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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of the Riesz-Herglotz theorem on representing measures

Author: Peter A. Loeb
Journal: Proc. Amer. Math. Soc. 71 (1978), 65-68
MSC: Primary 31D05
MathSciNet review: 0588522
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Abstract: A simple construction is given that obtains maximal representing measures for positive harmonic functions on a domain W as the $\mathrm {weak}^*$ limits of finite sums of point masses on ${[0, + \infty ]^W}$. This new standard result, new even for the unit disk, is established for very general elliptic differential equations and domains, in fact, for a Brelot harmonic space, using nonstandard analysis.

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Keywords: Herglotz Theorem, representing measures, Brelot harmonic spaces, nonstandard analysis, <!– MATH $\mathrm {weak}^*$ –> <IMG WIDTH="58" HEIGHT="22" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathrm {weak}^*$"> convergence
Article copyright: © Copyright 1978 American Mathematical Society