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**Weak compactness of vector measures on topological spaces.**
*(English)*
Zbl 1164.60002

Let \(X\) be a completely regular Hausdorff space and \(E\) be a quasi-complete locally convex space. In the paper under review, the author studies the weak compactness in the spaces of tight, \(\tau\)-smooth and \(\sigma\)-smooth \(E\)-valued measures on \(X\). A typical result (which answers in the positive a conjecture raised by J. Kawabe [Georgian Math. J. 8, No. 2, 283–295 (2001; Zbl 1023.28008)]): Let \(C_b(X)\) be the space of bounded, scalar valued, continuous functions on \(X\) and \(S\) be its closed unit ball. An \(E\)-valued measure \(\mu\) is called tight if for any uniformly bounded net \(\{f_{\alpha}\}\subset C_b(X)\), \(f_{\alpha}\to 0\) uniformly on compact subsets of \(X\), we have \(\mu(f_{\alpha})\to 0\). Denote by \(M_t(X,E)\) the space of tight measures. Then for a subset \(H\subset M_t(X,E)\) there is an absolutely convex compact subset \(B\) of \(E\) such that \(\mu(S)\subset B\), \(\forall\mu\in H\). If \(H\) is weakly uniformly tight, then it is uniformly tight.

Reviewer: Anatolij M. Plichko (Krakow)

### MSC:

60B05 | Probability measures on topological spaces |

46G10 | Vector-valued measures and integration |

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

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |