Difference between revisions of "Sigma-algebra"
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* [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]] | * [[Sigma-algebra generated by|{{Sigma|algebra}} generated by]] | ||
* [[Sigma-ring|{{Sigma|ring}}]] | * [[Sigma-ring|{{Sigma|ring}}]] | ||
| + | * [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]] | ||
==Notes== | ==Notes== | ||
Revision as of 16:01, 28 August 2015
A Sigma-algebra of sets, or [ilmath]\sigma[/ilmath]-algebra is very similar to a [ilmath]\sigma[/ilmath]-ring of sets.
Like how ring of sets and algebra of sets differ, the same applies to [ilmath]\sigma[/ilmath]-ring compared to [ilmath]\sigma[/ilmath]-algebra
Contents
Definition
A non empty class of sets [ilmath]S[/ilmath] is a [ilmath]\sigma[/ilmath]-algebra[Note 1] if[1][2]
- if [math]A\in S[/math] then [math]A^c\in S[/math]
- if [math]\{A_n\}_{n=1}^\infty\subset S[/math] then [math]\cup^\infty_{n=1}A_n\in S[/math]
That is it is closed under complement and countable union.
Immediate consequences
Among other things immediately we see that:
- [ilmath]\mathcal{A} [/ilmath] is closed under set subtraction
- As [ilmath]A-B=(A^c\cup B)^c[/ilmath] and a [ilmath]\sigma[/ilmath]-algebra is closed under complements and unions, this shows it is closed under set subtraction too
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed (furthermore, that [ilmath]\mathcal{A} [/ilmath] is in fact [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed - that is closed under countable intersections)
- [ilmath]\emptyset\in\mathcal{A} [/ilmath]
- [ilmath]\forall A\in\mathcal{A} [/ilmath] we have [ilmath]A-A\in\mathcal{A} [/ilmath] (by closure under set subtraction), as [ilmath]A-A=\emptyset[/ilmath], [ilmath]\emptyset\in\mathcal{A} [/ilmath]
- [ilmath]X\in\mathcal{A} [/ilmath][Note 2]
- As [ilmath]\emptyset\in\mathcal{A} [/ilmath] and it is closed under complement we see that [ilmath]\emptyset^c\in\mathcal{A} [/ilmath] (by closure under complement) and [ilmath]\emptyset^c=X[/ilmath] - the claim follows.
- [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-algebra [ilmath]\implies[/ilmath] [ilmath]\mathcal{A} [/ilmath] is a [ilmath]\sigma[/ilmath]-ring
- To prove this we must check:
- [ilmath]\mathcal{A} [/ilmath] is closed under countable union
- True by definition of [ilmath]\sigma[/ilmath]-algebra
- [ilmath]\mathcal{A} [/ilmath] is closed under set subtraction
- We've already shown this, so this is true too.
- [ilmath]\mathcal{A} [/ilmath] is closed under countable union
- This completes the proof.
- To prove this we must check:
Important theorems
The intersection of [ilmath]\sigma[/ilmath]-algebras is a [ilmath]\sigma[/ilmath]-algebra
TODO: Proof - see PTACC page 5, also in Halmos AND in that other book
Common [ilmath]\sigma[/ilmath]-algebras
See also: Index of common [ilmath]\sigma[/ilmath]-algebras
- [ilmath]\sigma[/ilmath]-algebra generated by
- Trace [ilmath]\sigma[/ilmath]-algebra
- Pre-image [ilmath]\sigma[/ilmath]-algebra
See also
- Types of set algebras
- [ilmath]\sigma[/ilmath]-algebra generated by
- [ilmath]\sigma[/ilmath]-ring
- Properties of a class of sets closed under set subtraction
Notes
- ↑ Some books (notably Measures, Integrals and Martingales) give [ilmath]X\in\mathcal{A} [/ilmath] as a defining property of [ilmath]\sigma[/ilmath]-algebras, however the two listed are sufficient to show this (see the immediate consequences section)
- ↑ Measures, Integrals and Martingales puts this in the definition of [ilmath]\sigma[/ilmath]-algebras
References
- ↑ Halmos - Measure Theory - page 28 - Springer - Graduate Texts in Mathematics - 18
- ↑ Measures, Integrals and Martingales - Rene L. Schilling
