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{{MSC|03E15|28A33}}
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[[Category:Descriptive set theory]]
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[[Category:Classical measure theory]]
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{{TEX|done}}
  
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====Algebra of sets====
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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
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* $A\in\mathcal{A}\Rightarrow X\setminus A\in \mathcal{A}$;
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* $A,B\in \mathcal{A}\Rightarrow A\cup B\in\mathcal{A}$;
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* $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.
 +
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 procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$
 +
as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections
 +
of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.
 +
 +
====$\sigma$-Algebra====
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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 $\mathcal{B}$. A
 +
construction can be given using
 +
[[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
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\[
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\bigcup_{\beta<\alpha} \mathcal{A}_\beta\, .
 +
\]
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$\mathcal{A}$ is the union of the classes $\mathcal{A}_\alpha$ where the index $\alpha$ runs over all countable ordinals.
 +
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====Relations to measure theory====
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Algebras (respectively $\sigma$-algebras) are the  natural domain of definition of finitely-additive ($\sigma$-additive)  measures.
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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 (so-called finite-cofinite algebra). The collection of subsets
 +
of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).
 +
 +
2) The collection of finite unions of intervals of the type
 +
\[
 +
\{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where $-\infty \leq a <b\leq \infty$}
 +
\]
 +
is an algebra.
 +
 +
3) If $X$ is a topological space, the elements of the $\sigma$-algebra generated by the open sets are called [[Borel set|Borel sets]].
 +
 +
4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (so-called Lebesgue σ-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 $\mathbb R^T$ (so-called cylindrical algebra).
 +
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====
 +
{|
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|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 19:14, 31 July 2012

2020 Mathematics Subject Classification: Primary: 03E15 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 procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$ as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.

$\sigma$-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 $\mathcal{B}$. A construction can be given using 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 (so-called finite-cofinite algebra). The collection of subsets of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).

2) The collection of finite unions of intervals of the type \[ \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where '"`UNIQ-MathJax47-QINU`"'} \] 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 (so-called Lebesgue σ-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 $\mathbb R^T$ (so-called cylindrical algebra). 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=27303
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article