Difference between revisions of "Hereditary sigma-ring"
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==Definition== | ==Definition== | ||
A ''hereditary {{sigma|ring}}'', {{M|\mathcal{H} }}, is a system of sets that is both [[hereditary (measure theory)|hereditary]] and a [[sigma-ring|{{sigma|ring}}]]{{rMTH}}. This means {{M|\mathcal{H} }} has the following properties: | A ''hereditary {{sigma|ring}}'', {{M|\mathcal{H} }}, is a system of sets that is both [[hereditary (measure theory)|hereditary]] and a [[sigma-ring|{{sigma|ring}}]]{{rMTH}}. This means {{M|\mathcal{H} }} has the following properties: | ||
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Ideally another references, more properties. Additionally the "use" section requires expansion. Comment on power-set and sigma-algebra special case. Maybe note a synonym: sigma-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}[A-B\in\mathcal{H}][/ilmath] - hereditary-ness 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 pre-measure to an outer-measure.
See also
References
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