Additivity of quasi-measures

We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are $\aleph _1$-additive. Finally, we construct an example that places strong restrictions on other forms of additivity.