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.
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abNULL absolute Borel space absolute measurable space absolute null space assume BDG space bi-Lipschitzian bijection Borel class Borel measure space Borel set Cantor space card(X cardinal number class 1 map co-analytic collection compact complete Borel measure completely metrizable space continuous continuum hypothesis Corollary countable define definition denoted dimH dimX example exists finite Borel measure fixed FX(X Grzegorek h in HOMEO(X Hausdorff dimension Hausdorff measure Hence Hilbert cube homeomorphism Lebesgue measure Lebesgue-like Lemma Let f Let ” LetX Lipschitzian Lusin set m-convergence measurable map measure ” non-L nonempty open sets Oxtoby–Ulam theorem positive measure positive number proof Proposition proved real-valued function satisfies separable metrizable space sequence SierpiŽnski set singular sets statement subspace support(” Suppose thatX topological copy topological dimension totally imperfect univ universally measurable map universally measurable sets universally null set univM(X whence Zahorski set Zindulka ”-measurable