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Inner regular measure

## Summary

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,

${\displaystyle \mu (A)=\sup\{\mu (K)\mid {\text{compact }}K\subseteq A\}.}$

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 finite 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.

## Examples

When the real line R is given its usual Euclidean topology,

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

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.{{cite book}}: CS1 maint: multiple names: authors list (link)
2. ^ Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. xii+276. ISBN 0-8218-3889-X. MR2169627