Aljabar sigma: Béda antarrépisi

Dina [[matematika]], '''aljabar σσ''' (atawa '''widang σσ''') ''X'' pikeun sasét ''S'' hartina anggota [[subsét]] ''S'' nu katutup ku sét operasi-operasi nu bisa diitung; aljabar σσ utamana dipaké pikeun nangtukeun [[ukuran]] ''S''. Ieu konsép penting dina [[analisis matematika]] jeung [[téori probabilitas]].

Sacara formal, ''X'' kaasup aljabar σσ mun jeung ukur mun (''jika dan hanya jika'', ''if and only if'') miboga pasipatan di handap ieu:

From 1 and 2 it follows that ''S'' is in ''X''; from 2 and 3 it follows that the σσ-algebra is also closed under countable intersections (via [[De Morgan's laws]]).

An ordered pair (''S'', ''X''), where ''S'' is a set and ''X'' is a σσ-algebra over ''S'', is called a '''measurable space'''.

Mun ''S'' mangrupa sét naon baé, then the family consisting only of the empty set and ''S'' is a σσ-algebra over ''S'', the so-called ''trivial σσ-algebra''. Another σσ-algebra over ''S'' is given by the full [[power set]] of ''S''.

If {''X''_a} is a family of σσ-algebras over ''S'', then the intersection of all ''X''_a is also a σσ-algebra over ''S''.

If ''U'' is an arbitrary family of subsets of ''S'' then we can form a special σσ-algebra from ''U'', called the ''σσ-algebra generated by U''. We denote it by σσ(''U'') and define it as follows.

First note that there is a σσ-algebra over ''S'' that contains ''U'', namely the power set of ''S''.

Let ΦΦ be the family of all σσ-algebras over ''S'' that contain ''U'' (that is, a σσ-algebra ''X'' over ''S'' is in ΦΦ if and only if ''U'' is a subset of ''X''.)

Then we define σσ(''U'') to be the intersection of all σσ-algebras in ΦΦ. σσ(''U'') is then the smallest σσ-algebra over ''S'' that contains ''U''.

This leads to the most important example: the [[Borel algebra]] over any [[topological space]] is the σσ-algebra generated by the [[open set]]s (or, equivalently, by the [[closed set]]s).

Note that this σσ-algebra is not, in general, the whole power set.

On the [[Euclidean space]] '''R'''^''n'', another σσ-algebra is of importance: that of all [[Lebesgue measure|Lebesgue measurable]] sets. This σσ-algebra contains more sets than the Borel algebra on '''R'''^''n'' and is preferred in [[Integral|integration]] theory.

[[CategoryKategori:Aljabar]]

[[sk:Sigma -algebra]]