This book addresses the need for an accessible comprehensive exposition
of the theory of uniform measures; the need that became more critical
when recently uniform measures reemerged in new results in abstract
harmonic analysis. Until now, results about uniform measures have been
scattered through many papers written by a number of authors, some
unpublished, written using a variety of definitions and notations.
Uniform measures are certain functionals on the space of bounded
uniformly continuous functions on a uniform space. They are a common
generalization of several classes of measures and measure-like
functionals studied in abstract and topological measure theory,
probability theory, and abstract harmonic analysis. They offer a natural
framework for results about topologies on spaces of measures and about
the continuity of convolution of measures on topological groups and
semitopological semigroups. The book is a reference for the theory of
uniform measures. It includes a self-contained development of the theory
with complete proofs, starting with the necessary parts of the theory of
uniform spaces. It presents diverse results from many sources organized
in a logical whole, and includes several new results. The book is also
suitable for graduate or advanced undergraduate courses on selected
topics in topology and functional analysis. The text contains a number
of exercises with solution hints, and four problems with suggestions for
further research.
