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 σ-ring, H, is a system of sets that is both hereditary and a σ-ring[1]. This means H has the following properties:
- ∀A∈H∀B∈P(A)[B∈H] - hereditary - all subsets of any set in H are in H.
- ∀(An)∞n=1⊆H[⋃∞n=1An∈H] - σ-∪-closed, closed under countable union.
Immediate properties
- H is closed under set subtraction
- That is: ∀A,B∈H[A−B∈H] - hereditary-ness is sufficient for this.
- ∅∈H
TODO: Format these using inline theorem boxes, proofs are so easy that the "requires proof" tag would be overkill
Use
Hereditary σ-rings are used when going from a pre-measure to an outer-measure.
See also
References
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