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 &Sigma; be a &sigma;-algebra on X that contains the topology T (so that every open set is a measurable set, and &Sigma; is at least as fine as the Borel &sigma;-algebra on X). Then a measure &mu; on the measurable space (X, &Sigma;) is called inner regular if, for every set A in &Sigma;,


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

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

Some authors 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 &mu; is inner regular if and only if, for all &epsilon; &gt; 0, there is some compact subset K of X such that &mu;(X \ K) < &epsilon;. This is precisely the condition that the singleton collection of measures {&mu;} 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.
 * The Lebesgue measure on R is inner regular; and
 * The Gaussian measure (the normal distribution on R) is an inner regular probability measure.