Hereditary [ilmath]\sigma[/ilmath]ring
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Ideally another references, more properties. Additionally the "use" section requires expansion. Comment on powerset and sigmaalgebra special case. Find out about related term, [ilmath]\sigma[/ilmath]ideal
Definition
A hereditary [ilmath]\sigma[/ilmath]ring, [ilmath]\mathcal{H} [/ilmath], is a system of sets that is both hereditary and a [ilmath]\sigma[/ilmath]ring^{[1]}. This means [ilmath]\mathcal{H} [/ilmath] has the following properties:
 [ilmath]\forall A\in\mathcal{H}\forall B\in\mathcal{P}(A)[B\in\mathcal{H}][/ilmath]  hereditary  all subsets of any set in [ilmath]\mathcal{H} [/ilmath] are in [ilmath]\mathcal{H} [/ilmath].
 [ilmath] \forall ({ A_n })_{ n = 1 }^{ \infty }\subseteq \mathcal{H} [\bigcup_{n=1}^\infty A_n\in\mathcal{H}] [/ilmath]  [ilmath]\sigma[/ilmath][ilmath]\cup[/ilmath]closed, closed under countable union.
Immediate properties
 [ilmath]\mathcal{H} [/ilmath] is closed under set subtraction
 That is: [ilmath]\forall A,B\in\mathcal{H}[AB\in\mathcal{H}][/ilmath]  hereditaryness is sufficient for this.
 [ilmath]\emptyset\in\mathcal{H} [/ilmath]
TODO: Format these using inline theorem boxes, proofs are so easy that the "requires proof" tag would be overkill
Use
Hereditary [ilmath]\sigma[/ilmath]rings are used when going from a premeasure to an outermeasure.
See also
References
