InDina [[mathematicsmatematika]], a '''aljabar σ-algebra''' (oratawa '''widang σ-field''') ''X'' overpikeun a [[set]]sasét ''S'' ishartina a family ofanggota [[subset|subsetssubsét]] of ''S'' whichnu iskatutup closedku undersét [[countable]]operasi-operasi nu setbisa operationsdiitung; aljabar σ-algebras are mainly usedinutamana orderdipaké topikeun definenangtukeun [[measure (mathematics)|measuresukuran]] on ''S''. The conceptIeu iskonsép importantpenting indina [[mathematicalanalisis analysismatematika]] andjeung [[probabilitytéori theoryprobabilitas]].
FormallySacara formal, ''X'' iskaasup aaljabar σ-algebra mun jeung ukur mun (''jika dan hanya jika'', ''if and only if'') itmiboga haspasipatan thedi followinghandap propertiesieu:
# The [[empty set]] is in ''X'',
Baris ka-12:
An ordered pair (''S'', ''X''), where ''S'' is a set and ''X'' is a σ-algebra over ''S'', is called a '''measurable space'''.
== ExamplesConto ==
IfMun ''S'' ismangrupa anysét setnaon baé, then the family consisting only of the empty set and ''S'' is a σ-algebra over ''S'', the so-called ''trivial σ-algebra''. Another σ-algebra over ''S'' is given by the full [[power set]] of ''S''.▼
▲If ''S'' is any set, then the family consisting only of the empty set and ''S'' is a σ-algebra over ''S'', the so-called ''trivial σ-algebra''. Another σ-algebra over ''S'' is given by the full [[power set]] of ''S''.
If {''X''<sub>a</sub>} is a family of σ-algebras over ''S'', then the intersection of all ''X''<sub>a</sub> is also a σ-algebra over ''S''.
