Notes:Hereditary sigma-ring/Facts

1. An hereditary system is a sigma-ring $\iff$ it is closed under countable unions.
   • Thus $\sigma_R(\mathcal{H}(S))$ is just $\mathcal{H}(S)$ with the additional property:
     • $\forall(A_n)_{n=1}^\infty\subseteq\mathcal{H}(S)\left[\bigcup_{n=1}^\infty A_n\in\sigma_R(\mathcal{H}(S))\right]$
2. $\mathcal{H}(\mathcal{R})$ is a $\sigma$-ring (for any $\sigma$-ring, $\mathcal{R}$)
   • This means $\sigma_R(\mathcal{H}(\mathcal{R}))=\mathcal{H}(\mathcal{R})$
   • It also means $\mathcal{H}(\sigma_R(S))$ is a $\sigma$-ring
3. $\sigma_R(\mathcal{H}(S))$ is just $\mathcal{H}(S)$ closed under countable union.
4. $\sigma_R(\mathcal{H}(S))$ is hereditary