Sigma-algebra: Difference between revisions
Jump to navigation
Jump to search
(Major content update to the page, reflecting new material learned and polishing of old.) |
(Syntax fixes) |
||
| Line 1: | Line 1: | ||
A '''<math>\sigma</math>-algebra''' is an [[algebra]] that is closed under countable unions. Thus a <math>\sigma</math>-algebra is a nonempty collection ''A'' of subsets of a nonempty set ''X'' closed under countable unions and complements. | A '''<math>\sigma</math>-algebra''' is an [[algebra]] that is closed under countable unions. Thus a <math>\sigma</math>-algebra is a nonempty collection ''A'' of subsets of a nonempty set ''X'' closed under countable unions and complements. | ||
<ref name=" | <ref name="Folland1">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.2 </ref> | ||
==<math>\sigma</math>-algebra Generation== | ==<math>\sigma</math>-algebra Generation== | ||
| Line 6: | Line 6: | ||
The intersection of any number of <math>\sigma</math>-algebras on a set <math>X</math> is a <math>\sigma</math>-algebra. The <math>\sigma</math>-algebra generated by a collection of subsets of <math>X</math> is the smallest <math>\sigma</math>-algebra containing <math>X</math>, which is unique by the above. | The intersection of any number of <math>\sigma</math>-algebras on a set <math>X</math> is a <math>\sigma</math>-algebra. The <math>\sigma</math>-algebra generated by a collection of subsets of <math>X</math> is the smallest <math>\sigma</math>-algebra containing <math>X</math>, which is unique by the above. | ||
The <math>\sigma</math>-algebra generated by <math>E \subseteq 2^X</math> is denoted as <math>M(E)< | The <math>\sigma</math>-algebra generated by <math>E \subseteq 2^X</math> is denoted as <math>M(E)</math>. | ||
If <math>E</math> and <math>F</math> are subsets of <math>2^X</math> and <math>E \subseteq M(F)</math> then <math>M(E) \subseteq M(F)</math>. This result is commonly used to simplify proofs of containment in <math>\sigma</math>-algebras. | If <math>E</math> and <math>F</math> are subsets of <math>2^X</math> and <math>E \subseteq M(F)</math> then <math>M(E) \subseteq M(F)</math>. This result is commonly used to simplify proofs of containment in <math>\sigma</math>-algebras. | ||
An important common example is the Borel <math>\sigma</math>-algebra on <math>X</math>, the <math>\sigma</math>-algebra generated by the open sets of <math>X</math>. | An important common example is the Borel <math>\sigma</math>-algebra on <math>X</math>, the <math>\sigma</math>-algebra generated by the open sets of <math>X</math>. | ||
| Line 15: | Line 14: | ||
==Product <math>\sigma</math>algebras== | ==Product <math>\sigma</math>-algebras== | ||
If <math>A</math> is a countable set, then <math>\otimes_{\alpha \in A} M_{\alpha}</math> is the <math>\sigma</math>algebra generated by <math>\{\ | If <math>A</math> is a countable set, then <math>\otimes_{\alpha \in A} M_{\alpha}</math> is the <math>\sigma</math>-algebra generated by <math>\{\Pi_{\alpha \in A} E_{\alpha} : E_{\alpha} \in M_{\alpha}\}</math>. | ||
<ref name=" | <ref name="Folland1">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.2 </ref> This is called the product <math>\sigma</math>-algebra. | ||
| Line 27: | Line 26: | ||
*If <math>X</math> is uncountable, the set of countable and co-countable subsets of <math>X</math> is a <math>\sigma</math>-algebra. | *If <math>X</math> is uncountable, the set of countable and co-countable subsets of <math>X</math> is a <math>\sigma</math>-algebra. | ||
*By Carathéodory's Theorem, if <math>\mu^*</math> is an outer measure on <math>X</math>, the collection of <math>\mu^*</math>-measurable sets is a <math>\sigma</math>-algebra. <ref name=" | *By Carathéodory's Theorem, if <math>\mu^*</math> is an outer measure on <math>X</math>, the collection of <math>\mu^*</math>-measurable sets is a <math>\sigma</math>-algebra. <ref name="Folland3">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.4 </ref> | ||
==Non-examples== | ==Non-examples== | ||
* The algebra of finite and cofinite subsets of a nonempty set <math>X</math> may no longer be a <math>\sigma</math>-algebra. Let <math>X = \mathbb{Z}</math>, then every set of the form <math>\{ 2n \}</math> for <math>n \in \mathbb{Z}</math> is finite, but their countable union <math>\bigcup\limits_{n \in \mathbb{Z}} \{ 2n \} = 2\mathbb{Z}</math> is neither finite nor cofinite. | * The algebra of finite and cofinite subsets of a nonempty set <math>X</math> may no longer be a <math>\sigma</math>-algebra. Let <math>X = \mathbb{Z}</math>, then every set of the form <math>\{ 2n \}</math> for <math>n \in \mathbb{Z}</math> is finite, but their countable union <math>\bigcup\limits_{n \in \mathbb{Z}} \{ 2n \} = 2\mathbb{Z}</math> is neither finite nor cofinite. | ||
==References== | |||
Revision as of 22:53, 13 November 2020
A -algebra is an algebra that is closed under countable unions. Thus a -algebra is a nonempty collection A of subsets of a nonempty set X closed under countable unions and complements. [1]
-algebra Generation
The intersection of any number of -algebras on a set is a -algebra. The -algebra generated by a collection of subsets of is the smallest -algebra containing , which is unique by the above.
The -algebra generated by is denoted as .
If and are subsets of and then . This result is commonly used to simplify proofs of containment in -algebras.
An important common example is the Borel -algebra on , the -algebra generated by the open sets of .
Product -algebras
If is a countable set, then is the -algebra generated by . [1] This is called the product -algebra.
Other Examples of -algebras
- Given a set , then and are -algebras, called the indiscrete and discrete -algebras respectively.
- If is uncountable, the set of countable and co-countable subsets of is a -algebra.
- By Carathéodory's Theorem, if is an outer measure on , the collection of -measurable sets is a -algebra. [2]
Non-examples
- The algebra of finite and cofinite subsets of a nonempty set may no longer be a -algebra. Let , then every set of the form for is finite, but their countable union is neither finite nor cofinite.
