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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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