# 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 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,

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 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,

- 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.

## References

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