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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
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