Inner regular measure

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.

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

• Lebesgue measure on R is inner regular; and
• Gaussian measure (the normal distribution on R) is an inner regular probability measure.

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.

• Radon measure
• Regular measure