The Big Book of Real Analysis: From Numbers to Measures

Preface
	Overview
	Course Structure Plan
	Alternative Course Structure Plans
	Final Words
Contents
List of Figures
1 Logic and Sets
	1.1 Introduction to Logic
		And, Or, Not
		Conditional Statement
		Modus Ponens and Modus Tollens
	1.2 Proofs
	1.3 Sets
		Set Algebra
		Power Sets and Cartesian Product
	1.4 Quantifiers
	1.5 Functions
		Image and Preimage
		Injection, Surjection, Bijection
		Composite, Inverse, Restriction Functions
	Exercises
2 Integers
	2.1 Relations
		Equivalence Relation
	2.2 Natural Numbers N
		Algebra of Natural Numbers
		Principle of Mathematical Induction
	2.3 Ordering on N
		Factors and Divisors
	2.4 Integers Z
	2.5 Algebra on Z
	2.6 Ordering on Z
	Exercises
3 Construction of Real Numbers
	3.1 Rational Numbers Q
	3.2 Algebra on Q
	3.3 Ordering on Q
		Archimedean Property of Q
	3.4 Cardinality
		Cardinality of a Set
		Cardinality of Q
	3.5 Irrational Numbers
	3.6 Bounds, Supremum, and Infimum
		Bounds
		Supremum and Infimum
		Completeness Axiom
	3.7 Dedekind Cuts
	3.8 Algebra and Ordering of Dedekind Cuts
		The Real Numbers
	Exercises
4 Real Numbers
	4.1 Properties of Real Numbers R
		Supremum, Infimum, Minimum, Maximum
	4.2 Exponentiation
		Rational Exponents
		Irrational Exponents
	4.3 Logarithm
	4.4 Decimal Representation of the Real Numbers
		Decimal Representation for Rational Numbers
		Decimal Representation of Irrational Numbers
		Cardinality of R
	4.5 Topology on R
		Intervals
		Open and Closed Sets
		Compact Sets
	4.6 Real n-Space and Complex Numbers
		Real n-Space
		Complex Numbers
		Topology on Rn and C
	Exercises
5 Real Sequences
	5.1 Algebra of Real Sequences
	5.2 Limits and Convergence
		Bounded Sequences
		Convergent Sequences
	5.3 Blowing up to Infinity
	5.4 Monotone Sequences
	5.5 Subsequences
		Bolzano-Weierstrass Theorem
	5.6 Comparing Sequences
	5.7 Asymptotic Notations
		Big-O and Little-o Notations
	5.8 Cauchy Sequences
	5.9 Algebra of Limits
	5.10 Limit Superior and Limit Inferior
	Exercises
6 Some Applications of Real Sequences
	6.1 Circular Arclength
		Approximating Arclength
		Value of π
		Radians
	6.2 Limit Points and Topology
	6.3 Sequences in C and Rn
	6.4 Introduction to Metric Spaces
	Exercises
7 Real Series
	7.1 Partial Sums
	7.2 Convergent Series
		Algebra of Series
		Monotone Series
	7.3 Absolute and Conditional Convergence
	7.4 Alternating Series
	7.5 Comparison Tests
		Direct Comparison Test
		Limit Comparison Test
	7.6 Ratio and Root Tests
		Ratio Test
		Root Test
		Generalised Ratio and Root Tests
	7.7 Raabe\'s Test
	7.8 Dirichlet\'s and Abel\'s Tests
	Exercises
8 Additional Topics in Real Series
	8.1 Rearrangement of Series
	8.2 Bracketing of Series
	8.3 Cauchy Product
	Exercises
9 Functions and Limits
	9.1 Algebra of Real-Valued Functions
	9.2 Limit of a Function
	9.3 One-Sided Limits
	9.4 Blowing Up and Limits at Infinity
		Blowing Up to ∞
		Limits at ∞
	9.5 Algebra of Limits
	9.6 Asymptotic Notations
	Exercises
10 Continuity
	10.1 Continuous Functions
	10.2 Algebra of Continuous Functions
	10.3 One-Sided Continuity
	10.4 Intermediate Value Theorem
	10.5 Extreme Value Theorem
	10.6 Uniform and Lipschitz Continuity
		Uniform Continuity
		Lipschitz Continuity
		Relationship Between Different Types of Continuities
	Exercises
11 Functions Sequence and Series
	11.1 Pointwise Convergence
	11.2 Uniform Convergence
	11.3 Consequences of Uniform Convergence
	11.4 Functions Series
		Pointwise Convergence of Functions Series
		Uniform Convergence of Functions Series
		Dirichlet\'s and Abel\'s Tests for Functions Series
	Exercises
12 Power Series
	12.1 Convergence of Power Series
		Radius of Convergence
		Domain of Convergence
		Finding Radius of Convergence
	12.2 Continuity of Power Series
	12.3 Algebra of Power Series
	12.4 Exponentiation and Logarithm Revisited
	Exercises
13 Differentiation
	13.1 Derivatives
	13.2 Algebra of Derivatives
	13.3 Differentiable Functions
	13.4 Implicit Differentiation
	13.5 Extremum and Critical Points
	13.6 Rolle\'s Theorem and Mean Value Theorems
	13.7 Inverse Function Theorem
	Exercises
14 Some Applications of Differentiation
	14.1 Graph Sketching
		Monotonicity of Functions
		Convexity of Functions
		Graph Sketching
	14.2 Differentiation and Limits
		Differentiation of Function Sequence
		Differentiation of Functions Series
	14.3 L\'Hôpital\'s Rule
	14.4 Introduction to Differential Equations
		Antiderivatives
		Ordinary Differential Equations
		First Order ODEs
		Initial/Boundary Value Problem
		Second Order ODEs
	Exercises
15 Riemann and Darboux Integration
	15.1 Step Functions
	15.2 Riemann Integrals
	15.3 Darboux Integrals
		Lower and Upper Sums
		Darboux Integral
	15.4 Correspondence between Riemann and Darboux Integrals
	15.5 Properties of Riemann Integrals
	15.6 Some Sufficient Conditions for Riemann Integrability
	Exercises
16 Fundamental Theorem of Calculus
	16.1 Fundamental Theorem of Calculus
		Integration by Parts and by Change of Variable
	16.2 Lengths and Volumes
		Arclength
		Solids of Revolution
	16.3 Antiderivatives and Indefinite Integrals
	16.4 Improper Integrals
		Comparison Tests for Improper Riemann Integrals
		Integral Test for Real Series
	16.5 Integration and Limits
		Integrable Limit Theorem
		Monotone and Dominated Convergence Theorems
	Exercises
17 Taylor and Maclaurin Series
	17.1 Taylor Polynomial and Series
	17.2 Taylor Remainder
	17.3 Polynomial Approximation
	Exercises
18 Introduction to Measure
	18.1 Extended Real Numbers
	18.2 π-Systems and Semirings
		π-Systems
		Semirings
	18.3 Rings and Algebras
		Rings and Algebras
		σ-Rings and σ-Algebras
	18.4 Outer Measure
	18.5 Measure
	18.6 Carathéodory Extension Theorem
	18.7 Lebesgue and Borel σ-Algebra
		Lebesgue σ-Algebra
		Borel σ-Algebra
	18.8 Uniqueness of Carathéodory Extension Theorem
	18.9 Measurable Functions
		Limits of Measurable Functions
		Almost-Everywhere Property
	Exercises
19 Lebesgue Integration
	19.1 Simple Functions
	19.2 Integral of Simple Functions
	19.3 Lebesgue Integral of Non-negative Functions
	19.4 Monotone Convergence Theorem
		Fatou\'s Lemmas
		Lebesgue Integral of Non-negative Functions Series
	19.5 Lebesgue Integral
		Approximations of Measurable Functions
	19.6 Convergence Theorems
		Dominated and Bounded Convergence Theorems
		Lebesgue Integrals of Functions Series
	19.7 Comparison Between Lebesgue and Riemann Integrals
		Domain of Integration
		Fundamental Theorem of Calculus
		Equality of Riemann and Lebesgue Integrals
		Improper Integrals
		Limits of Integrals
	Exercises
20 Double Integrals
	20.1 Product Measure Space
		Product Measurable Space
		Product Measure
	20.2 Iterated Integrals
		Lebesgue Integral over X Y
		Sections and Section Functions
		Measurability of Section Functions
		Alternative Formulation of Product Measure
	20.3 Fubini\'s and Tonelli\'s Theorems
	20.4 Multiple Integrals
	Exercises
Hints for Exercises
	Chapter 1: Logic and Sets
	Chapter 2: Integers
	Chapter 3: Construction of Real Numbers
	Chapter 4: Real Numbers
	Chapter 5: Real Sequences
	Chapter 6: Some Applications of Real Sequences
	Chapter 7: Real Series
	Chapter 8: Additional Topics in Real Series
	Chapter 9: Functions and Limits
	Chapter 10: Continuity
	Chapter 11: Functions Sequence and Series
	Chapter 12: Power Series
	Chapter 13: Differentiation
	Chapter 14: Some Applications of Differentiation
	Chapter 15: Riemann and Darboux Integrals
	Chapter 16: Fundamental Theorem of Calculus
	Chapter 17: Taylor and Maclaurin Series
	Chapter 18: Introduction to Measure
	Chapter 19: Lebesgue Integration
	Chapter 20: Double Integrals
Reference
Index