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Quasi-measures and dimension theory. (English) Zbl 0842.28005
A normal space \(X\) is said to be an \(A\)-space (resp. \(MA\)-space) if every quasi-linear (multiplicative quasi-linear) functional on \(C_b(X)\), the space of bounded continuous real-valued functions on \(X\), is linear. The purpose of this paper is to give a sufficient condition for a normal space \(X\) to be an \(A\)-space or an \(MA\)-space in terms of classical notions of dimension theory such as the large inductive dimension \(\text{Ind}(X)\) or the Lebesgue covering dimension \(\dim(X)\). Concerning this, the author shows that if \(X\) is a normal space and \(\text{Ind}(X)\leq 1\) (resp. \(\dim(X)\leq 1\)), then \(X\) is an \(A\)-space (\(MA\)-space). In other words, it is shown that for normal spaces \(X\) with \(\text{Ind}(X)\leq 1\) (resp. \(\dim(X)\leq 1\)), every quasi-measure (\(\{0, 1\}\)-valued quasi-measure) on \(X\) admits a finitely additive extension to the smallest algebra of subsets of \(X\) which contains the closed sets.

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
54F45 Dimension theory in general topology
Full Text: DOI
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