More on the extension of linear operators on Riesz spaces

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone $E^+$ of a Riesz space $E$ taking values in an Archimedean Riesz space $F$, to the entire space $E$. We prove that, if $E$ has the principal projection property and $F$ is Dedekind $\sigma$-complete then for every $e \in E^+$ every positive finitely additive $F$-valued measure defined on the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ has a unique positive linear extension to the ideal $E_e$ of $E$ generated by $e$. If, moreover, the measure is $\tau$-continuous then the linear extension is order continuous.