Difference between revisions of "Sigma-algebra"
From Maths
m (Reverted edits by JessicaBelinda133 (talk) to last revision by Alec) |
m (→Immediate consequences) |
||
| Line 14: | Line 14: | ||
{{Begin Inline Proof}} | {{Begin Inline Proof}} | ||
:: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this shows it is closed under set subtraction too | :: As {{M|1=A-B=(A^c\cup B)^c}} and a {{sigma|algebra}} is closed under complements and unions, this shows it is closed under set subtraction too | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | {{Begin Inline Theorem}} | ||
| + | * {{M|\mathcal{A} }} is {{M|\cap}}-closed (furthermore, that {{M|\mathcal{A} }} is in fact {{M|\sigma}}-{{M|\cap}}-closed - that is closed under countable intersections) | ||
| + | {{Begin Inline Proof}} | ||
| + | : See [[Class of sets closed under set-subtraction properties|Properties of a class of sets closed under set subtraction]] | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
| Line 35: | Line 40: | ||
:: This completes the proof. | :: This completes the proof. | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
| + | |||
==Important theorems== | ==Important theorems== | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
Revision as of 16:00, 28 August 2015
A Sigma-algebra of sets, or σ-algebra is very similar to a σ-ring of sets.
Like how ring of sets and algebra of sets differ, the same applies to σ-ring compared to σ-algebra
Contents
[hide]Definition
A non empty class of sets S is a σ-algebra[Note 1] if[1][2]
- if A∈S then Ac∈S
- if {An}∞n=1⊂S then ∪∞n=1An∈S
That is it is closed under complement and countable union.
Immediate consequences
Among other things immediately we see that:
[Expand]
- A is ∩-closed (furthermore, that A is in fact σ-∩-closed - that is closed under countable intersections)
Important theorems
[Expand]
The intersection of σ-algebras is a σ-algebra
Common σ-algebras
See also: Index of common σ-algebras
See also
Notes
- Jump up ↑ Some books (notably Measures, Integrals and Martingales) give X∈A as a defining property of σ-algebras, however the two listed are sufficient to show this (see the immediate consequences section)
- Jump up ↑ Measures, Integrals and Martingales puts this in the definition of σ-algebras
References
- Jump up ↑ Halmos - Measure Theory - page 28 - Springer - Graduate Texts in Mathematics - 18
- Jump up ↑ Measures, Integrals and Martingales - Rene L. Schilling
