Regular measure

Let (X, T) be a topological space and let Σ be a σ-algebra on X.

A measurable subset A of X is said to be inner regular if 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.