Signed quasi-measures and dimension theory

A quasi-linear functional on $C(X)$ is a real-valued function that is linear on each closed, singly generated subalgebra and is norm bounded. We show that if the covering dimension $\dim X\leq 1$, then every quasi-linear functional on $C(X)$ is, in fact, linear. We do this by considering an associated set function, called a quasi-measure, and ask when such a set function can be extended to be a measure.