Regular measure

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In mathematics, a regular measure on a topological space is one for which every measurable set is "approximately open" and "approximately closed".

1 Definition
2 Examples
3 Reference
4 See also

[edit] Definition

Let (X, T) be a topological space and let Σ be a σ-algebra on X that contains the topology T (so that all open and closed sets are measurable sets, and Σ is at least as fine as the Borel σ-algebra on X). A measure μ on (X, Σ) is called regular if, for every measurable set A in Σ and every δ > 0, there exists a closed set C and an open set U such that

$C \subseteq A \subseteq U$

and

$\mu (U \setminus C) < \delta.$

[edit] Examples

Lebesgue measure on the real line is a regular measure: see the regularity theorem for Lebesgue measure.
The trivial measure, which assigns measure zero to every measurable subset, is a regular measure.
A trivial example of a non-regular measure is the measure μ on the real line with its usual Borel topology that assigns measure zero to the empty set and infinite positive measure to any non-empty set.