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.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.