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**Order convergence of vector measures on topological spaces.**
*(English)*
Zbl 1199.28008

Summary: Let \(X\) be a completely regular Hausdorff space, \(E\) a boundedly complete vector lattice, \(C_{b}(X)\) the space of all, bounded, real-valued continuous functions on \(X\), \(\mathcal {F}\) the algebra generated by the zero-sets of \(X\), and \(\mu \: C_{b}(X) \to E\) a positive linear map. First, we give a new proof that \(\mu \) extends to a unique, finitely additive measure \(\mu \: \mathcal {F} \to E^{+}\) such that \(\nu \) 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 \(\mathcal {F}\) are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov theorem about convergent sequences of \(\sigma \)-additive measures is extended to the case of order convergence.

### MSC:

28A33 | Spaces of measures, convergence of measures |

28B15 | Set functions, measures and integrals with values in ordered spaces |

28C05 | Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures |

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

46G10 | Vector-valued measures and integration |