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Signed quasi-measures and dimension theory. (English) Zbl 0942.28011
Let \(X\) be a compact Hausdorff space. Then, the main result of this paper is that if the covering dimension \(\dim(X)\leq 1\), then every signed quasi-measure on \(X\) extends to a regular Borel signed measure on \(X\) (that is, every quasi-linear functional on \(C(X)\) is linear). In order to prove this, the author presents and uses the following interesting extension theorem of signed quasi-measures: Let \(\mu\) be a signed quasi-measure on \(X\) satisfying that \(\mu(U)+ \mu(V)= \mu(U\cup V)+ \mu(U\cap V)\) for every open subset \(U\) and \(V\) of \(X\). Then \(\mu\) extends to a regular Borel signed measure on \(X\).

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