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| In [[mathematics]], a nonempty collection of [[Set (mathematics)|sets]] <math>\mathcal{R}</math> is called a '''σ-ring''' (pronounced ''sigma-ring'') if it is [[closure (mathematics)|closed]] under countable [[union (set theory)|union]] and [[Complement (set theory)|relative complementation]]:
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| #<math>\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{R}</math> if <math>A_{n} \in \mathcal{R}</math> for all <math>n \in \mathbb{N}</math>
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| #<math>A \smallsetminus B \in \mathcal{R}</math> if <math>A, B \in \mathcal{R}</math>
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| From these two properties we immediately see that
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| :<math>\bigcap_{n=1}^{\infty} A_n \in \mathcal{R}</math> if <math>A_{n} \in \mathcal{R}</math> for all <math>n \in \mathbb{N}</math>
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| This is simply because <math>\cap_{n=1}^\infty A_n = A_1 \smallsetminus \cup_{n=1}^{\infty}(A_1 \smallsetminus A_n)</math>.
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| If the first property is weakened to closure under finite union (i.e., <math>A \cup B \in \mathcal{R}</math> whenever <math>A, B \in \mathcal{R}</math>) but not countable union, then <math>\mathcal{R}</math> is a [[Ring of sets|ring]] but not a σ-ring.
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| σ-rings can be used instead of [[Sigma-algebra|σ-fields]] in the development of [[Measure (mathematics)|measure]] and [[Integral|integration]] theory, if one does not wish to require that the [[Universe (mathematics)|universal set]] be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be a σ-field.
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| A σ-ring induces a [[Sigma-algebra|σ-field]]. If <math>\mathcal{R}</math> is a σ-ring over the set <math>X</math>, then define <math>\mathcal{A}</math> to be the collection of all subsets of X that are elements of <math>\mathcal{R}</math> or whose complements are elements of <math>\mathcal{R}</math>. We see that <math>\mathcal{A}</math> is a σ-field over the set
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| X.
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| == See also == | |
| *[[Delta ring]]
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| *[[Ring of sets]]
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| *[[Sigma-algebra|Sigma field]]
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| == References ==
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| * [[Walter Rudin]], 1976. ''Principles of Mathematical Analysis'', 3rd. ed. McGraw-Hill. Final chapter uses σ-rings in development of Lebesgue theory.
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| {{Mathanalysis-stub}}
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| [[Category:Measure theory]]
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| [[Category:Set families]]
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Hello and welcome. My name is Numbers Wunder. Bookkeeping is my profession. It's not a typical factor but what she likes doing is base jumping and now she is attempting to earn cash with it. California is our birth location.
Check out my page http://fankut.com/index.php?do=/profile-2572/info