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(Ebook PDF) An Introduction to Measure Theory 1st Edition by Terence Tao 0821869191 9780821869192 Full chapter

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An Introduction to Measure Theory 1st Edition Terence Tao Digital Instant Download

Author(s): Terence Tao
ISBN(s): 9781470466404, 1470466406
Edition: 1
File Details: PDF, 2.52 MB
Year: 2011
Language: english
SKU: EB-56490206 Category: Tag:

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Product details:

ISBN 10:         0821869191
ISBN 13: 978-0821869192
Author:          Terence Tao

This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini’s theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester’s worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood’s three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.
 

An Introduction to Measure Theory 1st Table of contents:

  1. Preliminaries

    • 1.1 Sets, Functions, and Sequences
    • 1.2 Basic Properties of Functions
    • 1.3 Relations and Equivalence Relations
    • 1.4 Countable and Uncountable Sets
    • 1.5 Cardinality and the Continuum Hypothesis

  2. Sigma-Algebras

    • 2.1 Definition of a Sigma-Algebra
    • 2.2 Examples of Sigma-Algebras
    • 2.3 Borel Sigma-Algebra
    • 2.4 Generated Sigma-Algebras
    • 2.5 Measurable Functions

  3. Measure Spaces

    • 3.1 Definition of a Measure
    • 3.2 Properties of Measures
    • 3.3 Examples of Measures
    • 3.4 Outer Measures and Carathéodory’s Criterion
    • 3.5 Measurable Sets

  4. Integration of Simple Functions

    • 4.1 Simple Functions and Their Integrals
    • 4.2 The Lebesgue Integral of Simple Functions
    • 4.3 Monotone Convergence Theorem for Simple Functions

  5. The Lebesgue Integral

    • 5.1 The Integral of Non-negative Measurable Functions
    • 5.2 The Integral of Bounded Measurable Functions
    • 5.3 The Lebesgue Integral of Complex Functions
    • 5.4 Dominated Convergence Theorem

  6. Convergence Theorems

    • 6.1 Fatou’s Lemma
    • 6.2 Monotone Convergence Theorem
    • 6.3 Dominated Convergence Theorem
    • 6.4 Lebesgue’s Differentiation Theorem

  7. Product Measures

    • 7.1 Product Sigma-Algebras
    • 7.2 Construction of Product Measures
    • 7.3 Fubini’s Theorem

  8. Lp Spaces

    • 8.1 Definition of Lp Spaces
    • 8.2 Norms and Banach Spaces
    • 8.3 Hölder’s Inequality
    • 8.4 Minkowski’s Inequality

  9. Measure and Probability

    • 9.1 Probability Spaces
    • 9.2 Independent Events and Random Variables
    • 9.3 Expectation and Variance
    • 9.4 Law of Large Numbers

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