Measure and Integration: An Introduction

Introduction
Contents
1 Preliminaries
	1.1 Sets
	1.2 Rings and Algebras of Sets
	1.3 upper F Subscript sigmaFσ and upper G Subscript deltaGδ-sets, Borel Sets
	1.4 Sequence of Sets
	1.5 Cartesian Product
	1.6 Completely Additive Set Function
	1.7 Construction of sigmaσ-Algebra
	1.8 Exercises
2 Lebesgue Measure on Real Line
	2.1 Outer Measure
	2.2 Measurable Sets and Their Properties
	2.3 Nonmeasurable Sets
	2.4 Further Properties of Measurable Sets
	2.5 Vitali\'s Covering
	2.6 Exercises
3 Measurable Functions
	3.1 Measurable Functions
	3.2 Properties of Measurable Function
	3.3 Class of Measurable Functions
	3.4 Further Properties of Measurable Functions. Simple Functions
	3.5 Convergence Theorems for Measurable Functions
	3.6 Exercises
4 More About Sets and Functions
	4.1 Cardinal Members
	4.2 The Cardinal Numbers ModifyingBelow a With quotation dashunderlinea and ModifyingBelow c With quotation dashunderlinec
	4.3 Properties of Set
	4.4 Cantor Sets
	4.5 Cantor Ternary Set
	4.6 Cardinality of the sigmaσ–Algebra of All Borel Sets
	4.7 Cardinality of the sigmaσ–Algebra of All Lebesgue Measurable Sets
	4.8 Cardinality of the Class of All Nonmeasurable Sets
	4.9 Cantor Function
	4.10 Lebesgue Function
	4.11 Properties of Cantor Function
	4.12 Associated Cantor Function and Its Inverse Function and Some Consequences
	4.13 Functions Similar to Cantor Function
	4.14 Exercises
5 The Lebesgue Integral
	5.1 Lebesgue Integral of Bounded Functions
	5.2 Properties of the Lebesgue Integral of Bounded Functions
	5.3 Lebesgue\'s Criterion for Riemann Integrability
	5.4 Riemann Integrability of Bounded Derivatives. Volterra Function
	5.5 Lebesgue Integral of Unbounded Function
	5.6 Properties of Lebesgue Integral
	5.7 Convergence Theorems for Lebesgue Integral
	5.8 Lebesgue Integral of Functions on Sets of Infinite Measure
	5.9 Improper Riemann Integral and Lebesgue Integral
	5.10 Newton Integral and Lebesgue Integral
	5.11 Conclusion
	5.12 Exercises
6 Differentiation of Functions
	6.1 Limits of a Function and Their Properties
	6.2 Derivates of a Function and Their Properties
	6.3 Measurablity of Dini Derivates
	6.4 Differentiability of Monotone Functions
	6.5 Functions of Bounded Variation and Their Properties
	6.6 Absolutely Continuous Functions
	6.7 Monotonicity Theorems and Their Consequences
	6.8 The Indefinite Lebesgue Integral
	6.9 Characterization of Indefinite Lebesgue Integral and Indefinite Riemann Integral
	6.10 Integration by Parts for Lebesgue Integral
	6.11 Change of Variable for Lebesgue Integral
	6.12 Lebesgue Set
	6.13 Singular Function
	6.14 Points of Density and Approximate Continuity
	6.15 Properties of Approximately Continuous Function
	6.16 Exercises
7 Lebesgue Measure and Integration in double struck upper R Superscript upper NmathbbRN
	7.1 Structure of Open Sets in double struck upper R squaredmathbbR2
	7.2 Lebesgue Outer Measure and Measure in double struck upper R Superscript upper NmathbbRN
	7.3 Lebesgue Measurable Function in double struck upper R Superscript upper NmathbbRN
	7.4 Lebesgue Integral in double struck upper R Superscript upper NmathbbRN
	7.5 Exercises
8 General Measure and Outer Measure
	8.1 Measure
	8.2 Outer Measure
	8.3 Outer Measure Induced by a Measure
	8.4 Extension of Measure. Interplay Between Measure and Outer Measure
	8.5 Construction of Outer Measure in double struck upper RmathbbR
	8.6 Lebesgue Stieltje\'s Outer Measure and Measure
	8.7 Hausdorff Measure on the Real Line
	8.8 Measurable Function and Integration
	8.9 Measure and Integration in a Product Space
	8.10 Product Measure and Lebesgue Measure in double struck upper R Superscript upper NmathbbRN
	8.11 Integration of Complex Valued Functions
	8.12 Exercises
9 Function Spaces
	9.1 Metric Space and Linear Space
	9.2 The upper L Superscript pLp-spaces
	9.3 Counting Measure and Application
	9.4 Completeness of the upper L Superscript pLp-spaces
	9.5 Other Properties of upper L Superscript pLp-spaces
	9.6 Space of Measurable Functions
	9.7 Spaces of Continuous Functions and Riemann Integrable Functions
	9.8 Convergence Theorems (2)
	9.9 Exercises
10 Signed Measure and Complex Measure
	10.1 Signed Measure
	10.2 Hahn and Jordan Decomposition
	10.3 Integration with Respect to Signed Measure
	10.4 Absolute Continuity of Measures
	10.5 Radon-Nikodym Theorems
	10.6 Application of Radon-Nikodym Theorem
	10.7 Radon Nikodym Derivative
	10.8 Complex Measure and Integration
	10.9 Point Wise Differentiation of Measures
	10.10 Exercises
Appendix  References