Inner regular measure

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In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.

[edit] Definition

Let <math>(X, \mathcal{T})</math> be a Hausdorff topological space and let <math>\mathcal{F}</math> be a σ-algebra on <math>X</math> that contains the topology <math>\mathcal{T}</math> (so that every open set is a measurable set). Then a measure <math>\mu</math> on the measurable space <math>(X, \mathcal{F})</math> is called inner regular if

<math>\mu (A) = \sup \{ \mu (K) | \mathrm{compact \,} K \subseteq A \}</math> for all <math>A \in \mathcal{F}.</math>

Some authors<ref name=AGS>L. Ambrosio, N. Gigli & G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)</ref> use the term tight as a synonym for inner regular. This use of the term is not to be confused with tightness of a family of measures.