Affine images of compact convex sets and maximal measures
Let \(X\) and \(Y\) be compact convex subsets of real locally convex Hausdorff topological spaces. Then \(\text{ext\,}X\) denotes the set of extreme points of \(X\). Let \(\varphi: X\to Y\) be a continuous affine mapping, let \({\mathcal M}^1(X)\) denote the set of probability Radon measures on \(X\), and let \(\varphi:{\mathcal M}^1(X)\to{\mathcal M}^1(Y)\) be the mapping induced by \(\varphi\). Then \({\mathcal M}^1_{\max}\) denotes the subset of those measures in \({\mathcal M}^1(X)\) which are maximal or boundary measures. The paper considers the following two questions: {\parindent5mm \begin{itemize}\item[1.] Does the condition \(\varphi(\text{ext\,}X)\subseteq\text{ext\,} Y\) imply that \(\varphi_\sharp({\mathcal M}^1_{\max}(X))\subseteq{\mathcal M}^1_{\max}(Y)\)? \item[2.] Do the conditions \(\varphi(\text{ext\,} X)\subseteq\text{ext\,} Y\), and \(\varphi\) is injective on \(\text{ext\,}X\), imply that \(\varphi_\sharp\) is injective on \({\mathcal M}^1_{\max}(X)\)? \end{itemize}} \textit{S.\,Teleman} [C.\ R.\ Acad.\ Sci., Paris, Sér.\ I 318, No.\,6, 525--528 (1994; Zbl 0830.46006)] stated as a theorem, without proof, that the answer to the first question is `yes', but the present paper gives an example, based on the space \([0,1]\times\{-1,0,1\}\) with a `porcupine' topology of \textit{E.\,Bishop} and \textit{K.\,de Leeuw} [Ann.\ Inst.\ Fourier 9, 305--331 (1959; Zbl 0096.08103)] for which the answer is `no'. Nonetheless, it is proved that the answer is `yes' if \(\text{ext\,}Y\) is a Lindelöf space, and a new proof is given that it is `yes' if \(Y\) is a simplex. It is also proved that the answer to Question 2 is `yes' if \(Y\) is a simplex, but an example is given, again using a porcupine space, for which the amswer is `no'.
scientific article; zbMATH DE number 5597498
scientific article