Solvable Lie algebra - Revision history

en>YohanN7: /* References */

2015-01-08T18:19:13Z

References

← Older revision		Revision as of 20:19, 8 January 2015
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	~~In [[mathematics]],~~ the '''Hahn–Kolmogorov theorem''' characterizes when a [[finitely additive]] [[function (mathematics)\|function]] with [[negative and positive numbers\|non-negative]] (possibly infinite) values can be extended to a ''bona fide'' [[measure (mathematics)\|measure]]. It is ~~named after the~~ [~~[Austria]]n [[mathematician]] [[Hans Hahn (mathematician)\|Hans Hahn]] and the [[Russia]]n~~/~~[[Soviet Union\|Soviet]] mathematician [[Andrey Kolmogorov]]~~.		The individual who wrote the post is [http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews authentic psychic readings] known as Jayson Hirano and he completely digs that title. What me and my family members love is performing ballet but I've been good psychic ([http://165.132.39.93/xe/visitors/372912 link homepage]) taking on new things recently. Office supervising is where my main income arrives from but I've usually wanted my personal business. I've always cherished residing in Kentucky but now I'm considering other choices.<br><br>Feel free to surf to my blog post; online psychics [[http://jplusfn.gaplus.kr/xe/qna/78647 http://jplusfn.gaplus.kr/xe/qna/78647]]

	~~==Statement of the theorem==~~
	~~Let <math>\Sigma_0<~~/~~math> be an [[Field of sets\|algebra of subset]]s of a [[Set (mathematics)\|set]] <math>X~~.~~</math> Consider a function~~

	~~:<math>\mu_0\colon \Sigma_0 \to[0,\infty]</math>~~

	~~which is ''finitely additive'', meaning that~~
	~~: <math>\mu_0(\bigcup_{n=1}^N A_n)~~=~~\sum_{n~~=~~1}^N \mu_0(A_n)</math>~~

	~~for any positive [[integer~~]~~] ''N''~~ and ~~<math>A_1, A_2, \dots, A_N</math> [[disjoint set]]s in <math>\Sigma_0</math>~~.

	~~Assume that this function satisfies the stronger~~ '~~'sigma additivity'' assumption~~

	~~:<math> \mu_0~~(~~\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>~~

	~~for any disjoint family <math>\{A_n~~:~~n\in \mathbb{N}\}<~~/~~math> of elements of <math>\Sigma_0<~~/~~math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>~~. ~~(Functions <math>\mu_0</math> obeying these two properties are known as [[pre-measures]]~~.~~) Then,~~
	~~<math>\mu_0<~~/~~math> extends to a measure defined on the [[sigma-algebra]] <math>\Sigma<~~/~~math> generated by <math>\Sigma_0<~~/~~math>; i.e., there exists a measure~~

	~~:<math>\mu \colon \Sigma \to[0,\infty~~]~~</math>~~

	~~such that its [[Restriction (mathematics~~)~~\|restriction]] to <math>\Sigma_0</math> coincides with <math>\mu_0~~.~~</math>~~

	~~If <math>\mu_0</math>~~ is ~~<math>\sigma</math>-finite, then the extension is unique.~~

	~~==Non-uniqueness of the extension==~~

	~~If <math>\mu_0</math> is not <math>\sigma</math>-finite then the extension need not be unique, even if the extension itself is <math>\sigma</math>-finite~~.

	~~Here is an example:~~

	~~We call ''rational closed-open interval~~''~~, any subset of <math>\mathbb{Q}</math> of the form <math>[a,b)</math>, where <math>a, b \in \mathbb{Q}</math>~~.

	~~Let~~ <~~math~~>X<~~/math~~> ~~be <math>\mathbb{Q}\cap~~[~~0,1)</math> and let <math>\Sigma_0</math> be the algebra of all finite union of rational closed-open intervals contained in <math>\mathbb{Q}\cap~~[~~0,1)<~~/~~math>. It is easy to prove that <math>\Sigma_0</math> is, in fact, an algebra. It is also easy to see that every non-empty set in <math>\Sigma_0<~~/~~math> is infinite~~.

	~~Let <math>\mu_0</math> be the counting set function (<math>\#</math>) defined in <math>\Sigma_0</math>~~.
	~~It is clear that <math>\mu_0<~~/~~math> is finitely additive and <math>\sigma<~~/~~math>-additive in <math>\Sigma_0<~~/~~math>. Since every non-empty set in <math>\Sigma_0<~~/~~math> is infinite, we have, for every non-empty set <math>A\in\Sigma_0</math>, <math>\mu_0(A)=+\infty<~~/~~math>~~

	~~Now, let <math>\Sigma</math> be the <math>\sigma</math>-algebra generated by <math>\Sigma_0</math>~~. It is easy to see that <math>\Sigma</math> is the Borel <math>\sigma</math>-algebra of subsets of <math>X</math>, and both <math>\#</math> and <math>2\#</math> are measures defined on <math>\Sigma</math> and both are extensions of <math>\mu_0</math>.

	~~==Comments==~~
	This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. ~~The proof of this theorem is not trivial, since it requires extending <math>\mu_0<~~/~~math> from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if <math>\mu_0<~~/~~math> is <math>\sigma<~~/~~math>-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.~~

	~~== See also ==~~
	* [[Carathéodory's extension theorem]]
	* [[pre-measure]]

	~~{{PlanetMath attribution\|id=5409\|title=Hahn–Kolmogorov theorem}}~~

	~~{{DEFAULTSORT:Hahn-Kolmogorov theorem}}~~
	~~[[Category:Theorems in measure theory~~]]

en>Stefan.Groote: Undid revision 588951632 by Stefan.Groote (talk)

2014-01-17T14:07:22Z

Undid revision 588951632 by Stefan.Groote (talk)

← Older revision		Revision as of 16:07, 17 January 2014
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	~~Jayson Berryhill is how I~~'~~m known as~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~Office supervising~~ is ~~where her main income arrives from~~. ~~To play lacross is one~~ of the ~~things she loves most~~. I'~~ve usually loved living~~ in ~~Mississippi~~.<br><br>~~Also visit my website~~ ... [~~http~~://~~galab~~-~~work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl~~=~~1489804 psychics online~~]		In [[mathematics]], the '''Hahn–Kolmogorov theorem''' characterizes when a [[finitely additive]] [[function (mathematics)\|function]] with [[negative and positive numbers\|non-negative]] (possibly infinite) values can be extended to a ''bona fide'' [[measure (mathematics)\|measure]]. It is named after the [[Austria]]n [[mathematician]] [[Hans Hahn (mathematician)\|Hans Hahn]] and the [[Russia]]n/[[Soviet Union\|Soviet]] mathematician [[Andrey Kolmogorov]].

			==Statement of the theorem==
			Let <math>\Sigma_0</math> be an [[Field of sets\|algebra of subset]]s of a [[Set (mathematics)\|set]] <math>X.</math> Consider a function

			:<math>\mu_0\colon \Sigma_0 \to[0,\infty]</math>

			which is ''finitely additive'', meaning that
			: <math>\mu_0(\bigcup_{n=1}^N A_n)=\sum_{n=1}^N \mu_0(A_n)</math>

			for any positive [[integer]] ''N'' and <math>A_1, A_2, \dots, A_N</math> [[disjoint set]]s in <math>\Sigma_0</math>.

			Assume that this function satisfies the stronger ''sigma additivity'' assumption

			:<math> \mu_0(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty \mu_0(A_n)</math>

			for any disjoint family <math>\{A_n:n\in \mathbb{N}\}</math> of elements of <math>\Sigma_0</math> such that <math>\cup_{n=1}^\infty A_n\in \Sigma_0</math>. (Functions <math>\mu_0</math> obeying these two properties are known as [[pre-measures]].) Then,
			<math>\mu_0</math> extends to a measure defined on the [[sigma-algebra]] <math>\Sigma</math> generated by <math>\Sigma_0</math>; i.e., there exists a measure

			:<math>\mu \colon \Sigma \to[0,\infty]</math>

			such that its [[Restriction (mathematics)\|restriction]] to <math>\Sigma_0</math> coincides with <math>\mu_0.</math>

			If <math>\mu_0</math> is <math>\sigma</math>-finite, then the extension is unique.

			==Non-uniqueness of the extension==

			If <math>\mu_0</math> is not <math>\sigma</math>-finite then the extension need not be unique, even if the extension itself is <math>\sigma</math>-finite.

			Here is an example:

			We call ''rational closed-open interval'', any subset of <math>\mathbb{Q}</math> of the form <math>[a,b)</math>, where <math>a, b \in \mathbb{Q}</math>.

			Let <math>X</math> be <math>\mathbb{Q}\cap[0,1)</math> and let <math>\Sigma_0</math> be the algebra of all finite union of rational closed-open intervals contained in <math>\mathbb{Q}\cap[0,1)</math>. It is easy to prove that <math>\Sigma_0</math> is, in fact, an algebra. It is also easy to see that every non-empty set in <math>\Sigma_0</math> is infinite.

			Let <math>\mu_0</math> be the counting set function (<math>\#</math>) defined in <math>\Sigma_0</math>.
			It is clear that <math>\mu_0</math> is finitely additive and <math>\sigma</math>-additive in <math>\Sigma_0</math>. Since every non-empty set in <math>\Sigma_0</math> is infinite, we have, for every non-empty set <math>A\in\Sigma_0</math>, <math>\mu_0(A)=+\infty</math>

			Now, let <math>\Sigma</math> be the <math>\sigma</math>-algebra generated by <math>\Sigma_0</math>. It is easy to see that <math>\Sigma</math> is the Borel <math>\sigma</math>-algebra of subsets of <math>X</math>, and both <math>\#</math> and <math>2\#</math> are measures defined on <math>\Sigma</math> and both are extensions of <math>\mu_0</math>.

			==Comments==
			This theorem is remarkable for it allows one to construct a measure by first defining it on a small algebra of sets, where its sigma additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. The proof of this theorem is not trivial, since it requires extending <math>\mu_0</math> from an algebra of sets to a potentially much bigger sigma-algebra, guaranteeing that the extension is unique (if <math>\mu_0</math> is <math>\sigma</math>-finite), and moreover that it does not fail to satisfy the sigma-additivity of the original function.

			== See also ==
			* [[Carathéodory's extension theorem]]
			* [[pre-measure]]

			{{PlanetMath attribution\|id=5409\|title=Hahn–Kolmogorov theorem}}

			{{DEFAULTSORT:Hahn-Kolmogorov theorem}}
			[[Category:Theorems in measure theory]]

en>Paolo328: Put two hyperlinks

2011-11-30T14:17:51Z

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