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Jensen measures and boundary values of plurisubharmonic functions. (English) Zbl 1021.32014
For a bounded domain $$\Omega$$, let $${\mathcal J}_z^c$$ and $${\mathcal J}_z$$ denote the collections of all Jensen measures with barycentre $$z\in\overline\Omega$$ for continuous (resp., upper bounded) plurisubharmonic functions in $$\Omega$$. It is shown that $${\mathcal J}_z$$ can be a proper subset of $${\mathcal J}_z^c$$, however these two classes coincide for all B-regular domains $$\Omega$$. The proof of the latter result is based on an approximation theorem for upper bounded plurisubharmonic functions in a B-regular domain $$\Omega$$ by continuous plurisubharmonic functions on $$\overline\Omega$$. Conversely, if a bounded hyperconvex domain $$\Omega$$ is such that $${\mathcal J}_z ={\mathcal J}_z^c$$ for all $$z$$, then $$\Omega$$ possesses this approximation property. The author gives also an exact characterization, in terms of Jensen measures, of those continuous functions on $$\partial\Omega$$ that are boundary values of continuous plurisubharmonic functions.

##### MSC:
 32U05 Plurisubharmonic functions and generalizations 46E27 Spaces of measures 31C10 Pluriharmonic and plurisubharmonic functions
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##### References:
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