A generalization of the Riesz-Herglotz theorem on representing measures

Loeb, Peter A.

doi:10.1090/S0002-9939-1978-0588522-2

A generalization of the Riesz-Herglotz theorem on representing measures
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Proc. Amer. Math. Soc. 71 (1978), 65-68 Request permission

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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Weak limits of measures and the standard part map

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