Class of sets closed under complements properties

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Theorem statement

If [ilmath]\mathcal{A} [/ilmath] is a system of subsets of [ilmath]\Omega[/ilmath] such that[1]:

  • [math]\forall A\in\mathcal{A}[A^c\in\mathcal{A}][/math] where denotes the complement of [ilmath]A[/ilmath] - That is to say "[ilmath]\mathcal{A} [/ilmath] is closed under complements"

Then we have:[Note 1]

  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed


TODO: see [1] page 2 if help needed - basically is just De Morgan's laws

See also


  1. See Index of properties under "closed" for the exact meanings of these


  1. 1.0 1.1 Probability Theory - A comprehensive course - Achim Klenke