A generalization of Lyapounov's theorem
Let \((\Omega,{\mathcal F})\) be a measurable space and let \(\vec\mu=(\mu_ 1,\dots,\mu_ n)\) be a vector measure on \(\mathcal F\) with non-negative finite coordinates. Define \({\mathcal M}{\mathcal R}_ k(\vec\mu)\) and \({\mathcal P}{\mathcal R}(\vec\mu)\) as \[ \Bigl\{(\mu_ i(P_ j))^{n\;\;\;k}_{i=1\;j=1}: \{P_ j\}^ k_{j=1}\text{ is a measurable partition of }\Omega\Bigr\}, \] \[ \Bigl\{(\mu_ 1(P_ 1),\dots,\mu_ n(P_ n)): \{P_ j\}^ n_{j=1}\text{ is a measurable partition of }\Omega\Bigr\}, \] respectively. Assuming that the atomic parts of \(\mu_ i\), \(i=1,2,\dots\), are mutually singular, the author gives necessary and sufficient conditions in order that \({\mathcal M}{\mathcal R}_ k(\vec\mu)\) respectively \({\mathcal P}{\mathcal R}(\vec\mu)\) be convex (Theorems 2.7 and 2.8). The proofs are based on a classical result due to \textit{A. Dvoretzky}, \textit{A. Wald}, and \textit{J. Wolfowitz} [Pac. J. Math. 1, 59-74 (1951; Zbl 0044.150)] which asserts that \({\mathcal M}{\mathcal R}_ k(\vec\mu)\) is convex (and compact) if \(\vec\mu\) is non-atomic. \{Reviewer's remarks: (1) Theorem 2.5 and Lemma 2.6 become obvious if one introduces \(\nu=\mu_ 1+\cdots+ \mu_ n\) and uses \(d\mu_ i/d\nu\) to define the measures and sets in question. Also, the proofs of the necessary parts of Theorems 2.7 and 2.8 can be simplified by using \(d\mu_ i/d\nu\). (2) The proofs of the sufficiency parts of Theorems 2.7 and 2.8 can be simplified with the help of the formulas \[ \begin{aligned} {\mathcal M}{\mathcal R}_ k(\vec\mu\bigl|_{F_ 2\cup F_ 2}) &={\mathcal M}{\mathcal R}_ k(\vec\mu\bigr|_{F_ 1})+{\mathcal M}{\mathcal R}_ k(\vec\mu\bigl|_{F_ 2}),\\ {\mathcal P}{\mathcal R}(\vec\mu\bigl|_{F_ 1\cup F_ 2}) &= {\mathcal P}{\mathcal R}(\vec\mu\bigr|_{F_ 1})+{\mathcal P}{\mathcal R}(\vec\mu\bigl|_{F_ 2}),\end{aligned} \] where \(F_ 1\) and \(F_ 2\) are arbitrary disjoint measurable sets. (3) For additional literature on the subject see Zbl 0765.28010\}.
scientific article; zbMATH DE number 469228
scientific article