SOME APPLICATIONS OF THE THEORY OF COMPACTIFICATIONS OF TOPOLOGICAL SPACES AND MEASURE THEORY

Abstract

CONTENTS Introduction Chapter I. Preliminary information from the compactification theory of topological spaces § 1. The concept of a topological compactification § 2. Some methods of obtaining compactifications § 3. Some special compactifications Chapter II. The compactification theory of topological spaces and the integral representation of linear functionals § 1. The Riesz representation theorem for compact and locally compact spaces § 2. A functional representation by measures on compactifications § 3. An integral representation of functionals on the space of all continuous functions § 4. An integral representation of functionals on spaces of continuous functions of bounded support § 5. An integral representation of functionals on spaces without a topology § 6. The problem of extending measures with values in vector lattices Chapter III. Invariant Banach limits § 1. The classical statement of the problem § 2. The problem in a more general topological setting § 3. The support of the representative measure of an invariant permanent functional Chapter IV. Summability theory § 1. The classical statement of the problem § 2. The Stone-Čech compactification and multiplicative summability methods § 3. The Stone-Čech compactification and compatibility criteria § 4. The Stone-Čech compactification of non-discrete countable spaces and summability § 5. Summability problems in the more general setting of compactification theory and measure theory References