Sigma-algebra: Difference between revisions
Jump to navigation
Jump to search
(More beginning of edits toward making a full page.) |
No edit summary |
||
| Line 5: | Line 5: | ||
==Examples of <math>\sigma</math>-algebras== | ==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. | * Given a set <math>X</math>, then <math>2^X</math> and <math>\{\emptyset,X\}</math> are <math>\sigma</math>-algebras. | ||
==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. | |||
Revision as of 18:51, 8 October 2020
This page is under construction.
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.
Examples of -algebras
- Given a set , then and are -algebras.
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.
