{\mathbb R} $ | | + | | @#55EE55: postulate | @#55EE55: $ \mu(E)\ge 0$ | |
| | @#55EE55: postulate | @#55EE55: $ \mu(\emptyset)=0 $ | | | @#55EE55: postulate | @#55EE55: $ \mu(\emptyset)=0 $ | | ||
| - | | @#55EE55: postulate | @#55EE55: $ \mu(E)\ge 0$ | | ||
| | @#55EE55: postulate | @#55EE55: $ \mu\left(\bigcup_{j=1}^\infty S_j\right)=\sum_{j=1}^\infty \mu(S_j) $ | | | @#55EE55: postulate | @#55EE55: $ \mu\left(\bigcup_{j=1}^\infty S_j\right)=\sum_{j=1}^\infty \mu(S_j) $ | | ||
| - | ==== Discussion ==== | + | ----- |
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Measure_%28mathematics%29|Measure]] | Wikipedia: [[http://en.wikipedia.org/wiki/Measure_%28mathematics%29|Measure]] | ||
| - | ==== Parents ==== | + | |
| + | ----- | ||
| + | === Context === | ||
| + | [[σ-algebra]], [[Extended real number line]], [[Infinite series]] | ||
| === Equivalent to === | === Equivalent to === | ||
| [[Measure space]] | [[Measure space]] | ||
| - | === Context === | ||
| - | [[σ-algebra]], [[Extended real number line]], [[Infinite series]] | ||
