## Arkiv för Matematik

• Ark. Mat.
• Volume 46, Number 1 (2008), 153-182.

### Extreme Jensen measures

Sylvain Roy

#### Abstract

Let Ω be an open subset of Rd, d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫udμ≤u(x) for every superharmonic function u on Ω. Denote by Jx(Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(Jx(Ω)), the set of extreme elements of Jx(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains.

This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53.

As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {αn}n=1 and a continuous function $M\colon\textbf{C}\rightarrow\textbf{R}^+$, there exists an entire function f≢0 satisfying fn)=0 for all n, and |f(z)|≤M(z) for all zC.

#### Article information

Source
Ark. Mat., Volume 46, Number 1 (2008), 153-182.

Dates
Received: 30 March 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907019

Digital Object Identifier
doi:10.1007/s11512-007-0054-9

Mathematical Reviews number (MathSciNet)
MR2379689

Zentralblatt MATH identifier
1182.31004

Rights
2007 © Institut Mittag-Leffler

#### Citation

Roy, Sylvain. Extreme Jensen measures. Ark. Mat. 46 (2008), no. 1, 153--182. doi:10.1007/s11512-007-0054-9. https://projecteuclid.org/euclid.afm/1485907019

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