Absolute Measurable Spaces

Absolute measurable space and absolute null space are very old topological notions, developed from well-known facts of descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition are the action of the group of homeomorphisms of a space on Borel measures, the Oxtoby-Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem and recent advances on homeomorphic Borel probability measures on the Cantor space, are among the many topics discussed. A brief discussion of set-theoretic results on absolute null space is given, and a four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. First book on absolute measurable space to emphasize topological, geometrical and analytical properties of such spaces. Four-part appendix aids the reader with topics such as topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. Presents complete proofs of the Oxtaby-Ulam theorem, and of several theorems on the existence of uncountable absolute null space.