Pathologies of Compact Nonmetric Spaces

Recall that a topological space is called locally compact iff every point in it is contained in some open set with compact closure. For some time, locally compact spaces were considered the most natural spaces on which to study general measures and integrals. The regularity extension, treated in §7.3, appeared as a primary advantage of measures on compact or locally compactspaces. Local compactness also fits well with group structures. A topological group is a group G with a Hausdorff topology  for which the group operations 〈g, h〉 ↦ gh and g ↦ g −1 are continuous. A locally compact Hausdorff topological group has left and right Haar measures , which are Radon measures (finite on compact sets) invariant under all left or right translations g ↦ hg or g ↦ gh , respectively. The theory of measures on locally compact spaces, particularly groups, occupied the last three of the twelve chapters of the classic text of Halmos (1950). Bourbaki (1952–1969) put even more emphasis on locally compact spaces. The main purpose of this appendix is to indicate why locally compact spaces have had less attention in this book than in those of Halmos and Bourbaki. Another class of spaces for measures theory is complete separable metric spaces, or topological spaces which can be metrized to be separable and complete (Polish spaces). Among such spaces, for example, are separable, infinite-dimensional Banach spaces, none of which is locally compact.