Hereditary [ilmath]\sigma[/ilmath]-ring

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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:

  1. [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].
  2. [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


Hereditary [ilmath]\sigma[/ilmath]-rings are used when going from a pre-measure to an outer-measure.

See also


  1. Measure Theory - Paul R. Halmos