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Complete topologies on spaces of Baire measure


Author: R. B. Kirk
Journal: Trans. Amer. Math. Soc. 184 (1973), 1-29
MSC: Primary 28A32; Secondary 60B05
MathSciNet review: 0325913
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Abstract: Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let $ {M_\sigma }$ denote the space of Baire measures on X. The following is proved: If $ {M_\sigma }$ is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of $ {C^b}(X)$, then $ {M_\sigma }$ is a complete locally convex space in which L is dense and whose dual is $ {C^b}(X)$, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let $ {M_C}$ be the subspace of $ {M_\sigma }$ consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If $ {M_C}$ is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of $ C(X)$, then $ {M_C}$ is a complete locally convex space in which L is dense and whose dual is $ C(X)$, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of $ {C^b}(X)$. The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of $ {C^b}(X)$, then M is a complete locally convex space in chich L is dense. It is also proved that $ {M_\sigma }$ is metrizable if and only if X is discrete and that the metrizability of either $ {M_C}$ or M is equivalent to X being finite. Finally the following is proved: If $ {M_C}$ has the Mackey topology for the pair $ ({M_C},C(X))$, then $ {M_C}$ is complete and L is dense in $ {M_C}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0325913-9
Keywords: Measures in topological spaces, D-spaces, equicontinuous families, complete locally convex topologies, approximation of measures
Article copyright: © Copyright 1973 American Mathematical Society