Inner regular measure

From Wikipedia, the free encyclopedia

In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.

[edit] Definition

Let (X, \mathcal{T}) be a Hausdorff topological space and let \mathcal{F} be a σ-algebra on X that contains the topology \mathcal{T} (so that every open set is a measurable set). Then a measure μ on the measurable space (X, \mathcal{F}) is called inner regular if

\mu (A) = \sup \{ \mu (K) | \mathrm{compact \,} K \subseteq A \} for all A \in \mathcal{F}.

Some authors[1] use the term tight as a synonym for inner regular. This use of the term is not to be confused with tightness of a family of measures.

[edit] Reference

  1. ^ L. Ambrosio, N. Gigli & G. Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel. (2005)

[edit] See also

Retrieved from "http://en.wikipedia.org../../../i/n/n/Inner_regular_measure.html"