essential boundary

Let E𝐑n be a measurable setMathworldPlanetmath. We define the essential boundary of E as


where || is the Lebesgue measureMathworldPlanetmath.

Compare the definition of *E with the definition of the topological boundary E which can be written as


Hence one clearly has *EE.

Notice that the essential boundary does not depend on the Lebesgue representative of the set E, in the sense that if |EF|=0 then *E=*F. For example if E=𝐐n𝐑n is the set of points with rational coordinates, one has *E= while E=𝐑n.

Nevertheless one can easily prove that *E is always a closed setPlanetmathPlanetmath (in the usual sense).

Title essential boundary
Canonical name EssentialBoundary
Date of creation 2013-03-22 15:01:54
Last modified on 2013-03-22 15:01:54
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 9
Author paolini (1187)
Entry type Definition
Classification msc 49-00