Sigma-algebra: Difference between revisions
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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="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.2 </ref> | |||
==<math>\sigma</math>-algebra Generation== | |||
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)<\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. | |||
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>. | |||
==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>\{\Uppi_{\aplha \in A} E_{\alpha} : E_{\alpha} \in M_{\alpha}\}</math>. | |||
<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, Second Edition'', §1.2 </ref> This is called the product <math>\sigma</math>-algebra. | |||
==Other Examples of <math>\sigma</math>-algebras== | |||
* Given a set <math>X</math>, then <math>2^X</math> and <math>\{\emptyset,X\}</math> are <math>\sigma</math>-algebras, called the indiscrete and discrete <math>\sigma</math>-algebras respectively. | |||
*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="Folland">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. | ||
Revision as of 22:44, 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 Failed to parse (unknown function "\math"): {\displaystyle M(E)<\math>. If <math>E} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\Uppi_{\aplha \in A} E_{\alpha} : E_{\alpha} \in M_{\alpha}\}} . [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. [1]
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.
