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smallest_generated_σ-algebra [2013/08/12 20:23] ben |
smallest_generated_σ-algebra [2013/08/14 16:52] nikolaj |
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| ===== Smallest generated σ-algebra ===== | ===== Smallest generated σ-algebra ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | @#88DDEE: $A\subseteq X$ | | + | | @#88DDEE: $X$ | |
| - | | @#55EE55: $\sigma_X(A)\equiv \bigcap \{B\ |\ A\subseteq B\land B\subseteq \mathrm{SigmaAlgebra}(X)\} $ | | + | | @#55EE55: $\sigma: \mathcal P(X) \to \mathcal P(X) $ | |
| + | | @#55EE55: $\sigma(A):= \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, $A$. | ||
