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Measure theory. Mathematical integration --- Measure theory --- Functions of real variables --- Integrals, Generalized --- Integrals, Generalized. --- Measure theory. --- Functions of real variables. --- Fonctions d'une variable réelle --- Calcul intégral --- Fonctions d'une variable reelle --- Calcul integral

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This book sets out to restructure certain fundamentals in measure and integration theory, and thus to fee the theory from some notorious drawbacks. It centers around the ubiquitous task of producing appropriate contents and measures from more primitive data, in order to extend elementary contents and to represent elementary integrals. This task has not been met with adequate unified means so far. The traditional main tools, the Carathéodory and Daniell-Stone theorems, are too restrictive and had to be supplemented by other ad-hoc procedures. Around 1970 a new approach emerged, based on the notion of regularity, which in traditional measure theory is linked to topology. The present book develops the new approach into a systematic theory. The theory unifies the entire context and is much more powerful than the former means. It has striking implications all over measure theory and beyond. Thus it extends the Riesz representation theorem in terms of Randon measures from locally compact to arbitrary Hausdorff topological spaces. It furthers the methodical unification with non-additive set functions, as shown in natural extensions of the Choquet capacitability theorem. The presentation of this research monograph is self-contained, and starts from the beginning. It is addressed to research workers in mathematical analysis and in applications like mathematical economics, and in particular for university teachers in measure and integration theory. The corrected, second printing includes required corrections and appropriate small alterations of the text and a list of the subsequent articles by the author. .

Measure theory. Mathematical integration --- Measure theory. --- Integrals, Generalized. --- Mathematics. --- Measure theory --- Integrals, Generalized --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Functions of real variables. --- Functional analysis. --- Real Functions. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Real variables --- Functions of complex variables --- Mesure, Théorie de la --- Mesure et integration

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The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Per

Henstock-Kurzweil integral. --- Lebesgue integral. --- Calculus, Integral. --- Integral calculus --- Differential equations --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Integrals, Generalized --- Measure theory --- Gauge integral --- Generalized Riemann integral --- Henstock integrals --- HK integral --- Kurzweil-Henstock integral --- Kurzweil integral --- Riemann integral, Generalized --- Integrals, Generalized

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Functional analysis --- Measure theory. --- Integrals, Generalized. --- Analyse fonctionnelle --- Mesure, Théorie de la --- Intégrales généralisées --- Mathematical analysis. --- 517.518.1 --- Measure. Integration. Differentiation --- Functional analysis. --- 517.518.1 Measure. Integration. Differentiation --- Mesure, Théorie de la --- Intégrales généralisées --- Integrals, Generalized --- Measure theory --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Measure algebras --- Rings (Algebra) --- Calculus, Integral --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Number Theory --- Ergodic theory. --- Number theory. --- Dynamics. --- Measure theory. --- Dynamical Systems and Ergodic Theory. --- Number Theory. --- Measure and Integration. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Number study --- Numbers, Theory of --- Algebra --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics