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Reduced functions and Jensen measures


Authors: Wolfhard Hansen and Ivan Netuka
Journal: Proc. Amer. Math. Soc. 146 (2018), 153-160
MSC (2010): Primary 31B05, 31D05, 35J15, 60J45, 60J60, 60J75
DOI: https://doi.org/10.1090/proc/13688
Published electronically: July 10, 2017
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Abstract: Let $ \varphi $ be a locally upper bounded Borel measurable function on a Greenian open set $ \Omega $ in $ \mathbb{R}^d$ and, for every $ x\in \Omega $, let $ v_\varphi (x)$ denote the infimum of the integrals of $ \varphi $ with respect to Jensen measures for $ x$ on $ \Omega $. Twenty years ago, B.J.Cole and T.J.Ransford proved that $ v_\varphi $ is the supremum of all subharmonic minorants of $ \varphi $ on $ X$ and that the sets $ \{v_\varphi <t\}$,
$ t\in \mathbf {R}$, are analytic. In this paper, a different method leading to the inf-sup-result establishes at the same time that, in fact, $ v_\varphi $ is the minimum of $ \varphi $ and a subharmonic function, and hence Borel measurable. This is presented in the generality of harmonic spaces, where semipolar sets are polar, and the key tools are measurability results for reduced functions on balayage spaces which are of independent interest.


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Additional Information

Wolfhard Hansen
Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
Email: hansen@math.uni-bielefeld.de

Ivan Netuka
Affiliation: Faculty of Mathematics and Physics, Charles University, Mathematical Institute, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: netuka@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/proc/13688
Keywords: Reduced function, Jensen measure, axiom of polarity
Received by editor(s): November 5, 2016
Received by editor(s) in revised form: January 18, 2017
Published electronically: July 10, 2017
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2017 American Mathematical Society

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