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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}}
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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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| − | * $A,B\in \mathcal{A}\Rightarrow A\cap B\in\mathcal{A}$.
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| − | Indeed it is sufficient to assume that $\mathcal{A}$ satisfies the first two properties to conclude that also
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| − | the third holds.
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| − | The algebra generated by a family $\mathcal{B}$ of subsets of $X$ is defined as the smallest algebra $\mathcal{A}$ of subsets
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| − | of $X$ containing $\mathcal{B}$. A simple procedure to construct $\mathcal{A}$ is the following. Define $\mathcal{A}_0$
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| − | as the set of all elements of $\mathcal{B}$ and their complements. Define $\mathcal{A}_1$ as the elements which are intersections
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| − | of finitely many elements of $\mathcal{A}_0$. $\mathcal{A}$ consists then of finite unions of arbitrary elements of $\mathcal{A}_1$.
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| − | ====$\sigma$-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
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| − | under countable intersections. As above, given a collection $\mathcal{B}$ of subsets of $X$, the $\sigma$-algebra generated
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| − | by $\mathcal{B}$ is defined as the smallest $\sigma$-algebra of subsets of $X$ containing $\mathcal{B}$. A
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| − | construction can be given using
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| − | [[Transfinite number|transfinite numbers]]. As above, $\mathcal{A}_0$consists of all elements of $\mathcal{B}$ and their complements.
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| − | Given a countable ordinal $\alpha$, $\mathcal{A}_\alpha$ consists of those sets which are countable unions or countable intersections
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| − | of elements belonging to
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| − | \[
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| − | \bigcup_{\beta<\alpha} \mathcal{A}_\beta\, .
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| − | \]
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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]].
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| − | 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 $A$.
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| − | ====Examples.====
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| − | 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
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| − | of $X$ which are at most countable and of their complements is a $\sigma$-algebra (so-called countable-cocountable σ-algebra).
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| − |
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| − | 2) The collection of finite unions of intervals of the type
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| − | \[
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| − | \{x\in\mathbb R : a\leq x <b\} \qquad \mbox{where $-\infty \leq a <b\leq \infty$}
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| − | \]
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| − | is an algebra.
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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]].
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| − | 4) The Lebesgue measurable sets of $\mathbb R^k$ form a $\sigma$ algebra (so-called Lebesgue σ-algebra, see [[Lebesgue measure]]).
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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 $\mathbb R$).
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| − | Let $A$ be the class of sets of the type
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| − | \[
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| − | \{\omega\in \mathbb R^T: (\omega (t_1), \ldots,\omega t_k)\in E\}
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| − | \]
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| − | where $k$ is an arbitrary natural number, $E$ an arbitrary Borel subset of $\mathbb R^k$ and $t_1,\ldots, t_k$
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| − | an arbitrary collection of distinct elements of $T$. $A$ is an algebra of subsets of $\mathbb R^T$ (so-called cylindrical algebra).
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| − | In the theory of random processes a [[Probability measure|probability measure]]
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| − | is often originally defined only on an algebra of this type, and then subsequently extended to the $\sigma$-algebra generated by $A$.
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| − | ====References====
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| − | {|
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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}}
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| − | |valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory", '''1''', Interscience (1958) {{MR|0117523}}
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| − | |valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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| − | |valign="top"|{{Ref|Ne}}|| J. Neveu, "Bases mathématiques du calcul des probabilités", Masson (1970)
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| − | |}
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