Difference between revisions of "Sigma-algebra"
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* 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 esd Theorem}} |
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{{Begin Inline Theorem}} | {{Begin Inline Theorem}} | ||
* {{M|\emptyset\in\mathcal{A} }} | * {{M|\emptyset\in\mathcal{A} }} | ||
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{{Begin Inline Proof}} | {{Begin Inline Proof}} | ||
:: To prove this we must check: | :: To prove this we must check: | ||
| − | ::# {{M|\ | + | ::# {{M|\mathcalyes} is closed under [[Set subtraction|set subtraction]] |
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::#* 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}} | ||
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==Important theorems== | ==Important theorems== | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
Revision as of 07:46, 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∈Sthen Ac∈S
- if {An}∞n=1⊂Sthen ∪∞n=1An∈S
That is it esd Theorem}}
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
