Regular measure

From Wikipedia, the free encyclopedia

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

1 Definition
2 Examples
3 References
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). Let μ be a measure on (X, Σ). A measurable subset A of X is said to be μ-regular if

$\mu (A) = \sup \{ \mu (F) | F \subseteq A, F \mbox{ closed} \}$

and

$\mu (A) = \inf \{ \mu (G) | G \supseteq A, F \mbox{ open} \}.$

Equivalently, A is a μ-regular set if and only if, for every δ > 0, there exists a closed set F and an open set G such that

$F \subseteq A \subseteq G$

and

$\mu (G \setminus F) < \delta.$

If every measurable set is regular, then the measure μ is said to be a regular measure.

[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.
Any Borel probability measure on any metric space is a regular measure.

[edit] References

Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc.. ISBN 0-471-19745-9.
Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI, pp.xii+276. ISBN 0-8218-3889-X. MR 2169627 (See chapter 2)