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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.

##### MSC:
 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
##### Keywords:
solid state-measure; quasi-measure; solid set-function
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