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A non-empty collection of subsets of some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114001.png" /> that is closed under the set-theoretic operations (of union, intersection, taking complements), carried out a finite number of times. In order for a class of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114002.png" /> to be an algebra of sets, it is necessary (and sufficient) for it to be closed under finite unions and taking the complement. An algebra of sets that is closed under countable unions is known as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114004.png" />-algebra of sets. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114005.png" />-algebra of sets is closed under the set-theoretic operations carried out a countable number of times.
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{{MSC|03A15|28A33}}
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[[Category:Descriptive set theory]]
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[[Category:Classical measure theory]]
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{{TEX|done}}
  
===Examples.===
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====Algebra of sets====
 +
A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of union, intersection and taking complements, i.e. such that
 +
* $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
 +
* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
 +
* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.
 +
Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also
 +
the third holds.
  
 +
The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets
 +
of $X$ containing $\mathcal{B}$. A simple inductive procedure allows to "construct" $\mathcal{A}$ as follows. $\mathcal{A}_0$
 +
consists of all elements of $\mathcal{B}$ and their complements. For any
 +
$n\in\mathbb N\setminus \{0\}$ we define $\mathcal{A}_n$ as the collection of those sets which are finite unions or finite
 +
intersections of elements of $\mathcal{A}_{n-1}$. Then $\mathcal{A}=\bigcup_{n\in\mathbb N} \mathcal{A}_n$.
  
1) The collection of finite subsets of an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114006.png" /> and their complements is an algebra of sets; the collection consisting of the at most countable subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114007.png" /> and their complements is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114008.png" />-algebra of sets.
+
====$\sigma$-Algebra====
 +
An algebra of sets that is also closed under countable unions. As a corollary a $\sigma$-algebra is also closed
 +
under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated
 +
by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$. The explicit
 +
construction given above for the algebra generated by $\mathcal{B}$ can be extended to $\sigma$-algebras with the aid of
 +
[[Transfinite number|transfinite numbers]]. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements.
 +
Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections
 +
of elements belonging to
 +
\[
 +
\bigcup_{\beta<\alpha} \mathcal{A}_\beta\, .
 +
\]
 +
$\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals.  
  
2) The collection of finite unions of intervals of the type
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====Relations to measure theory====
 +
Algebras (respectively $\sigma$-algebras) are the  natural domain of definition of finitely-additive ($\sigma$-additive)  measures.
 +
Therefore $\sigma$-algebras play a central role in measure theory, see for instance [[Measure space]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a0114009.png" /></td> </tr></table>
+
According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure,  defined on an algebra A, can be uniquely  extended to a $\sigma$-additive measure  defined on the $\sigma$-algebra generated  by $A$.
  
forms an algebra of sets.
+
====Examples.====
 +
1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets. The collection of subsets
 +
of $X$ which are at most countable and of their complements is a $\sigma$-algebra.  
  
3) let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140010.png" /> be a topological space; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140011.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140012.png" /> of sets generated by the open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140013.png" /> (in other words, the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140014.png" />-algebra of sets containing all open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140015.png" />) is known as the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140017.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140018.png" />, while the sets belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140019.png" /> are known as Borel sets.
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2) The collection of finite unions of intervals of the type
 
+
\[
4) let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140021.png" /> is an arbitrary set (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140022.png" /> is the set of all real functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140023.png" />); the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140024.png" /> of sets of the type
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\{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where $-\infty \leq a <b\leq \infty$}
 
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\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140025.png" /></td> </tr></table>
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is an algebra
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140026.png" /> is a Borel subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140027.png" />, is an algebra of sets; in the theory of random processes a [[Probability measure|probability measure]] is often originally defined only on an algebra of this type, and is their subsequently extended to a wider class of sets (to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140028.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140029.png" />).
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3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called [[Borel set|Borel sets]].
  
5) The collection of Lebesgue-measurable subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140030.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140031.png" />-algebra of sets.
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4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (see [[Lebesgue measure]]).
  
Algebras (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140032.png" />-algebras) are the natural domain of definition of finitely-additive (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140033.png" />-additive) measures. According to the theorem of extension of measures, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140034.png" />-finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140035.png" />-additive measure, defined on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140036.png" />, can be uniquely extended to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140037.png" />-additive measure defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140038.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011400/a01140039.png" />.
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5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$).
 +
Let $A$ be the class of sets of the type
 +
\[
 +
\{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega t_k)\in E\}
 +
\]
 +
where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$
 +
an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$.
 +
In the theory of random processes a [[Probability measure|probability measure]]
 +
is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Neveu,  "Bases mathématiques du calcul des probabilités" , Masson  (1970)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|Bo}}||      N. Bourbaki, "Elements of mathematics. Integration" ,  Addison-Wesley    (1975) pp. Chapt.6;7;8 (Translated from French)  {{MR|0583191}}    {{ZBL|1116.28002}} {{ZBL|1106.46005}}  {{ZBL|1106.46006}}    {{ZBL|1182.28002}} {{ZBL|1182.28001}}  {{ZBL|1095.28002}}    {{ZBL|1095.28001}} {{ZBL|0156.06001}}
 +
|-
 +
|valign="top"|{{Ref|DS}}||    N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,   '''1''' , Interscience (1958) {{MR|0117523}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory" , v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|Ne}}|| J. Neveu,  "Bases mathématiques du calcul des probabilités" , Masson  (1970)
 +
|-
 +
|}

Revision as of 14:51, 31 July 2012

2020 Mathematics Subject Classification: Primary: 03A15 Secondary: 28A33 [MSN][ZBL]

Algebra of sets

A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed under the set-theoretic operations of union, intersection and taking complements, i.e. such that

  • $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
  • $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.

Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also the third holds.

The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets of $X$ containing $\mathcal{B}$. A simple inductive procedure allows to "construct" $\mathcal{A}$ as follows. $\mathcal{A}_0$ consists of all elements of $\mathcal{B}$ and their complements. For any $n\in\mathbb N\setminus \{0\}$ we define $\mathcal{A}_n$ as the collection of those sets which are finite unions or finite intersections of elements of $\mathcal{A}_{n-1}$. Then $\mathcal{A}=\bigcup_{n\in\mathbb N} \mathcal{A}_n$.

$\sigma$-Algebra

An algebra of sets that is also closed under countable unions. As a corollary a $\sigma$-algebra is also closed under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$. The explicit construction given above for the algebra generated by $\mathcal{B}$ can be extended to $\sigma$-algebras with the aid of transfinite numbers. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements. Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections of elements belonging to \[ \bigcup_{\beta<\alpha} \mathcal{A}_\beta\, . \] $\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals.

Relations to measure theory

Algebras (respectively $\sigma$-algebras) are the natural domain of definition of finitely-additive ($\sigma$-additive) measures. Therefore $\sigma$-algebras play a central role in measure theory, see for instance Measure space.

According to the theorem of extension of measures, any $\sigma$-finite, $\sigma$-additive measure, defined on an algebra A, can be uniquely extended to a $\sigma$-additive measure defined on the $\sigma$-algebra generated by $A$.

Examples.

1) Let $X$ be an arbitrary set. The collection of finite subsets of $X$ and their complements is an algebra of sets. The collection of subsets of $X$ which are at most countable and of their complements is a $\sigma$-algebra.

2) The collection of finite unions of intervals of the type \[ \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where '"`UNIQ-MathJax49-QINU`"'} \] is an algebra

3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called Borel sets.

4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (see Lebesgue measure).

5) Let $T$ be an arbitrary set and consider $X = \mathbb R^T$ (i.e. the set of all real-valued functions on $\mathbb R$). Let $A$ be the class of sets of the type \[ \{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega t_k)\in E\} \] where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$ an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$. In the theory of random processes a probability measure is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.

References

[Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Ha] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ne] J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)
How to Cite This Entry:
Algebra of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebra_of_sets&oldid=27292
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article