Contents

Preface

Author

Note to the Reader

1Review of Riemann Integral

1.1Definition and Some Characterizations

1.2Advantages and Some Disadvantages

1.3Notations and Conventions

2Lebesgue Measure

2.1Lebesgue OuterMeasure

2.2LebesgueMeasurable Sets

2.3Problems

3Measure and Measurable Functions

3.1Measure on an Arbitrary σ-Algebra

3.1.1Lebesgue measure on ℝ^k

3.1.2Generated σ-algebra and Borel σ-algebra

3.1.3Restrictions of σ-algebras and measures

3.1.4Complete measure space and the completion

3.1.5General outer measure and induced measure

3.2Some Properties of Measures

3.3Measurable Functions

3.3.1Probability space and probability distribution

3.3.2Further properties of measurable functions

3.3.3Sequences and limits of measurable functions

3.3.4Almost everywhere properties

3.4Simple Measurable Functions

3.4.1Measurability using simple measurable functions

3.4.2Incompleteness of Borel σ-algebra

3.5Problems

4Integral of Positive Measurable Functions

4.1Integral of Simple Measurable Functions

4.2Integral of Positive Measurable Functions

4.2.1Riemann integral as Lebesgue integral

4.2.2Monotone convergence theorem(MCT)

4.2.3Radon-Nikodym theorem

4.2.4Conditional expectation

4.3Appendix: Proof of the Radon-Nikodym Theorem

4.4Problems

5Integral of Complex Measurable Functions

5.1Integrability and Some Properties

5.1.1Riemann integral as Lebesgue integral

5.1.2Dominated convergence theorem(DCT)

5.2L^p Spaces

5.2.1Hölder’s and Minkowski’s inequalities

5.2.2Completeness of L^p(μ)

5.2.3Denseness of C_c(Ω) in L^p(Ω) for 1 ≤ p < ∞

5.3Fundamental Theorems

5.3.1Indefinite integral and its derivative

5.3.2Fundamental theorems of Lebesgue integration

5.4Appendix

5.5Problems

6Integration on Product Spaces

6.1Motivation

6.2Product σ-algebra and Product Measure

6.3Fubini’s Theorem

6.4Counter Examples

6.4.1σ-finiteness condition cannot be dropped

6.4.2Product of complete measures need not be complete

6.5Problems

7Fourier Transform

7.1Fourier Transform on L¹(ℝ)

7.1.1Definition and some basic properties

7.1.2Fourier transform as a linear operator

7.1.3Fourier inversion theorem

7.2Fourier-Plancherel Transform

7.3Problems

Bibliography

Index