Regular measure

In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

Definition

Let (X, T) be a topological space and let Σ be a σ-algebra on X. Let μ be a measure on (X, Σ). A measurable subset A of X is said to be inner regular if

\mu (A) = \sup \{ \mu (F) | F \subseteq A, F \mbox{ compact and measurable} \}

and said to be outer regular if

\mu (A) = \inf \{ \mu (G) | G \supseteq A, G \mbox{ open and measurable} \}

Examples

Regular measures

Inner regular measures that are not outer regular

Outer regular measures that are not inner regular

Measures that are neither inner nor outer regular

See also

References

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