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	also [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed^{[Theorem 1]} Is a Dynkin system^[3]^{[Note 8]}
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

↑ ^1.0 ^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
↑
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

↑ Closed under finite Set subtraction
↑ Closed under finite Union
↑ As given [ilmath]A\in\mathcal{A} [/ilmath] we must have [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath]
↑ closed under finite or countably infinite union
↑ 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.
↑ As we are closed under set subtraction we see [ilmath]A-A=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{A} [/ilmath]
↑ 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]
↑ Trivial - satisfies the definitions
↑ As [ilmath]\Omega^c=\emptyset[/ilmath] by being closed of complements, [ilmath]\emptyset\in\mathcal{A} [/ilmath]

References

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

[.5C-closed-consequences-7] 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] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed

[3] [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed[ilmath]\implies[/ilmath][ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed

[complement-closed-consequences-11] 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

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

[6] [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

[.5C-closed-3] Closed under finite Set subtraction

[U-closed-4] Closed under finite Union

[5] As given [ilmath]A\in\mathcal{A} [/ilmath] we must have [ilmath]A-A\in\mathcal{A} [/ilmath] and [ilmath]A-A=\emptyset[/ilmath]

[sigma-U-closed-6] sed under finite or countably infinite union

[8] 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.

[9] As we are closed under set subtraction we see [ilmath]A-A=\emptyset[/ilmath] so [ilmath]\emptyset\in\mathcal{A} [/ilmath]

[10] 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]

[13] Trivial - satisfies the definitions

[14] As [ilmath]\Omega^c=\emptyset[/ilmath] by being closed of complements, [ilmath]\emptyset\in\mathcal{A} [/ilmath]

[PTACC-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 Probability Theory - A comprehensive course - second edition - Achim Klenke

[MTH-2] 2.0 ^2.1 ^2.2 ^2.3 Measure Theory - Paul R. Halmos

[MIM-12] 3.0 ^3.1 Measures Integrals and Martingales - Rene L. Schilling

[1]

[2]

[Note 1]

[Note 2]

[Note 3]

[Note 4]

[Theorem 1]

[Note 5]

[Note 6]

[Note 7]

[Theorem 2]

[3]

[Note 8]

[Note 9]

Types of set algebras/Type table

Contents

Type table

Theorems

Notes

References

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