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Grothendieck measures. (English) Zbl 0681.46030
Let X be a completely regular Hausdorff space, \(C_ b(X)\) the space of all scalar-valued continuous bounded functions on X. Denote by \(t_ b\) the topology of pointwise convergence on subsets of \(C_ b(X)\) and by \(M_ g(X)\) the space of all Grothendieck measures on X, i.e. \(\mu \in M_ g(X)\) iff \(\mu\) is a continuous linear functional on \((C_ b(X),\| \|_{\infty})\) and the restriction of \(\mu\) to each absolutely convex \(t_ p\)-compact subset of \(C_ b(X)\) is \(t_ b\)- continuous.
The authors introduce and examine a locally solid linear topology \(\beta_ g\) on \(C_ b(X)\) such that \(M_ g(X)\) is the continuous dual of \((C_ b(X),\beta_ g)\). Some results are extended to \(C_ b(X,E)\), where E is a Banach space.
Reviewer: H.Weber

MSC:
46E27 Spaces of measures
46E40 Spaces of vector- and operator-valued functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
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