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.
{{R from merge}}
==Definition==
{{R to section}}
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) | \mbox{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 | id=ISBN 3-10071-2424-7 }}</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
:<math>\mu \left( X \setminus K \right) < \varepsilon.</math>
This is precisely the condition that the [[singleton (mathematics)|singleton]] collection of measures {''μ''} is tight.
