Introduction to Measurement and Integration
Authors
Description
The book is the result of the author’s lecture notes over several years teaching the Measure Theory course in IMPA’s graduate program. The presentation of the material is constructive and self-sufficient. A course in analysis in several variables is sufficient as a prerequisite. The text is presented in a simple manner. The theory is illustrated with complementary examples and each chapter contains a list of exercises to reinforce the theory and often extend some previously developed concepts.
Target audience
Higher education
Name: Introduction to Measurement and Integration
Author(s): e Carlos Isnard
Pages: 314
Publication: IMPA, 2018
ISBN: 978-85-244-0457-3
Edition: 3
Introduction: Riemann x Lebesque
1 Measure
1.1 Conventions
1.2 Semi-Rings
1.3 Rings
1.4 Positive Measures
1.5 Regular Measures
1.6 Additive σ-Measures
1.7 Finite Sets and the Counting Measure (review)
1.8 Enumerable Sets (review)
1.9 Exercises
2 The Integral of Simple Functions and the Higher Integral
2.1 Notations
2.2 Simple Functions
2.3 Vector Spaces of Functions
2.4 Vector Reticulums
2.5 Properties of the Integral of Simple Functions
2.6 The Higher Integral
2.7 Exercises
3 The Lebesque Extension
3.1 Semi-Norms and Norms
3.2 The Quotient Normed Space
3.3 Convergence and Continuity
3.4 The Cauchy Criterion
3.5 Bounded Linear Operators
3.6 σ-Algebras
3.7 The σ-Lebesque Algebra
3.8 Exercises
4 Measurable Sets
4.1 Measure Spaces
4.2 An Example from Borel
4.3 Cantor’s Ternary Set
4.4 Exercises
5 Measurable Functions
5.1 Qualitative Properties
5.2 The Integral of Non-Negative Measurable Functions
5.3 Properties that Apply Almost Everywhere
5.4 Semi-Integrable or Integrable Functions
5.5 Monotone Convergence
5.6 The Indefinite Integral
5.7 Exercises
6 The Lebesque and Riemann Integrals
6.1 The Product Measure
6.2 The Lebesque Measure
6.3 Riemann-integrable functions
6.4 The Fundamental Theorem of Calculus
6.5 The Riemann Improper Integral
6.6 Exercises
7 The Dominated Convergence Theorem
7.1 Vector or Complex Functions
7.2 Fatou’s Lemma
7.3 Dominated Convergence
7.4 Dominated Derivation
7.5 Exercises
8 Borel and Lebesque σ-algebras
8.1 Generated σ-algebra
8.2 Borelians
8.3 The Measure Value
8.4 σ-finite sets
8.5 Completion of σ-algebra
8.6 The Uniqueness of Λ
8.7 Borel-measurable functions
8.8 Borelians in Subspaces
8.9 The Extension of μ: S → [0, +∞]
8.10 Exercises
9 The Transport of Measures and other Issues
9.1 Two Useful Lemmas
9.2 Integration in Measure Spaces
9.3 Transported Measure
9.4 Invariance by Translations and Permutations of Variables
9.5 Non-Measurable Sets
9.6 Borel’s σ-Algebra is not Complete
9.7 The Counting Measure
9.8 Exercises
10 Fubini’s Theorem
10.1 Definite Functions q.t.p.
10.2 The Measure Product
10.3 The Product of Measurable Sets
10.4 The Integral as a Measure of the Area between Graphs
10.5 Some Examples
10.6 Exercises
11 Changes of Variable
11.1 The Measure ν = ∫gdμ
11.2 Integration in Subspaces
11.3 The Lebesque-Stieltjes Measure
11.4 The Change of Variable Theorem
11.5 Polar Coordinates inR2
11.6 Spherical Coordinates
11.7 Exercises
12 Lp Spaces
12.1 The L∞Space
12.2 Convex Functions
12.3 Inequalities
12.4 The Lp Spaces
12.5 Particular Cases
12.6 Exercises
13 Lp is Complete
13.1 Metrics Invariant by Translations
13.2 Series
13.3 Subsequences
13.4 Complete Spaces
13.5 Convergence in Lp x convergence q.t.p
13.6 Dense Subspaces of Lp
13.7 Hilbert Spaces and Fourier Series
13.8 Exercises
14 Measures with Sign and the Radon-Nikodym Theorem
14.1 The Positive, Negative and Total Variances of the Measure
14.2 The Hahn Decomposition
14.3 The Radon-Nikodym Theorem
14.4 Singular Measures
14.5 The M(A) Space
14.6 Exercises
15 Convergence in Measurement
15.1 Quasi-uniform convergence
15.2 Convergence in Measurement
15.3 Vitali’s Convergence Theorem
15.4 Exercises
16 The Riesz Duality Theorem
16.1 Operator Spaces
16.2 Dual Space
16.3 Exercises
Bibliography
Index