Regular measure

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

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

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

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

:

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

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.

References

*
* cite book
last = Parthasarathy
first = K. R.
title = Probability measures on metric spaces
publisher = AMS Chelsea Publishing, Providence, RI
year = 2005
pages = pp.xii+276
isbn = 0-8218-3889-X MathSciNet|id=2169627 (See chapter 2)

ee also

* Borel regular measure
* Regularity theorem for Lebesgue measure