العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolINNER 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.
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,
:
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors[1] 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.
1. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Ambrosio, L., Gigli, N. & Savaré, G., , , ETH Zürich, Birkhäuser Verlag, 2005, ISBN 3-7643-2428-7
★ Radon measure
| Contents |
| Definition |
| Reference |
| See also |
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,
:
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors[1] 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.
Reference
1. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Ambrosio, L., Gigli, N. & Savaré, G., , , ETH Zürich, Birkhäuser Verlag, 2005, ISBN 3-7643-2428-7
See also
★ Radon measure
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
