Measure and Integration by Nair, M. Thamban -- Read -- Imperial Library of Trantor

Index

Cover Half Title Title Page Copyright Page Contents Preface Author Note to the Reader 1. Review of Riemann Integral

1.1 Definition and Some Characterizations 1.2 Advantages and Some Disadvantages 1.3 Notations and Conventions

2. Lebesgue Measure

2.1 Lebesgue Outer Measure 2.2 Lebesgue Measurable Sets 2.3 Problems

3. Measure and Measurable Functions

3.1 Measure on an Arbitrary σ-Algebra

3.1.1 Lebesgue measure on ℝk 3.1.2 Generated σ-algebra and Borel σ-algebra 3.1.3 Restrictions of σ-algebras and measures 3.1.4 Complete measure space and the completion 3.1.5 General outer measure and induced measure

3.2 Some Properties of Measures 3.3 Measurable Functions

3.3.1 Probability space and probability distribution 3.3.2 Further properties of measurable functions 3.3.3 Sequences and limits of measurable functions 3.3.4 Almost everywhere properties

3.4 Simple Measurable Functions

3.4.1 Measurability using simple measurable functions 3.4.2 Incompleteness of Borel σ-algebra

3.5 Problems

4. Integral of Positive Measurable Functions

4.1 Integral of Simple Measurable Functions 4.2 Integral of Positive Measurable Functions

4.2.1 Riemann integral as Lebesgue integral 4.2.2 Monotone convergence theorem (MCT) 4.2.3 Radon-Nikodym theorem 4.2.4 Conditional expectation

4.3 Appendix: Proof of the Radon-Nikodym Theorem 4.4 Problems

5. Integral of Complex Measurable Functions

5.1 Integrability and Some Properties

5.1.1 Riemann integral as Lebesgue integral 5.1.2 Dominated convergence theorem (DCT)

5.2 Lp Spaces

5.2.1 Hölder’s and Minkowski’s inequalities 5.2.2 Completeness of Lp(μ) 5.2.3 Denseness of Cc(Ω) in Lp(Ω) for 1 ≤ p < ∞

5.3 Fundamental Theorems

5.3.1 Indefinite integral and its derivative 5.3.2 Fundamental theorems of Lebesgue integration

5.4 Appendix 5.5 Problems

6. Integration on Product Spaces

6.1 Motivation 6.2 Product σ-algebra and Product Measure 6.3 Fubini’s Theorem 6.4 Counter Examples

6.5 Problems

7. Fourier Transform

7.1 Fourier Transform on L1(ℝ)

7.1.1 Definition and some basic properties 7.1.2 Fourier transform as a linear operator 7.1.3 Fourier inversion theorem

7.2 Fourier-Plancherel Transform 7.3 Problems

Bibliography Index