Difference between revisions of "Caratheodory extension theorem"

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In the branch of [[mathematics]] known as [[measure theory]], the '''Caratheodory extension theorem''' states that a countably additive non-negative set function on an algebra of subsets of a set can be extended to be a [[measure]] on the [[sigma algebra]] generated by that algebra. Measure in this context specifically refers to a non-negative measure.  
 
In the branch of [[mathematics]] known as [[measure theory]], the '''Caratheodory extension theorem''' states that a countably additive non-negative set function on an algebra of subsets of a set can be extended to be a [[measure]] on the [[sigma algebra]] generated by that algebra. Measure in this context specifically refers to a non-negative measure.  
  
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<blockquote>(Caratheodory extension theorem) Let ''X'' be a set and <math>\mathcal{F}_0</math> be an algebra of subsets of X. Let <math>\mu_0</math> be a countably additive non-negative set function on <math>\mathcal{F}_0</math>. Then there exists a measure <math>\mu</math> on the <math>\sigma</math>-algebra <math>\mathcal{F}=\sigma(\mathcal{F}_0)</math> (i.e., the smallest sigma algebra containing <math>\mathcal{F}_0</math>) such that <math>\mu(A)=\mu_0(A)</math> for all <math>A \in \mathcal{F}_0</math>. Furthermore, if <math>\mu(X)=\mu_0(X)<\infty</math> then the extension is unique.</blockquote>
 
<blockquote>(Caratheodory extension theorem) Let ''X'' be a set and <math>\mathcal{F}_0</math> be an algebra of subsets of X. Let <math>\mu_0</math> be a countably additive non-negative set function on <math>\mathcal{F}_0</math>. Then there exists a measure <math>\mu</math> on the <math>\sigma</math>-algebra <math>\mathcal{F}=\sigma(\mathcal{F}_0)</math> (i.e., the smallest sigma algebra containing <math>\mathcal{F}_0</math>) such that <math>\mu(A)=\mu_0(A)</math> for all <math>A \in \mathcal{F}_0</math>. Furthermore, if <math>\mu(X)=\mu_0(X)<\infty</math> then the extension is unique.</blockquote>
  
<math>\mathcal{F}</math> is also referred to as the ''sigma algebra generated by'' <math>\mathcal{F}_0</math>. The term "algebra of subsets" in the theorem refers to a collection of subsets of a set ''X'' which contains ''X'' itself and is closed under the operation of taking complements, finite unions and finite intersections in ''X''. That is, any algebra <math>\mathcal{A}</math> of subsets of ''X'' satisfies the following requirements:
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<math>\mathcal{F}</math> is also referred to as the ''sigma algebra generated by'' <math>\mathcal{F}_0</math>. The term "algebra of subsets" in the theorem refers to a collection of subsets of a set ''X'' which contains ''X'' itself and is closed under the operation of taking [[complement (set theory)|complement]]s, finite [[union]]s and finite [[intersection]]s in ''X''. That is, any algebra <math>\mathcal{A}</math> of subsets of ''X'' satisfies the following requirements:
 
#<math>X \in \mathcal{A}</math>
 
#<math>X \in \mathcal{A}</math>
 
#If <math>A \in \mathcal{A}</math> then <math>X-A \in \mathcal{A} </math>  
 
#If <math>A \in \mathcal{A}</math> then <math>X-A \in \mathcal{A} </math>  
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==References==
 
==References==
#D. Williams, ''Probability with Martingales'', Cambridge : Cambridge University Press, 1991.
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#D. Williams, ''Probability with Martingales'', Cambridge : Cambridge University Press, 1991.
 
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[[Category:CZ Live]]
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In the branch of mathematics known as measure theory, the Caratheodory extension theorem states that a countably additive non-negative set function on an algebra of subsets of a set can be extended to be a measure on the sigma algebra generated by that algebra. Measure in this context specifically refers to a non-negative measure.

Statement of the theorem

(Caratheodory extension theorem) Let X be a set and be an algebra of subsets of X. Let be a countably additive non-negative set function on . Then there exists a measure on the -algebra (i.e., the smallest sigma algebra containing ) such that for all . Furthermore, if then the extension is unique.

is also referred to as the sigma algebra generated by . The term "algebra of subsets" in the theorem refers to a collection of subsets of a set X which contains X itself and is closed under the operation of taking complements, finite unions and finite intersections in X. That is, any algebra of subsets of X satisfies the following requirements:

  1. If then
  2. For any positive integer n, if then

The last two properties imply that is also closed under the operation of taking finite intersections of elements of .

References

  1. D. Williams, Probability with Martingales, Cambridge : Cambridge University Press, 1991.