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smallest_generated_σ-algebra [2013/08/12 20:23] ben |
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| ===== Smallest generated σ-algebra ===== | ===== Smallest generated σ-algebra ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $A\subseteq X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
| - | | @#55EE55: $\sigma_X(A)\equiv \bigcap \{B\ |\ A\subseteq B\land B\subseteq \mathrm{SigmaAlgebra}(X)\} $ | | + | | @#55EE55: postulate | @#55EE55: $\sigma(A):\equiv \bigcap \{C\ |\ A\subseteq C\land C\subseteq \mathrm{SigmaAlgebra}(X)\} $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| The smallest σ-algebra over $X$ generated by, and hence containing, $A$. | The smallest σ-algebra over $X$ generated by, and hence containing, $A$. | ||
| Line 9: | Line 9: | ||
| ==== Reference ==== | ==== Reference ==== | ||
| Wikipedia: [[http://de.wikipedia.org/wiki/%CE%A3-Algebra#Bedeutung|σ-Operator (german)]] | Wikipedia: [[http://de.wikipedia.org/wiki/%CE%A3-Algebra#Bedeutung|σ-Operator (german)]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Element of === | === Element of === | ||
| [[σ-algebra]] | [[σ-algebra]] | ||
