This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction.Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodym theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell's and Caratheodory's methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures.The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction.Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodym theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell's and Caratheodory's methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures.The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction. Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodým theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell’s and Carathéodory’s methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures. The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book originates from the graduate course Theory and Methods of Optimisation taught at the University of Pisa and is primarily intended for students seeking a rigorous yet accessible introduction to optimisation techniques. While designed with graduate students in mind, the text is largely self-contained and may also be approached by motivated undergraduates with a solid foundation in mathematical analysis, linear algebra, and the basic topology of Euclidean spaces. Key results from differential calculus and topology are recalled throughout, ensuring that the material remains accessible without compromising mathematical depth.Structured in three parts, the text offers a coherent progression from foundational theory to algorithmic methods. The first part provides an introduction to convex analysis; the second covers the theory of linear and nonlinear programming; and the third presents key classical algorithms, including the simplex method and gradient-based techniques. Each chapter builds on previous material, with methods presented in detail, including pseudocode and full convergence proofs.Throughout, the book combines theoretical rigour with applied insight. Every result is proved, and numerous worked examples illustrate the methods in action. This dual emphasis gives the work the character of both a rigorous theoretical text and a practical guide to mathematical optimisation.The book serves both as an introduction and as a comprehensive reference for those interested in applying mathematical models to real-world problems. It will be especially valuable to young researchers in applied mathematics looking to understand the theoretical underpinnings of optimisation methods as well as to those working on the practical implementation of such techniques.

This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction.Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodym theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell's and Caratheodory's methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures.The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction.Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodym theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell's and Caratheodory's methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures.The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction. Beyond standard treatments, Measure Theory and Integration features several distinctive elements: some classical results, such as Radon-Nikodým theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed at clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a complete account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell’s and Carathéodory’s methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and shows two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures. The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.

This book originates from the graduate course Theory and Methods of Optimisation taught at the University of Pisa and is primarily intended for students seeking a rigorous yet accessible introduction to optimisation techniques. While designed with graduate students in mind, the text is largely self-contained and may also be approached by motivated undergraduates with a solid foundation in mathematical analysis, linear algebra, and the basic topology of Euclidean spaces. Key results from differential calculus and topology are recalled throughout, ensuring that the material remains accessible without compromising mathematical depth.Structured in three parts, the text offers a coherent progression from foundational theory to algorithmic methods. The first part provides an introduction to convex analysis; the second covers the theory of linear and nonlinear programming; and the third presents key classical algorithms, including the simplex method and gradient-based techniques. Each chapter builds on previous material, with methods presented in detail, including pseudocode and full convergence proofs.Throughout, the book combines theoretical rigour with applied insight. Every result is proved, and numerous worked examples illustrate the methods in action. This dual emphasis gives the work the character of both a rigorous theoretical text and a practical guide to mathematical optimisation.The book serves both as an introduction and as a comprehensive reference for those interested in applying mathematical models to real-world problems. It will be especially valuable to young researchers in applied mathematics looking to understand the theoretical underpinnings of optimisation methods as well as to those working on the practical implementation of such techniques.