More on the extension of linear operators on Riesz spaces - Fotiy, O.G.; Gumenchuk, A.I.; Popov, M.M. (Carpathian Mathematical Publications, 2022)
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 σ-complete then for every e E^+ every positive finitely additive F-valued measure defined on the Boolean algebra 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 τ-continuous then the linear extension is order continuous.