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 be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T. Then a measure μ on the measurable space is called inner regular if, for every set A in Σ,
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 μ is inner regular if and only if, for all ε > 0, there is some compact subset K of X such that μ < ε. This is precisely the condition that the singleton collection of measures 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, 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.