Reduced functions and Jensen measures
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- by Wolfhard Hansen and Ivan Netuka PDF
- Proc. Amer. Math. Soc. 146 (2018), 153-160 Request permission
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.References
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Additional Information
- Wolfhard Hansen
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
- MR Author ID: 194850
- 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
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3723129