Complete topologies on spaces of Baire measure

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 ${M_C}$.