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| In [[mathematical analysis]], a '''σ-algebra''' (also '''sigma-algebra''', '''σ-field''', '''sigma-field''') on a set is a collection of [[Subset|subsets]] satisfying certain properties. The main use of σ-algebras is in the definition of [[measure (mathematics)|measures]]; specifically, the collection of those subsets for which a given measure is defined is a σ-algebra. This concept is important in [[mathematical analysis]] as the foundation for [[Lebesgue integration]], and in [[probability theory]], where it is interpreted as the collection of events which can be assigned probabilities.
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| The definition is that a σ-algebra over a set ''X'' is a nonempty collection Σ of [[subset]]s of ''X'' that is [[Closure (mathematics)|closed]] under the [[complement (set theory)|complement]] and [[countable]] [[union (set theory)|union]]s of its members and contains ''X'' itself.
| | シューという、ストレート吸引息が、それにおびえ [http://www.dmwai.com/webalizer/kate-spade-4.html ケイトスペード バッグ 新作]。<br><br>「私は次のステップは、あなたがフーGuoshengを入れて、右、彼の側に連れて行ってくれました、彼と私が満たしていることを偶然を製造すべきだと思う？ [http://www.dmwai.com/webalizer/kate-spade-3.html 財布 kate spade] '私は、道路の罪徐Pingqiu大きな驚きを見て、彼は微笑んで誇らしげにガラガラと述べ：古い福が私に対処するために持っているが、状況にあなたがより寛大与えるよりも、美しさ、ああと車に割り当てられたユニットを、私は私が最も私たちことを彼に告げ、拒否した時に刑務所から釈放された [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ アウトレット] 'それは、簡単だったさて、私たちLaotouはあなたと私の前に立って、私は助けることができると思いながらも、率直に言って？右、あなたは非常に失望しなければならないことを、お約束します。に実行していない、あなたはいけない。 [http://www.dmwai.com/webalizer/kate-spade-2.html ケイトスペード トートバッグ] '<br>より多くのラウンドを成長<br>徐Pingqiu目、より多くの私は、口をむき出し笑って涙が速く流出し笑った大きな罪を怖がって、彼はル·フーGuoshengから、長い間、この瞬間を待っていた彼が待っているとき、これまで副は徐Pingqiu落ち込んで孤独表情を見た [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード バッグ 激安]。<br><br>一度選択し、学校でdumbfounding先生を見ているように、彼は選択する必要はありませんが、呪いを破るしたとして罪の上に誇らしげに、クロツラヘラサギの上司を見て、長い間笑った |
| | | 相关的主题文章： |
| That is, a σ-algebra is an [[algebra of sets]], [[Completeness (order theory)|completed]] to include [[countably infinite]] operations. The pair (''X'', Σ) is also a [[field of sets]], called a '''measurable space'''.
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| If {{nowrap|1=X = {''a'', ''b'', ''c'', ''d''}}}, one possible sigma algebra on ''X'' is {{nowrap|1=Σ = {∅, {''a'', ''b''}, {''c'', ''d''}, {''a'', ''b'', ''c'', ''d''}}}}, where ∅ is the empty set. A more useful example is the set of subsets of the [[real line]] formed by starting with all [[open interval]]s and adding in all countable unions, countable intersections, and relative complements and continuing this process until the relevant closure properties are achieved (a construction known as the [[Borel set|Borel σ-algebra]]).
| | <li>[http://www.cuiter.net/home.php?mod=space&uid=47073 http://www.cuiter.net/home.php?mod=space&uid=47073]</li> |
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| ==Motivation==
| | <li>[http://j-links.sakura.ne.jp/hitozuma-ch/bbs005/j-links.cgi http://j-links.sakura.ne.jp/hitozuma-ch/bbs005/j-links.cgi]</li> |
| A [[Measure (mathematics)|measure]] on ''X'' is a [[function (mathematics)|function]] which assigns a non-negative [[real number]] to subsets of ''X''; this can be thought of as making precise a notion of "size" or "volume" for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets.
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| | | <li>[http://www.carmtcpj.com/sns/space.php?uid=2522&do=blog&id=7943 http://www.carmtcpj.com/sns/space.php?uid=2522&do=blog&id=7943]</li> |
| One would like to assign a size to ''every'' subset of ''X'', but in many natural settings, this is not possible. For example the [[axiom of choice]] implies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the [[Vitali set]]s. For this reason, one considers instead a smaller collection of privileged subsets of ''X''. These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.
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| | | </ul> |
| The collection of subsets of ''X'' that form the σ-algebra is usually denoted by Σ, the capital Greek letter [[sigma]]. The pair (''X'', Σ) is an [[algebra of sets]] and also a [[field of sets]], called a measurable space. If the subsets of ''X'' in Σ correspond to numbers in [[elementary algebra]], then the two set operations [[union (set theory)|union (symbol ∪)]] and [[intersection (set theory)|intersection (∩)]] correspond to addition and multiplication. The collection of sets Σ is [[Completeness (order theory)|completed]] to include [[countably infinite]] operations.
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| ==Definition and properties==
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| Let ''X'' be some set, and let 2<sup>''X''</sup> represent its [[power set]]. Then a subset {{nowrap|Σ ⊂ 2<sup>''X''</sup>}} is called a '''''σ''-algebra''' if it satisfies the following three properties:<ref>{{cite book | last=Rudin | first=Walter | authorlink=Walter Rudin | title=Real & Complex Analysis | publisher=[[McGraw-Hill]] | year=1987 | isbn=0-07-054234-1}}</ref>
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| # Σ is ''non-empty'': There is at least one ''A ''⊂'' X'' in Σ.
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| # Σ is ''closed under complementation'': If ''A'' is in Σ, then so is its [[complement (set theory)|complement]], {{nowrap|''X'' \ ''A''}}.
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| # Σ is ''closed under countable unions'': If ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, ... are in Σ, then so is ''A'' = ''A''<sub>1</sub> ∪ ''A''<sub>2</sub> ∪ ''A''<sub>3</sub> ∪ … .
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| From these axioms, it follows that the σ-algebra is also closed under countable [[intersection (set theory)|intersections]] (by applying [[De Morgan's laws]]).
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| It also follows that the set ''X'' itself and the [[empty set]] are both in Σ, since by '''(1)''' Σ is non-empty, so some particular ''A ∈ Σ'' may be chosen, and by '''(2)''', ''X'' \ ''A'' is also in Σ. By '''(3)''' ''A'' ∪ (''X'' \ ''A'') = ''X'' is in Σ. And finally, since ''X'' is in Σ, '''(2)''' asserts that its complement, the empty set, is also in Σ.
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| Elements of the ''σ''-algebra are called [[measurable set]]s. An ordered pair {{nowrap|(''X'', Σ)}}, where ''X'' is a set and Σ is a ''σ''-algebra over ''X'', is called a '''measurable space'''. A function between two measurable spaces is called a [[measurable function]] if the [[preimage]] of every measurable set is measurable. The collection of measurable spaces forms a [[category (mathematics)|category]], with the [[measurable function]]s as [[morphism]]s. [[Measure (mathematics)|Measures]] are defined as certain types of functions from a ''σ''-algebra to [0, ∞].
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| === Relation to σ-ring ===
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| A ''σ''-algebra Σ is just a [[Sigma-ring|''σ''-ring]] that contains the universal set ''X''.<ref name=Vestrup2009>{{cite book|last=Vestrup|first=Eric M.|title=The Theory of Measures and Integration|year=2009|publisher=John Wiley & Sons|isbn=9780470317952|page=12}}</ref> A ''σ''-ring need not be a ''σ''-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a ''σ''-ring, but not a ''σ''-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a [[Ring of sets|ring]] but not a ''σ''-ring, since the real line can be obtained by their countable union yet its measure is not finite.
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| === Typographic note ===
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| ''σ''-algebras are sometimes denoted using [[calligraphic]] capital letters, or the [[Fraktur (typeface)|Fraktur typeface]]. Thus {{nowrap|(''X'', Σ)}} may be denoted as <math>\scriptstyle(X,\,\mathcal{F})</math> or <math>\scriptstyle(X,\,\mathfrak{F})</math>. This is handy to avoid situations where the letter Σ may be confused for the [[summation]] operator.
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| === σ-algebra generated by a family of sets ===
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| Let ''F'' be an arbitrary family of subsets of ''X''. Then there exists a unique smallest ''σ''-algebra which contains every set in ''F'' (even though ''F'' may or may not itself be a ''σ''-algebra). This ''σ''-algebra is denoted ''σ''(''F'') and called '''the ''σ''-algebra generated by ''F'''.
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| To see that such a ''σ''-algebra always exists, let
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| :Φ ={ ''E'' ⊆ 2<sup>''X''</sup> : ''E'' is a σ-algebra which contains ''F'' }. | |
| The ''σ''-algebra generated by ''F'' will therefore be the smallest element in Φ. Indeed, such a smallest element exists: First, Φ is not empty because the power set 2<sup>''X''</sup> is in Φ. Consequently, let ''σ*'' denote the intersection of all elements in Φ. This intersection will be nonempty, moreover, it will also be a ''σ'' algebra, because each element in Φ is a ''σ''-algebra and the arbitrary intersection of σ-algebras is a σ-algebra (observe that if every element in Φ has the three properties of a σ-algebra, then the intersection of Φ will as well). Because each element in Φ contains ''F'', the intersection ''σ*'' will also contain ''F''. Hence, because ''σ*'' is a ''σ''-algebra which contains ''F'', ''σ*'' is in Φ, and because it is the intersection of all sets in Φ, ''σ*'' is indeed the ''smallest'' set in Φ by definition, which in turn implies that {{nowrap|1=''σ*'' = ''σ''(''F'')}}, the ''σ''-algebra generated by ''F''.
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| For a simple example, consider the set ''X'' = {1, 2, 3}. Then the ''σ''-algebra generated by the single subset {1} is ''σ''(<nowiki>{{1}}</nowiki>) = {∅, {1}, {2,3}, {1,2,3}}. By an [[abuse of notation]], when a collection of subsets contains only one element, ''A'', one may write ''σ''(''A'') instead of ''σ''({''A''}); in the prior example ''σ''({1}) instead of ''σ''(<nowiki>{{1}}</nowiki>). Also when that subset contains only one element, ''a'', one may write ''σ''(''a'') instead of ''σ''(''A'')=''σ''({''a''}); in the prior example ''σ''(1) instead of ''σ''({1}).
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| === σ-algebra generated by a function===
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| If ''f'' is a function from a set ''X'' to a set ''Y'' and ''B'' is a σ-algebra of subsets of ''Y'', then the '''σ-algebra generated by the function ''f''''', denoted by ''σ(f)'', is the collection of all inverse images ''f<sup>−1</sup>(S)'' of the sets ''S'' in ''B''. i.e.
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| :<math> \sigma (f) = \{ f^{-1}(S) \, | \, S\in B \}. </math>
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| Clearly, a function ''f'' from a set ''X'' to a set ''Y'' is [[Measurable function|measurable]] with respect to a σ-algebra ''Σ'' of subsets of ''X'' if and only if ''σ(f)'' is a subset of ''Σ''.
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| One common situation, and understood by default if ''B'' is not specified explicitly, is when ''Y'' is a [[metric space|metric]] or [[topological space]] and ''B'' are the [[Borel set]]s on ''Y''.
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| ==Examples==
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| Let ''X'' be any set, then the following are σ-algebras over ''X'':
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| * The family consisting only of the empty set and the set ''X'', called the minimal or '''trivial ''σ''-algebra''' over ''X''.
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| * The [[power set]] of ''X'', called the '''discrete ''σ''-algebra'''.
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| * The collection of subsets of ''X'' which are countable or whose complements are countable (which is distinct from the power set of ''X'' if and only if ''X'' is uncountable). This is the ''σ''-algebra generated by the [[Singleton (mathematics)|singletons]] of ''X''.
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| * If {Σ<sub>''λ''</sub>} is a family of ''σ''-algebras over ''X'' indexed by ''λ'' then the intersection of all Σ<sub>''λ''</sub>'s is a ''σ''-algebra over ''X''.
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| ===Examples for generated algebras===
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| An important example is the [[Borel algebra]] over any [[topological space]]: the ''σ''-algebra generated by the [[open set]]s (or, equivalently, by the [[closed set]]s). Note that this ''σ''-algebra is not, in general, the whole power set. For a non-trivial example, see the [[Vitali set]].
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| On the [[Euclidean space]] '''R'''<sup>''n''</sup>, another ''σ''-algebra is of importance: that of all [[Lebesgue measure|Lebesgue measurable]] sets. This ''σ''-algebra contains more sets than the Borel ''σ''-algebra on '''R'''<sup>''n''</sup> and is preferred in [[Integral|integration]] theory, as it gives a [[complete measure|complete measure space]].
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| ==See also== | |
| *[[Join (sigma algebra)]]
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| *[[Measurable function]]
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| *[[Sample space]]
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| *[[Separable sigma algebra]]
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| *[[Sigma ring]]
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| *[[Sigma additivity]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{springer|title=Algebra of sets|id=p/a011400}}
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| *[[PlanetMath:950|Sigma Algebra]] from PlanetMath.
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| <!-- http://planetmath.org/encyclopedia/SigmaAlgebra.html -->
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| [[Category:Measure theory]]
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| [[Category:Probability theory]]
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| [[Category:Set families]]
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| [[Category:Boolean algebra]]
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| {{Link GA|fr}}
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