zbMATH — the first resource for mathematics

Probability and quasi-measures – a new interpretation. (English) Zbl 0967.28014
Let \(X\) be a compact Hausdorff space, let \({\mathcal C}\) and \({\mathcal O}\) denote the families of closed and open subsets of \(X\), respectively, and set \({\mathcal A}={\mathcal C}\cup{\mathcal O}\). Let \({\mathcal A}_s\) be the family of all sets \(A\in{\mathcal A}\) such that both \(A\) and its complement \(A^c\) are connected, and put \({\mathcal C}_s={\mathcal A}_s\cap{\mathcal C}\) and \({\mathcal O}_s={\mathcal A}_s\cap{\mathcal O}\). A function \(\nu:{\mathcal A}\to \mathbb{R}^+\) is said to be a quasi-measure if: (i) \(\nu(A)\leq\nu(B)\) whenever \(A,B\in{\mathcal A}\) and \(A\subset B\); (ii) \(\nu(A)= \sum^n_{i=1} \nu(A_i)\) for every \(A\in{\mathcal A}\) and every partition \(\{A_1,\dots, A_n\}\subset{\mathcal A}\) of \(A\); (iii) \(\nu(U)= \sup\{\nu(C):U\supset C\in{\mathcal C}\}\) for any \(U\in{\mathcal O}\). A function \(\mu:{\mathcal A}_s\to \mathbb{R}^+\) is said to be a solid state-measure if: (i) \(\mu(A)+ \mu(A^c)= \mu(X)\) for each \(A\in{\mathcal A}_s\); (ii) \(\sum^n_{i=1}\mu(C_i)\leq \mu(C)\) whenever \(\{C_1,\dots, C_n\}\) is a family of pairwise disjoint sets in \({\mathcal C}_s\) such that \(\bigcup^n_{i=1} C_i\subset C\in{\mathcal C}_s\); (iii) \(\mu(U)= \sup\{\mu(C): U\supset C\in{\mathcal C}_s\}\) for any \(U\in{\mathcal O}_s\).
The main theorem is as follows: Let \(f: [0,1]\to [0,1]\) be a right continuous function such that \(f(0)= 0\), \(f(x-)+ f(x)= 1\), \(0< x\leq 1\), and \(\sum^n_{i=1} f(x_i)\leq f(\sum^n_{i=1} x_i)\) whenever \(x_1,\dots, x_n\in [0,1]\) and \(\sum^n_{i=1} x_i< 1\). Let \(\nu\) be a quasi-measure such that \(\nu(X)= 1\), and define \(\mu:{\mathcal A}_s\to \mathbb{R}^+\) by \(\mu(C)= f(\nu(C))\), \(C\in{\mathcal C}_s\), and \(\mu(U)= 1- \mu(U^c)\), \(U\in{\mathcal O}_s\). If either \(\nu\) is non-splitting (i.e., there are no disjoint sets \(C_1,C_2\in{\mathcal C}_2\) such that \(\nu(C_1)> 0\), \(\nu(C_2)> 0\) and \(\nu(C_1)+ \nu(C_2)= 1\)) or \(f\) is continuous, then \(\mu\) is a solid set-function.

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
60B05 Probability measures on topological spaces
28A10 Real- or complex-valued set functions
Full Text: DOI