Properties of classes of sets closed under set-subtraction
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Theorem statement
If [ilmath]\mathcal{A} [/ilmath] is a class of subsets of [ilmath]\Omega[/ilmath] such that[1]
- [math]\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}][/math] - that is closed under set-subtraction (or [ilmath]\backslash[/ilmath]-closed)
Then we have:
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[ilmath]\implies[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed
- Any countable union of sets in [ilmath]\mathcal{A} [/ilmath] can be expressed as a countable disjoint union of sets in [ilmath]\mathcal{A} [/ilmath][Note 1]
Proof:
Notes
- ↑ Note that this doesn't require [ilmath]\mathcal{A} [/ilmath] to be closed under union, we can still talk about unions we just cannot know that the result of a union is in [ilmath]\mathcal{A} [/ilmath]
References
