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σ-algebra [2013/08/18 23:43] nikolaj |
σ-algebra [2016/06/20 18:46] nikolaj |
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| ===== σ-algebra ===== | ===== σ-algebra ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $\Sigma$ in it | | |
| - | | @#55EE55: $ \Sigma\in \mathrm{SigmaAlgebra}(X) $ | | + | | @#55EE55: postulate | @#55EE55: $\Sigma\subseteq \mathcal P(X)$ | |
| - | + | | @#55EE55: postulate | @#55EE55: $ \Sigma\ne\emptyset $ | | |
| - | | @#88DDEE: $E\in\Sigma$ | | + | | forall | $E\in\Sigma$ | |
| - | | $A\in\mathrm{Sequence}(\Sigma)$ | | + | | @#55EE55: postulate | @#55EE55: $ X\smallsetminus E \in \Sigma $ | |
| - | | @#DDDDDD: $n\in \mathbb N$ | | + | | forall | $A\in\mathrm{Sequence}(\Sigma)$ | |
| - | + | | forall | $n\in \mathbb N$ | | |
| - | | @#55EE55: $\Sigma\subseteq \mathcal P(X)$ | | + | | @#55EE55: postulate | @#55EE55: $ \bigcup_{i=1}^n A_i \in \Sigma $ | |
| - | | @#55EE55: $ \Sigma\ne\emptyset $ | | + | |
| - | | @#55EE55: $ X\smallsetminus E \in \Sigma $ | | + | |
| - | | @#55EE55: $ \bigcup_{i=1}^n A_i \in \Sigma $ | | + | |
| ==== Ramifications ==== | ==== Ramifications ==== | ||
| ==== Reference ==== | ==== Reference ==== | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Sigma-algebra|Sigma-algebra]] | Wikipedia: [[http://en.wikipedia.org/wiki/Sigma-algebra|Sigma-algebra]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| [[Power set]] | [[Power set]] | ||
| - | === Image of === | + | === Equivalent to === |
| [[Measurable space]] | [[Measurable space]] | ||
| - | === Parents === | + | === Related === |
| [[Countable union]] | [[Countable union]] | ||
