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Concept
Regular measure
Summary
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.
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
and said to be outer regular if
A measure is called inner regular if every measurable set is inner regular. Some authors use a different definition: a measure is called inner regular if every open measurable set is inner regular.
A measure is called outer regular if every measurable set is outer regular.
A measure is called regular if it is outer regular and inner regular.
Lebesgue measure on the real line is a regular measure: see the regularity theorem for Lebesgue measure.
Any Baire probability measure on any locally compact σ-compact Hausdorff space is a regular measure.
Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology, or compact metric space, or Radon space, is regular.
An example of a measure on the real line with its usual topology that is not outer regular is the measure μ where , , and for any other set .
The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure. A variation of this example is a disjoint union of an uncountable number of copies of the real line with Lebesgue measure.
An example of a Borel measure μ on a locally compact Hausdorff space that is inner regular, σ-finite, and locally finite but not outer regular is given by as follows. The topological space X has as underlying set the subset of the real plane given by the y-axis of points (0,y) together with the points (1/n,m/n2) with m,n positive integers. The topology is given as follows. The single points (1/n,m/n2) are all open sets.
Official source
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