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Additivity of quasi-measures. (English) Zbl 0907.28007
The main result of this paper is the following theorem: Let \(\mu\) be a quasi-measure on a compact Hausdorff space \(X\), and let \(\{C_n\}_{n\geq 1}\) be a pairwise disjoint collection of closed subsets of \(X\) with \(C= \bigcup_{n\geq 1}C_n\) closed. Then \(\mu(C)= \sum_{n\geq 1}\mu(C_n)\), which induces the result (as its corollary): Every quasi-measure on \(X\) is countably additive (that is, \(\aleph_0\)-additive). In order to show this, the authors present a decomposition theorem of quasi-measures: Every quasi-measure on \(X\) can be decomposed uniquely into the sum of a measure and a proper quasi-measure. They also show that the statement “every quasi-measure on \(X\) is \(\aleph_1\)-additive” is consistent with and independent of ZFC, and construct an example indicating that certain other natural types of additivity of quasi-measures do not hold in general.

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
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