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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.
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| ==Definition==
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| 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,
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| :<math>\mu (A) = \sup \{ \mu (K) \mid \text{compact } K \subseteq A \}.</math>
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| This property is sometimes referred to in words as "approximation from within by compact sets."
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| 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
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| | last = Parthasarathy
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| | first = K. R.
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| | title = Probability measures on metric spaces
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| |publisher = AMS Chelsea Publishing, Providence, RI
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| | year = 2005
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| | isbn = 0-8218-3889-X
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| | page = xii+276
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| | nopp = true
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| }} {{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.
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| ==Examples==
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| When the [[real line]] '''R''' is given its usual Euclidean topology,
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| * [[Lebesgue measure]] on '''R''' is inner regular; and
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| * [[Gaussian measure]] (the [[normal distribution]] on '''R''') is an inner regular [[probability measure]].
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| 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.
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| ==References==
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| <references />
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| ==See also==
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| * [[Radon measure]]
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| * [[Regular measure]]
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| [[Category:Measures (measure theory)]]
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