Difference between revisions of "Sigma-algebra"
From Maths
(→Definition) |
m (Reverted edits by JessicaBelinda133 (talk) to last revision by Alec) |
||
| Line 7: | Line 7: | ||
* if <math>A\in S</math> then <math>A^c\in S</math> | * 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> | * if <math>\{A_n\}_{n=1}^\infty\subset S</math> then <math>\cup^\infty_{n=1}A_n\in S</math> | ||
| − | That is it | + | That is it is closed under [[Complement|complement]] and [[Countable|countable]] [[Union|union]].<br/> |
| + | ===Immediate consequences=== | ||
| + | Among other things immediately we see that: | ||
| + | {{Begin Inline Theorem}} | ||
| + | * {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]] | ||
| + | {{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 | ||
| + | {{End Proof}}{{End Theorem}} | ||
{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
* {{M|\emptyset\in\mathcal{A} }} | * {{M|\emptyset\in\mathcal{A} }} | ||
| Line 22: | Line 29: | ||
{{Begin Inline Proof}} | {{Begin Inline Proof}} | ||
:: To prove this we must check: | :: To prove this we must check: | ||
| − | ::# {{M|\ | + | ::# {{M|\mathcal{A} }} is closed under countable union |
| + | ::#* True by definition of {{Sigma|algebra}} | ||
| + | ::# {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]] | ||
::#* We've already shown this, so this is true too. | ::#* We've already shown this, so this is true too. | ||
:: 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 16:26, 23 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:
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
