Grothendieck measures. (English) Zbl 0681.46030

Khurana, Surjit Singh; Othman, Sadoon Ibrahim

J. Lond. Math. Soc., II. Ser. 39, No. 3, 481-486 (1989).

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

Keywords:

completely regular Hausdorff space; topology of pointwise convergence; space of all Grothendieck measures; locally solid linear topology