Inner regular measure

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

Definition

Let (X, T) be a Hausdorff topological space and let Σ be a 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 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,

\mu (A) = \sup \{ \mu (K) | \mbox{compact } K \subseteq A \}.

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

Some authors[1][2] use the term tight as a synonym for inner regular. This use of the term is closely related to 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 subset K of X such that μ(X \ K) < ε. This is precisely the condition that the singleton collection of measures {μ} is tight.

References

  1. ^ Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-7643-2428-7. 
  2. ^ Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. xii+276. ISBN 0-8218-3889-X.  MR2169627

See also