Signed Quasi-Measures

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, $\sum |\mu (U_\alpha )|\leq M$, 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.