Hereditary $\sigma$-ring

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Ideally another references, more properties. Additionally the "use" section requires expansion. Comment on power-set and sigma-algebra special case. Find out about related term, $\sigma$-ideal

Definition

A hereditary $\sigma$-ring, $\mathcal{H}$, is a system of sets that is both hereditary and a $\sigma$-ring. This means $\mathcal{H}$ has the following properties:

1. $\forall A\in\mathcal{H}\forall B\in\mathcal{P}(A)[B\in\mathcal{H}]$ - hereditary - all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$.
2. $\forall ({ A_n })_{ n = 1 }^{ \infty }\subseteq \mathcal{H} [\bigcup_{n=1}^\infty A_n\in\mathcal{H}]$ - $\sigma$-$\cup$-closed, closed under countable union.

Immediate properties

• $\mathcal{H}$ is closed under set subtraction
  • That is: $\forall A,B\in\mathcal{H}[A-B\in\mathcal{H}]$ - hereditary-ness is sufficient for this.
• $\emptyset\in\mathcal{H}$

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TODO: Format these using inline theorem boxes, proofs are so easy that the "requires proof" tag would be overkill

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Use

Hereditary $\sigma$-rings are used when going from a pre-measure to an outer-measure.

See also

• Hereditary $\sigma$-ring generated by
• Outer-measure