Liapounov's convexity theorem for topological measures
type: Item
instance of: scholarly article
title: Liapounov's convexity theorem for topological measures
zbMATH Open document ID: 0854.28004
DOI: 10.1007/BF01197598
author: Hermann Render
author: Heinrich Stroetmann
published in: Archiv der Mathematik
publication date: 1996-10-27T00:00:00Z
review text: Let \(\mu_i\) be non-atomic Borel measures on a separable metric space for \(i = 1,\dots, n\). Then the restricted range \(\{(\mu_1(U),\dots,\mu_n(U))\): \(U\) open\} is a convex set which is in general not closed. On the other side, the range is closed if the measures \(\mu_i\), defined over an open or closed interval, possess real-analytic densities for an appropriate dominating non-atomic measure \(\nu\). If the interval is compact there exists a natural number \(N\) such that for each Borel set \(B\) there exist intervals \(I_1,\dots, I_m\), \((m \leq N)\) such that \(\mu_i(B) = \mu_i(\bigcup^m_{k = 1} I_k)\) for all \(i = 1,\dots,n\).
reviewed by: Hermann Render
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46A22
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 938610
zbMATH Keywords: topological measures
zbMATH Keywords: vector measures
zbMATH Keywords: Liapunov's convexity theorem
zbMATH Keywords: non-atomic Borel measures
zbMATH Keywords: separable metric space
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01197598
OpenAlex ID: W2032739830
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description: scientific article; zbMATH DE number 938610