Difference between revisions of "Notes:Hereditary sigma-ring/Facts"
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| − | # {{M|1=\sigma_R(\mathcal{H}(S))}} is just {{M|\mathcal{H}(S)}} closed under countable union. | + | # {{M|1=\sigma_R(\mathcal{H}(S))}} is just {{M|\mathcal{H}(S)}} closed under countable union.<!-- |
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| + | FACT 4 | ||
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| + | # {{M|\sigma_R(\mathcal{H}(S))}} is hereditary | ||
Latest revision as of 02:24, 8 April 2016
- An hereditary system is a sigma-ring [ilmath]\iff[/ilmath] it is closed under countable unions.
- Thus [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is just [ilmath]\mathcal{H}(S)[/ilmath] with the additional property:
- [ilmath]\forall(A_n)_{n=1}^\infty\subseteq\mathcal{H}(S)\left[\bigcup_{n=1}^\infty A_n\in\sigma_R(\mathcal{H}(S))\right][/ilmath]
- Thus [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is just [ilmath]\mathcal{H}(S)[/ilmath] with the additional property:
- [ilmath]\mathcal{H}(\mathcal{R})[/ilmath] is a [ilmath]\sigma[/ilmath]-ring (for any [ilmath]\sigma[/ilmath]-ring, [ilmath]\mathcal{R} [/ilmath])
- This means [ilmath]\sigma_R(\mathcal{H}(\mathcal{R}))=\mathcal{H}(\mathcal{R})[/ilmath]
- It also means [ilmath]\mathcal{H}(\sigma_R(S))[/ilmath] is a [ilmath]\sigma[/ilmath]-ring
- [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is just [ilmath]\mathcal{H}(S)[/ilmath] closed under countable union.
- [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is hereditary
