Notes:Hereditary sigma-ring/Facts

1. 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]
2. [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
3. [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is just [ilmath]\mathcal{H}(S)[/ilmath] closed under countable union.
4. [ilmath]\sigma_R(\mathcal{H}(S))[/ilmath] is hereditary