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Let X be a completely regular Hausdorff space, E a boundedly complete vector lattice, Cb(X) the space of all, bounded, real-valued continuous functions on X, F the algebra generated by the zero-sets of X, and μ : Cb(X) → E a positive linear map. First we give a new proof that μ extends to a unique, finitely additive measure μ : F → E such that ν is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of E-valued finitely additive measures on F are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of σ-additive measures is extended to the case of order convergence.

Order convergence of vector measures on topological spaces