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Signed quasi-measures. (English) Zbl 0876.28017
Summary: Let \(X\) be a compact Hausdorff space and let \(\mathcal A \) denote the subsets of \(X\) which are either open or closed. A quasi-linear functional is a map \(\rho :C(X)\rightarrow \mathbf R \) which is linear on singly generated subalgebras and such that \(|\rho(f)|\leq M|f|\) for some \(M<\infty\). There is a one-to-one correspondence between the quasi-linear functional on \(C(X)\) and the set functions \(\mu :\mathcal A \rightarrow \mathbf R \) such that i) \(\mu(\emptyset)=0\), ii) If \(A,B,A\cup B\in \mathcal A \) with \(A\) and \(B\) disjoint, then \(\mu(A\cup B)=\mu(A)+\mu(B)\), iii) There is an \(M<\infty\) such that whenever \(\{U_\alpha \}\) are disjoint open sets, \(\displaystyle \sum |\mu(U_\alpha)|\leq M\), and iv) if \(U\) is open and \(\varepsilon >0\), there is a compact \(K\subseteq U\) such that whenever \(V\subseteq U\setminus K\) is open, then \(|\mu(V)|<\varepsilon\). The space of quasi-linear functionals is investigated and quasi-linear maps between two \(C(X)\) spaces are studied.

28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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