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.


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


Zbl 1023.28008