Illinois Journal of Mathematics

Representing measures in potential theory and an ideal boundary

Peter A. Loeb

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Abstract

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.

Article information

Source
Illinois J. Math., Volume 54, Number 4 (2010), 1451-1461.

Dates
First available in Project Euclid: 24 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1348505536

Digital Object Identifier
doi:10.1215/ijm/1348505536

Mathematical Reviews number (MathSciNet)
MR2981855

Zentralblatt MATH identifier
1254.28015

Subjects
Primary: 03H10: Other applications of nonstandard models (economics, physics, etc.) 28E05: Nonstandard measure theory [See also 03H05, 26E35] 31C35: Martin boundary theory [See also 60J50]

Citation

Loeb, Peter A. Representing measures in potential theory and an ideal boundary. Illinois J. Math. 54 (2010), no. 4, 1451--1461. doi:10.1215/ijm/1348505536. https://projecteuclid.org/euclid.ijm/1348505536


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