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Extreme topological measures. (English) Zbl 1116.28011
The author proves that under certain conditions, extreme topological measures are dense in the set of all topological measures on a compact Hausdorff space. The main terminology and notations are: $$X$$ is a connected and locally connected compact Hausdorff space. $$A \subset X$$ is solid if $$A$$ and $$X \setminus A$$ are connected; $$X$$ has genus $$0$$ if $$U \setminus C$$ is connected for any open connected U and solid closed set $$C \subset U$$. $$X$$ is assumed to have genus $$0$$. $$\mathcal{A}_{s}$$ denotes the collection of solid open and solid closed sets in $$X$$. A compact, connected and locally connected space having genus $$0$$ is called a $$q$$ space. All topological measures $$\mu$$ are assumed to be normalized, that is $$\mu(X)=1$$. A topological measure $$\mu$$ is said to be finitely defined if there is a finite set $$F$$ such that for any $$\{A, A_{1}, A_{2}, \dots, A_{n} \} \subset \mathcal{A}_{s}$$ with $$\{ A_{i} \cap F \}$$ mutually disjoint and $$\bigcup (A_{i} \cap F) \subset A$$, we get $$\sum_{i=1}^{n} \mu(A_{i}) \leq \mu(A)$$. A family $$\mathcal{F}$$ of non-empty subsets of $$X$$ has order $$m$$ if $$\mathcal{F}$$ contains a collection of $$m$$ disjoint sets and no collection of $$m+1$$ disjoint sets. For a collection $$\mathcal{C}= \{ I_{1}, I_{2}, \dots, I_{k} \}$$ of distinct sets and $$m \leq k$$, $$\mathcal{C}$$ is called $$m$$-disjoint $$k$$-chain if $$\mathcal{C}$$ has order $$m$$ and for each $$i= 1, 2, \dots, k$$, the sets $$\{ I_{i+1}, I_{i+2}, \dots, I_{i+m} \}$$ (indices are mod $$k$$) are disjoint. An $$m$$-disjoint $$(2m+1)$$-chain is called an $$m$$-chain. If $$\mathcal{S}= \{ S_{1}, \dots, S_{2m+1} \}$$ is a disjoint family of sets, the $$m$$-chain $$\{ J_{i}=S_{2i-1} \cup S_{2i}: i=1, 2, \dots, 2m+1 (\text{ mod }2m+1) \}$$ is called basic $$m$$-chain. A finite set $$F$$ is solidifiable if, given an open connected set $$A \subset X$$, there exists a closed solid set $$C\subset X$$ such that $$F \cap A \subset C \subset A$$. An $$m$$-chain $$\mathcal{C}= \{ C_{1}, C_{2}, \dots, C_{2m+1} \}$$ is called a solid extension of $$m$$-chain $$\mathcal{J}= \{ J_{1}, J_{2}, \dots, J_{2m+1} \}$$ of finite sets if $$C_{i}$$ are solid closed sets and $$C_{i} \cap (\bigcup \mathcal{J}) = J_{i}, \; \forall i$$.
The main result is: Let $$X$$ be a $$q$$-space and $$Y$$ a dense open subset of $$X$$ such that any finite set of $$Y$$ is solidifiable and every basic chain of finite sets of $$Y$$ has a solid extension. Then extreme topological measures are dense in the set of all topological measures on $$X$$. As a corollary, it is proved that under the same conditions, extreme finitely defined topological measures are dense in the set of all topological measures on $$X$$.
##### MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 46E27 Spaces of measures 54F65 Topological characterizations of particular spaces
##### Keywords:
$$m$$-chains; solid sets; solidifiable sets
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