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 σ {M_\sigma } denote the space of Baire measures on X. The following is proved: If M σ {M_\sigma } is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of C b ( X ) {C^b}(X) , then M σ {M_\sigma } is a complete locally convex space in which L is dense and whose dual is C b ( X ) {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 {M_C} be the subspace of M σ {M_\sigma } consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If M C {M_C} is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of C ( X ) C(X) , then M C {M_C} is a complete locally convex space in which L is dense and whose dual is C ( X ) 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 ) {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 ) {C^b}(X) , then M is a complete locally convex space in chich L is dense. It is also proved that M σ {M_\sigma } is metrizable if and only if X is discrete and that the metrizability of either M C {M_C} or M is equivalent to X being finite. Finally the following is proved: If M C {M_C} has the Mackey topology for the pair ( M C , C ( X ) ) ({M_C},C(X)) , then M C {M_C} is complete and L is dense in M C {M_C} .