This is a set of lecture notes which present an economical development of measure theory and integration in locally compact Hausdorff spaces. We have tried to illuminate the more difficult parts of the subject. The Riesz-Markov theorem is established in a form convenient for applications in modern analysis, including Haar measure on locally compact groups or weights on C∗-algebras...though appl...

Let G be any Hausdorff topological group and let βGX denote the maximal G-compactification of a G-Tychonoff space X (i.e., a Tychonoff G-space possessing a G-compactification). Recall that a completely regular Hausdorff topological space is called pseudocompact if every continuous function f : X →R is bounded. In this paper, we prove that if X and Y are two G-Tychonoff spaces such that the prod...

By Gromov’s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Am...

We introduce a new quantum Gromov-Hausdorff distance between C∗-algebraic compact quantum metric spaces. Because it is able to distinguish algebraic structures, this new distance fixes a weakness of Rieffel’s quantum distance. We show that this new quantum distance has properties analogous to the basic properties of the classical Gromov-Hausdorff distance, and we give criteria for when a parame...

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

It is shown that the space X [0,1], of continuous maps [0, 1] → X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T1-space X, it follows that X [0,1] is locally compact if and only if X is locally compact and totally path-disconnected. AMS Classification: 54C35, 54E45, 55P35, 18B30, 18D15