Difference between revisions of "Properties of classes of sets closed under set-subtraction"

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Revision as of 21:01, 15 June 2015

Theorem statement

If A is a class of subsets of Ω such that[1]

  • A,BA[ABA]
    - that is closed under set-subtraction (or -closed)

Then we have:

  1. A is -closed
  2. A is σ--closed A is σ--closed
  3. Any countable union of sets in A can be expressed as a countable disjoint union of sets in A[Note 1]
[Expand]

Proof:

Notes

  1. Jump up Note that this doesn't require A to be closed under union, we can still talk about unions we just cannot know that the result of a union is in A

References

  1. Jump up to: 1.0 1.1 Probability Theory - A comprehensive course - Second Edition - Achim Klenke