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	~~In [[mathematics]], an '''inner regular measure''' is one for which the [[Measure (mathematics)\|measure]]~~ of ~~a set can be approximated from within by [[Compact space\|compact]] [[subset]]s.~~		The title of the author is Figures but it's not the most masucline title out there. My working day job is a librarian. Years in the past he moved to North Dakota and his family members loves it. To collect coins is a factor that I'm totally addicted to.<br><br>Also visit my weblog; home std test ([http://fastrolls.com/index.php?do=/profile-72113/info/ visit the following post])

	~~==Definition==~~

	~~Let (''X'', ''T'') be a [[Hausdorff space\|Hausdorff]] [[topological space]] and let Σ be a [[sigma algebra\|σ-algebra]] on ''X'' that contains~~ the ~~topology ''T'' (so that every [[open set]]~~ is ~~a [[measurable set]], and Σ is at least as fine as the [[Borel sigma algebra\|Borel σ-algebra]] on~~ '~~'X''). Then a measure ''μ'' on~~ the ~~[[measurable space]] (''X'', Σ) is called '''inner regular''' if, for every set ''A'' in Σ,~~

	~~:<math>\mu (A) = \sup \{ \mu (K) \mid \text{compact } K \subseteq A \}~~.~~</math>~~

	~~This property~~ is ~~sometimes referred to in words as "approximation from within by compact sets."~~

	~~Some authors<ref name="AGS">{{cite book \| author=Ambrosio, L~~.~~, Gigli, N. & Savaré, G. \| title=Gradient Flows~~ in ~~Metric Spaces and in the Space of Probability Measures \| publisher=ETH Zürich, Birkhäuser Verlag \| location=Basel \| year=2005 \| isbn=3-7643-2428-7 }}</ref><ref name="Par">{{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~~
	~~\| nopp = true~~
	~~}} {{MathSciNet\|id=2169627}}</ref> use~~ the ~~term '''tight''' as a [[synonym]] for inner regular. This use of the term is closely related~~ to ~~[[Tightness of measures\|tightness of a~~ family of measures]], since a measure ''μ'' is inner regular [[if and only if]], for all ''ε'' > 0, there is some [[compact space\|compact subset]] ''K'' of ''X'' such that ''μ''(''X'' \ ''K'') < ''ε''. ~~This~~ is ~~precisely the condition~~ that ~~the [[singleton (mathematics)\|singleton]] collection of measures {~~'~~'μ''} is tight~~.

	~~==Examples==~~

	~~When the [[real line]] '''R''' is given its usual Euclidean topology,~~
	* [[Lebesgue measure]] on '''R''' is inner regular; ~~and~~
	* [[Gaussian measure]] (~~the~~ [~~[normal distribution]] on '''R''') is an inner regular [[probability measure]]~~.
	~~However, if the topology on '''R''' is changed, then these measures can fail to be inner regular~~. ~~For example, if '''R''' is given the [[lower limit topology]] (which generates the same σ~~-~~algebra as the Euclidean topology), then both of~~ the ~~above measures fail to be inner regular, because compact sets in that topology are necessarily countable, and hence of measure zero.~~

	~~==References==~~

	~~<references />~~

	~~==See also==~~

	* [[Radon measure]]
	* [[Regular measure]]

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

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In [[mathematics]], an '''inner regular measure''' is one for which the [[Measure (mathematics)|measure]] of a set can be approximated from within by [[Compact space|compact]] [[subset]]s.

==Definition==

Let (''X'', ''T'') be a [[Hausdorff space|Hausdorff]] [[topological space]] and let Σ be a [[sigma algebra|σ-algebra]] on ''X'' that contains the topology ''T'' (so that every [[open set]] is a [[measurable set]], and Σ is at least as fine as the [[Borel sigma algebra|Borel σ-algebra]] on ''X''). Then a measure ''μ'' on the [[measurable space]] (''X'', Σ) is called '''inner regular''' if, for every set ''A'' in Σ,

:<math>\mu (A) = \sup \{ \mu (K) \mid \text{compact } K \subseteq A \}.</math>

This property is sometimes referred to in words as "approximation from within by compact sets."

Some authors<ref name="AGS">{{cite book | author=Ambrosio, L., Gigli, N. & Savaré, G. | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag | location=Basel | year=2005 | isbn=3-7643-2428-7 }}</ref><ref name="Par">{{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
| nopp = true
}} {{MathSciNet|id=2169627}}</ref> use the term '''tight''' as a [[synonym]] for inner regular. This use of the term is closely related to [[Tightness of measures|tightness of a family of measures]], since a measure ''μ'' is inner regular [[if and only if]], for all ''ε'' > 0, there is some [[compact space|compact subset]] ''K'' of ''X'' such that ''μ''(''X'' \ ''K'') < ''ε''. This is precisely the condition that the [[singleton (mathematics)|singleton]] collection of measures {''μ''} is tight.

==Examples==

When the [[real line]] '''R''' is given its usual Euclidean topology,
* [[Lebesgue measure]] on '''R''' is inner regular; and
* [[Gaussian measure]] (the [[normal distribution]] on '''R''') is an inner regular [[probability measure]].
However, if the topology on '''R''' is changed, then these measures can fail to be inner regular. For example, if '''R''' is given the [[lower limit topology]] (which generates the same σ-algebra as the Euclidean topology), then both of the above measures fail to be inner regular, because compact sets in that topology are necessarily countable, and hence of measure zero.

==References==

<references />

==See also==

* [[Radon measure]]
* [[Regular measure]]

[[Category:Measures (measure theory)]]

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71.93.66.169 at 19:25, 5 April 2014

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