# zbMATH — the first resource for mathematics

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.

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