Types of set algebras/Type table

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Type table

System Type Definition Deductions
Ring[1][2]
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\backslash[/ilmath]-closed[Note 1]
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed[Note 2]
  • [ilmath]\emptyset\in\mathcal{A} [/ilmath][Note 3]
[ilmath]\sigma[/ilmath]-ring[1][2]
  • [ilmath]\mathcal{A} [/ilmath] is a ring
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[Note 4]
  • [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed also[Theorem 1]
Algebra[1][2]
  • [ilmath]\mathcal{A} [/ilmath] is closed under complements
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\backslash[/ilmath]-closed[Note 5]
  • [ilmath]\emptyset\in\mathcal{A} [/ilmath][Note 6]
  • [ilmath]\Omega\in\mathcal{A} [/ilmath][Note 7]
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed[Theorem 2]
[ilmath]\sigma[/ilmath]-algebra[1][2]
  • [ilmath]\mathcal{A} [/ilmath] is an algebra
  • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed
Semiring[1]

TODO: Page 3 in[1]


Dynkin system[1][3]
  • [ilmath]\Omega\in\mathcal{A} [/ilmath]
  • [ilmath]\mathcal{A} [/ilmath] is closed under complements
  • [ilmath]\sigma[/ilmath]-[ilmath]\udot[/ilmath]-closed
  • [ilmath]\emptyset\in\mathcal{A} [/ilmath][Note 9]

Theorems

  1. 1.0 1.1 Using Class of sets closed under set-subtraction properties we know that if [ilmath]\mathcal{A} [/ilmath] is closed under Set subtraction then:
    • [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed
    • [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[ilmath]\implies[/ilmath][ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed
  2. Using Class of sets closed under complements properties we see that if [ilmath]\mathcal{A} [/ilmath] is closed under complements then:
    • [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

Notes

  1. Closed under finite Set subtraction
  2. Closed under finite Union
  3. As given [ilmath]A\in\mathcal{A} [/ilmath] we must have [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath]
  4. closed under finite or countably infinite union
  5. Note that [ilmath]A-B=A\cap B^c=(A^c\cup B)^c[/ilmath] - or that [ilmath]A-B=(A^c\cup B)^c[/ilmath] - so we see that being closed under union and complement means we have closure under set subtraction.
  6. As we are closed under set subtraction we see [ilmath]A-A=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{A} [/ilmath]
  7. As we are closed under set subtraction we see that [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath], so [ilmath]\emptyset\in\mathcal{A} [/ilmath] - but we are also closed under complements, so [ilmath]\emptyset^c\in\mathcal{A} [/ilmath] and [ilmath]\emptyset^c=\Omega\in\mathcal{A}[/ilmath]
  8. Trivial - satisfies the definitions
  9. As [ilmath]\Omega^c=\emptyset[/ilmath] by being closed of complements, [ilmath]\emptyset\in\mathcal{A} [/ilmath]

References

  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Probability Theory - A comprehensive course - second edition - Achim Klenke
  2. 2.0 2.1 2.2 2.3 Measure Theory - Paul R. Halmos
  3. 3.0 3.1 Measures Integrals and Martingales - Rene L. Schilling