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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.

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