Absorption wavemeter - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10667)

2015-01-04T07:56:21Z

WP:CHECKWIKI error fixes using AWB (10667)

← Older revision		Revision as of 09:56, 4 January 2015
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	~~In [[mathematics]], a '''regular measure''' on a [[topological space]]~~ is ~~a [[measure (mathematics)\|measure]] for which every [[measurable set]] can be approximated from above by an open measurable set and from below by a compact measurable set.~~		Alyson is what my husband enjoys to call me but I don't like when people use my complete name. Credit authorising is where my primary earnings comes from. What I adore performing is football but I don't have the time lately. Ohio is exactly where his house is and his family enjoys it.<br><br>Check out my web blog: free psychic readings ([http://brazil.amor-amore.com/irboothe http://brazil.amor-amore.com/])

	~~==Definition==~~

	~~Let (''X'', ''T'') be a topological space and let Σ be a [[sigma algebra\|σ-algebra]] on ''X''. Let ''μ'' be a measure on (''X'', Σ). A measurable subset ''A'' of ''X'' is said~~ to ~~be '''inner regular''' if~~

	~~:<math>\mu (A) = \sup \{ \mu (F) \| F \subseteq A, F \mbox{ compact and measurable} \}</math>~~

	~~and said to be '''outer regular''~~' if

	~~:<math>\mu (A) = \inf \{ \mu (G) \| G \supseteq A, G \mbox{ open and measurable} \}</math>~~

	*A measure is called inner regular if every measurable set is inner regular. Some authors use ~~a different definition: a measure is called inner regular if every '''open''' measurable set is inner regular~~.
	*A measure is ~~called outer regular if every measurable set is outer regular~~.
	*A measure is ~~called regular if it is outer regular and inner regular.~~

	~~==Examples==~~

	~~===Regular measures===~~

	* [[Lebesgue measure]] on the ~~[[real line]] is a regular measure: see the [[regularity theorem for Lebesgue measure]]~~.
	* Any Baire [[probability measure]] on any locally compact σ-compact Hausdorff space is a regular measure.
	*Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology, or compact metric space, or [[Radon space]], is regular.

	~~===Inner regular measures that are not outer regular===~~

	* An example of a measure on the real line with its usual topology that is not outer regular is ~~the measure ''μ''~~ where ~~<math>\mu(\emptyset) = 0</math>, <math>\mu\left( \{1\}\right) = 0\,\,</math>, and <math>\mu(A) = \infty\,\,</math> for any other set <math>A</math>.~~
	*The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure. A variation of this example is a disjoint union of an uncountable number of copies of the real line with Lebesgue measure.
	*An example of a Borel measure μ on a locally compact Hausdorff space that is ~~inner regular, σ-finite,~~ and ~~locally finite but not outer regular is given by {{harvtxt\|Bourbaki\|2004\|loc=Exercise 5 of section 1}} as follows~~. ~~The topological space ''X'' has as underlying set the subset of the real plane given by the ''y''-axis of points (0,''y'') together with the points (1/''n'',''m''/''n''~~<~~sup~~>2<~~/sup~~>~~) with ''m'',''n'' positive integers. The topology is given as follows. The single points~~ (1/~~''n'',''m''~~/~~''n''<sup>2</sup>) are all open sets~~. ~~A base of neighborhoods of the point (0,''y'') is given by wedges consisting of all points in ''X'' of the form (''u'',''v'') with \|''v''~~-~~''y''\|≤\|''u''\|≤1/''n'' for a positive integer ''n''~~. ~~This space ''X'' is locally compact. The measure μ is given by letting the ''y''-axis have measure 0 and letting the point (1~~/~~''n'',''m''/''n''<sup>2</sup>) have measure 1~~/~~''n''<sup>3<~~/~~sup>~~. ~~This measure is inner regular and locally finite, but is not outer regular as any open set containing the ''y''~~-~~axis has measure infinity~~.

	~~===Outer regular measures that are not inner regular===~~

	*If μ is the inner regular measure in the previous example, and ''M'' is the measure given by ''M''(''S'') = inf<sub>''U''&supe;''S''</sub> μ(''U'') where the inf is taken over all open sets containing the Borel set ''S'', then ''M'' is an outer regular locally finite Borel measure on a locally compact Hausdorff space that is not inner regular in the strng sense, though all open sets are inner regular so it is inner regular in the weak sense. The measures ''M'' and μ coincide on all open sets, all compact sets, and all sets on which ''M'' has finite measure. The ''y''-axis has infinite ''M''-measure though all compact subsets of it have measure 0.

	*A [[measurable cardinal]] with the discrete topology has a Borel probability measure such that every compact subset has measure 0, so this measure is outer regular but not inner regular. The existence of measurable cardinals cannot be proved in ZF set theory but (as of 2013) is thought to be consistent with it.

	~~===Measures that are neither inner nor outer regular===~~

	*The space of all ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by open intervals, is a compact Hausdorff space. The measure that assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinalls and assigns 0 to other Borel sets is a Borel probability measure that is neither inner regular nor outer regular.

	~~==See also==~~

	* [[Borel regular measure]]
	* [[Radon measure]]
	* [[Regularity theorem for Lebesgue measure]]

	~~==References==~~

	* {{cite book \| last=Billingsley \| first=Patrick \| title=Convergence of Probability Measures \| publisher=John Wiley & Sons, Inc. \| location=New York \| year=1999 \| isbn = 0-471-19745-9}}
	* {{cite book
	~~\| last = Parthasarathy~~
	~~\| first = K. R.~~
	~~\| title = Probability measures on metric spaces~~
	~~\|publisher = AMS Chelsea Publishing, Providence, RI~~
	~~\| year = 2005~~
	~~\| isbn = 0-8218-3889-X~~
	~~\| page = xii+276~~
	~~}} {{MathSciNet\|id=2169627}} (See chapter 2)~~

	*{{cite book
	~~\|first=R. M.~~
	~~\|last=Dudley~~
	~~\|author-link=Richard M. Dudley~~
	~~\|title=Real Analysis and Probability~~
	~~\|publisher=Chapman & Hall~~
	~~\|year=1989}}~~

	~~[[Category:Measures (measure theory~~)]]

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3820627

2013-03-14T23:48:42Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3820627

← Older revision		Revision as of 01:48, 15 March 2013
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	~~Hi there~~. ~~Allow me start by introducing~~ the ~~writer~~, ~~her title~~ is ~~Sophia Boon~~ but ~~she~~ by ~~no means really liked that name~~. ~~Mississippi~~ is the ~~only place I~~'~~ve been residing in but I will have to transfer~~ in a ~~yr or two~~. ~~Invoicing~~ is ~~my occupation~~. ~~I am truly fond of to go to karaoke~~ but I'~~ve been taking on new things lately~~.<br><br>~~Take~~ a ~~look~~ at ~~my web~~ page ~~:: certified psychics~~ (~~[http://black7~~.~~mireene~~.~~com/aqw/5741~~ link ~~webpage~~])		In [[mathematics]], a '''regular measure''' on a [[topological space]] is a [[measure (mathematics)\|measure]] for which every [[measurable set]] can be approximated from above by an open measurable set and from below by a compact measurable set.

			==Definition==

			Let (''X'', ''T'') be a topological space and let Σ be a [[sigma algebra\|σ-algebra]] on ''X''. Let ''μ'' be a measure on (''X'', Σ). A measurable subset ''A'' of ''X'' is said to be '''inner regular''' if

			:<math>\mu (A) = \sup \{ \mu (F) \| F \subseteq A, F \mbox{ compact and measurable} \}</math>

			and said to be '''outer regular''' if

			:<math>\mu (A) = \inf \{ \mu (G) \| G \supseteq A, G \mbox{ open and measurable} \}</math>

			*A measure is called inner regular if every measurable set is inner regular. Some authors use a different definition: a measure is called inner regular if every '''open''' measurable set is inner regular.
			*A measure is called outer regular if every measurable set is outer regular.
			*A measure is called regular if it is outer regular and inner regular.

			==Examples==

			===Regular measures===

			* [[Lebesgue measure]] on the [[real line]] is a regular measure: see the [[regularity theorem for Lebesgue measure]].
			* Any Baire [[probability measure]] on any locally compact σ-compact Hausdorff space is a regular measure.
			*Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology, or compact metric space, or [[Radon space]], is regular.

			===Inner regular measures that are not outer regular===

			* An example of a measure on the real line with its usual topology that is not outer regular is the measure ''μ'' where <math>\mu(\emptyset) = 0</math>, <math>\mu\left( \{1\}\right) = 0\,\,</math>, and <math>\mu(A) = \infty\,\,</math> for any other set <math>A</math>.
			*The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure. A variation of this example is a disjoint union of an uncountable number of copies of the real line with Lebesgue measure.
			*An example of a Borel measure μ on a locally compact Hausdorff space that is inner regular, σ-finite, and locally finite but not outer regular is given by {{harvtxt\|Bourbaki\|2004\|loc=Exercise 5 of section 1}} as follows. The topological space ''X'' has as underlying set the subset of the real plane given by the ''y''-axis of points (0,''y'') together with the points (1/''n'',''m''/''n''<sup>2</sup>) with ''m'',''n'' positive integers. The topology is given as follows. The single points (1/''n'',''m''/''n''<sup>2</sup>) are all open sets. A base of neighborhoods of the point (0,''y'') is given by wedges consisting of all points in ''X'' of the form (''u'',''v'') with \|''v''-''y''\|≤\|''u''\|≤1/''n'' for a positive integer ''n''. This space ''X'' is locally compact. The measure μ is given by letting the ''y''-axis have measure 0 and letting the point (1/''n'',''m''/''n''<sup>2</sup>) have measure 1/''n''<sup>3</sup>. This measure is inner regular and locally finite, but is not outer regular as any open set containing the ''y''-axis has measure infinity.

			===Outer regular measures that are not inner regular===

			*If μ is the inner regular measure in the previous example, and ''M'' is the measure given by ''M''(''S'') = inf<sub>''U''&supe;''S''</sub> μ(''U'') where the inf is taken over all open sets containing the Borel set ''S'', then ''M'' is an outer regular locally finite Borel measure on a locally compact Hausdorff space that is not inner regular in the strng sense, though all open sets are inner regular so it is inner regular in the weak sense. The measures ''M'' and μ coincide on all open sets, all compact sets, and all sets on which ''M'' has finite measure. The ''y''-axis has infinite ''M''-measure though all compact subsets of it have measure 0.

			*A [[measurable cardinal]] with the discrete topology has a Borel probability measure such that every compact subset has measure 0, so this measure is outer regular but not inner regular. The existence of measurable cardinals cannot be proved in ZF set theory but (as of 2013) is thought to be consistent with it.

			===Measures that are neither inner nor outer regular===

			*The space of all ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by open intervals, is a compact Hausdorff space. The measure that assigns measure 1 to Borel sets containing an unbounded closed subset of the countable ordinalls and assigns 0 to other Borel sets is a Borel probability measure that is neither inner regular nor outer regular.

			==See also==

			* [[Borel regular measure]]
			* [[Radon measure]]
			* [[Regularity theorem for Lebesgue measure]]

			==References==

			* {{cite book \| last=Billingsley \| first=Patrick \| title=Convergence of Probability Measures \| publisher=John Wiley & Sons, Inc. \| location=New York \| year=1999 \| isbn = 0-471-19745-9}}
			* {{cite book
			\| last = Parthasarathy
			\| first = K. R.
			\| title = Probability measures on metric spaces
			\|publisher = AMS Chelsea Publishing, Providence, RI
			\| year = 2005
			\| isbn = 0-8218-3889-X
			\| page = xii+276
			}} {{MathSciNet\|id=2169627}} (See chapter 2)

			*{{cite book
			\|first=R. M.
			\|last=Dudley
			\|author-link=Richard M. Dudley
			\|title=Real Analysis and Probability
			\|publisher=Chapman & Hall
			\|year=1989}}

			[[Category:Measures (measure theory)]]

en>ChrisGualtieri: /* HF and VHF */TypoScan Project, typos fixed: , → , using AWB

2012-06-05T16:44:03Z

HF and VHF: TypoScan Project, typos fixed: , → , using AWB

New page

Hi there. Allow me start by introducing the writer, her title is Sophia Boon but she by no means really liked that name. Mississippi is the only place I've been residing in but I will have to transfer in a yr or two. Invoicing is my occupation. I am truly fond of to go to karaoke but I've been taking on new things lately.<br><br>Take a look at my web page :: certified psychics ([http://black7.mireene.com/aqw/5741 link webpage])