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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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