Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Signed Quasi-Measures


Author: D. J. Grubb
Journal: Trans. Amer. Math. Soc. 349 (1997), 1081-1089
MSC (1991): Primary 28C05
MathSciNet review: 1407700
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Abstract: 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.


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Additional Information

D. J. Grubb
Email: grubb@math.niu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01902-8
Received by editor(s): August 20, 1995
Article copyright: © Copyright 1997 American Mathematical Society