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This classroom-tested text is intended for a one-semester course in Lebesgue’s theory.  With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students.  The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis. In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text.  The presentation is elementary, where σ-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book.

Lebesgue integral --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Lebesgue integral --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Measure theory. --- Functions of real variables. --- Measure and Integration. --- Real Functions. --- Analysis. --- Lebesgue integral. --- -Integrals, Generalized --- Measure theory --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Math --- Science --- 517.1 Mathematical analysis --- Mathematical analysis --- Real variables --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)

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This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis. Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra. Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

Functions of real variables. --- Measure theory. --- Sequences (Mathematics). --- Mathematical logic. --- Mathematical analysis. --- Analysis (Mathematics). --- Real Functions. --- Measure and Integration. --- Sequences, Series, Summability. --- Mathematical Logic and Foundations. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Mathematical sequences --- Numerical sequences --- Algebra --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Real variables --- Functions of complex variables --- Ordered fields. --- Topological fields

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This classroom-tested text is intended for a one-semester course in Lebesgue’s theory.  With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students.  The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis. In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text.  The presentation is elementary, where σ-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book.

Mathematics --- Differential geometry. Global analysis --- Mathematical analysis --- Mathematical physics --- differentiaalvergelijkingen --- analyse (wiskunde) --- statistiek --- wiskunde

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• Reference Manager
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This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis. Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra. Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

Mathematical logic --- Functional analysis --- Differential equations --- Mathematical analysis --- Mathematics --- Measuring methods in physics --- Mathematical physics --- differentiaalvergelijkingen --- analyse (wiskunde) --- reeksen (wiskunde) --- mathematische modellen --- meettechniek --- wiskunde --- logica

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Mathematical logic --- Functional analysis --- Differential equations --- Mathematical analysis --- Mathematics --- Measuring methods in physics --- Mathematical physics --- differentiaalvergelijkingen --- analyse (wiskunde) --- reeksen (wiskunde) --- mathematische modellen --- meettechniek --- wiskunde --- logica