Regular measure

The notion of regularity of dimensions used to characterize measures on topological spaces.

Definition

Be on a Hausdorff space and a σ - algebra containing the Borel σ - algebra. Then lie in particular all open, all closed and all compact subsets of in.

A measure is called

• Regularly from the inside, if for each:

• Regularly from the outside, if for each:

• Regular if it is regular from the inside and from the outside.

Sometimes calls to the inner regularity only for open sets ( in this sense, then the hair - degree regular) or prompts that it is the measure is a Borel measure.

Properties and examples

Regular measurements allow in many proofs Approximationsargumente. Often it is sufficient to prove certain statements or for compact open sets, and then to extend through the two formulas on measurable quantities. Many dimensions are regular.

• The Lebesgue measure on is regular.

• More generally: If a locally compact Hausdorff space which is countable union of compact sets, and is a Borel measure on which is available on all compact sets is finite, then is regular.

• A Borel measure on a Polish space is regular.