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,

:$mu\; (A)\; =\; sup\; \{\; mu\; (K)\; |\; mbox\{compact\; \}\; K\; subseteq\; A\; \}.$

This property is sometimes referred to in words as "approximation from within by compact sets."

Some authorscite book | author=Ambrosio, L., Gigli, N. & Savaré, G. | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag | location=Basel | year=2005 | id=ISBN 3-7643-2428-7 ] 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] 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.

References

ee also

* Radon measure
* Regular measure