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Introduction to Integration and Measure Theory [Kietas viršelis]

(Queen's University at Kingston, Ontario, Canada)
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A textbook for a 36-lecture senior or beginning graduate course for students of mathematics, economics, or engineering. Totally different from the two courses Nielson took as a student, which turned him off to the subject for 25 years. Begins with integration theory, and why it is necessary to replace the Riemann integral with either the Stieltjes or the Lebesgue integral. No bibliography. Annotation c. by Book News, Inc., Portland, Or.

This book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.
PART ONE LIMITATIONS OF THE RIEMANN INTEGRAL 1(16)
Chapter 1 Limits of Integrals and Integrability
3(8)
1.1 General Discussion of the Problem
3(3)
1.2 Examples of Nonintegrability
6(1)
1.3 Examples of Limits of Integrals
7(1)
1.4 Examples of Incompleteness of Norms
8(1)
Exercises
9(2)
Chapter 2 Expectations in Probability Theory
11(6)
2.1 Probabilities
11(1)
2.2 Distributions and Expectations
12(2)
2.3 Two Examples
14(1)
Exercises
15(2)
PART TWO RIEMANN-STIELTJES INTEGRALS 17(70)
Chapter 3 Riemann-Stieltjes Integrals: Introduction
19(19)
3.1 Partitions
19(1)
3.2 Riemann-Stieltjes and Darboux-Stieltjes Sums
20(2)
3.3 Riemann-Stieltjes Integrals
22(3)
3.4 Some Examples
25(2)
3.5 Properties of Riemann-Stieltjes Integrals
27(5)
3.6 Another Example
32(2)
Exercises
34(4)
Chapter 4 Characterization of Riemann-Stieltjes Integrability
38(16)
4.1 Oscillation of a Function
38(1)
4.2 Null Selts
39(2)
4.3 The Cantor Set and the Cantor Function
41(1)
4.4 The Characterization for Continuous Integrators
42(4)
4.5 The Characterization for General Integrators
46(4)
Exercises
50(4)
Chapter 5 Continuous Linear Functionals on C [ a,b]
54(22)
5.1 The Norms on C [ a,b]
54(2)
5.2 Positive Linear Functionals on C [ a,b]
56(4)
5.3 Continuous Linear Functionals on C [ a,b]
60(4)
5.4 Variation of a Function
64(2)
5.5 Functions of Bounded Variation
66(5)
Exercises
71(5)
Chapter 6 Riemann-Stieltjes Integrals: Further Properties
76(11)
6.1 Integration by Parts
76(3)
6.2 Fundamental Theorem of Calculus
79(2)
6.3 A Theorem About Continuous Integrators
81(1)
6.4 A Proof of Arzela's Theorem
82(4)
Exercises
86(1)
PART THREE LEBESGUE-STIELTJES INTEGRALS 87(34)
Chapter 7 The Extension of the Riemann-Stieltjes Integral
89(11)
7.1 The Extended Real Numbers
89(2)
7.2 The Space C (R) and Riemann-Stieltjes Integrals
91(1)
7.3 The First Extension of the Riemann-Stieltjes Integral
91(5)
7.4 Two Examples
96(2)
Exercises
98(2)
Chapter 8 Lebesgue-Stieltjes Integrals
100(21)
8.1 Lebesgue-Stieltjes Integrals and Summable Functions
100(2)
8.2 Two Examples
102(3)
8.3 Linearity and Lattice Operations
105(3)
8.4 Convergence Theorems
108(5)
8.5 Riemann-Stieltjes and Lebesgue-Stieltjes Integrals
113(4)
Exercises
117(4)
PART FOUR MEASURE THEORY 121(66)
Chapter 9 Algebras and Algebras of Sets
123(6)
9.1 Algebras of Sets
123(1)
9.2 The Borel Algebra
124(1)
9.3 Rings and Algebras of Sets
125(1)
Exercises
126(3)
Chapter 10 Measurable Functions
129(8)
10.1 Simple Functions
129(1)
10.2 Definition and Examples of Measurable Functions
130(2)
10.3 Properties of Measurable Functions
132(3)
10.4 Approximation by Simple Functions
135(1)
Exercises
135(2)
Chapter 11 Measures
137(23)
11.1 Definitions and Examples of Measures
137(1)
11.2 Measures in Probability Theory
138(1)
11.3 Elementary Properties of Measures
139(1)
11.4 Null Sets and Almost Everywhere
140(1)
11.5 Finite and Semifinite Measures
141(2)
11.6 Completion of Measure
143(2)
11.7 Outer Measures
145(4)
11.8 Measures on Rings and Algebras of Sets
149(4)
11.9 Atoms and Nonatomic Measures
153(3)
Exercises
156(4)
Chapter 12 Lebesgue-Stieltjes Measures
160(27)
12.1 The Algebras
160(2)
12.2 The Measures
162(1)
12.3 Measurability of Summable Functions
163(1)
12.4 The Integral in Terms of the Measure
164(1)
12.5 Translation Invariance
165(3)
12.6 The Role of Null Sets
168(2)
12.7 Regularity of Lebesgue-Stieltjes Measures
170(1)
12.8 Characterization of Null Sets
171(2)
12.9 Existence of Non-Borel Sets of the Real Line
173(1)
12.10 Lusin's Theorem
173(2)
12.11 Characterizations of Lebesgue-Stieltjes Measures
175(5)
Exercises
180(7)
PART FIVE THE ABSTRACT LEBESGUE INTEGRAL 187(254)
Chapter 13 The Integral Associated with a Measure Space
189(20)
13.1 The Space of Simple Functions
189(3)
13.2 Definition of the Abstract Lebesgue Integral
192(2)
13.3 Properties of the Abstract Lebesgue Integral
194(8)
13.4 Lebesgue-Stieltjes Measures
202(1)
13.5 Counting Measure
203(1)
13.6 A Pathological Example
204(1)
13.7 Complex-Valued Functions
204(2)
Exercises
206(3)
Chapter 14 The Lebesgue Spaces and Norms
209(38)
14.1 Pre-Lebesgue Space and Minkowski's Inequality
209(5)
14.2 Some Examples
214(3)
14.3 Definition of the Lebesgue Spaces
217(3)
14.4 Completeness of the Norm
220(3)
14.5 Holder's Inequality
223(3)
14.6 Applications of Holder's Inequality
226(1)
14.7 Density Theorems
227(3)
14.8 Inclusion Relations Amongst the Lebesgue Spaces
230(6)
14.9 Jensen's Inequality
236(4)
14.10 Convexity and Continuity of the Norm
240(2)
Exercises
242(5)
Chapter 15 Absolutely Continuous Measures
247(21)
15.1 The Product of a Function and a Measure
247(3)
15.2 Absolutely Continuous Measures
250(3)
15.3 A Proof of the Radon-Nikodym Theorem
253(5)
15.4 Conditional Expectations
258(4)
15.5 The Lebesgue Decomposition Theorem
262(2)
Exercises
264(4)
Chapter 16 Linear Functionals on the Lebesgue Spaces
268(14)
16.1 Definition of the Canonical Maps
268(2)
16.2 A Condition for the Canonical Maps to Be Isometric
270(1)
16.3 A Condition for the Canonical Maps to Be Onto: The Finite Case
271(4)
16.4 A Condition for the Canonical Maps to Be Onto: The General Case
275(4)
16.5 A Characterization of L p (X,,//, u)
279(1)
Exercises
280(2)
Chapter 17 Product Measures and Fubini's Theorem
282(20)
17.1 Iterated Integrals
282(1)
17.2 Product Algebras
283(4)
17.3 Product Measures
287(2)
17.4 Integrals of Characteristic Functions
289(4)
17.5 Fubini's and Tonelli's Theorems
293(2)
17.6 Three Counterexamples
295(2)
17.7 Products of More than Two Measure Spaces
297(1)
Exercises
298(4)
Chapter 18 Lebesgue Integration and Measure on R"
302(15)
18.1 Lebesgue Measure on R"
302(5)
18.2 An Analogue of Spherical Coordinates
307(5)
18.3 Convolution Products
312(2)
Exercises
314(3)
Chapter 19 Signed Measures and Complex Measures
317(24)
19.1 Definitions and Examples
317(3)
19.2 Elementary Properties
320(2)
19.3 Jordan and Hahn Decompositions
322(1)
19.4 Existence of the Jordan Decomposition
323(3)
19.5 Total Variation of Signed and Complex Measures
326(4)
19.6 Norms of Real and Complex Measures
330(2)
19.7 Lattice Operation on Signed Measures
332(5)
19.8 Absolute Continuity and the Radon-Nikodym Theorem
337(1)
Exercises
338(3)
Chapter 20 Differentiation
341(26)
20.1 Vitali's Covering Theorem
341(3)
20.2 Differentiation of Monotone Functions
344(4)
20.3 Differentiation of Indefinite Integrals
348(2)
20.4 Lebesgue Points and Points of Density
350(3)
20.5 Absolutely Continuous Functions
353(3)
20.6 Application to Lebesgue-Stieltjes Measures
356(4)
20.7 Differentiation of Functions on R"
360(4)
Exercises
364(3)
Chapter 21 Convergence of Sequences of Functions
367(14)
21.1 Definitions
367(1)
21.2 Convergence on a General Measure Space
368(6)
21.3 Convergence on a Finite Measure Space
374(1)
21.4 Dominated Convergence
375(2)
21.5 Summary
377(1)
Exercises
378(3)
Chapter 22 Measures on Locally Compact Spaces
381(19)
22.1 Locally Compact Spaces
381(2)
22.2 Regular Measures
383(3)
22.3 Linear Functionals on Cc(X) and Cx(X)
386(5)
22.4 Completion of the Proof of Theorem 22.8
391(5)
Exercises
396(4)
Chapter 23 Hausdorff Measures and Dimension
400(17)
23.1 A Property of Outer Measures
400(2)
23.2 Hausdorff Measures
402(4)
23.3 A Proof of Theorem 23.9
406(6)
23.4 Hausdorff Dimension
412(3)
Exercises
415(2)
Chapter 24 Lorentz Spaces
417(24)
24.1 Distribution Functions and Nonincreasing Rearrangements
417(6)
24.2 The Lorentz Spaces
423(3)
24.3 Norms on the Lorentz Spaces
426(7)
24.4 Subadditivity of the Maximal Operator
433(4)
Exercises
437(4)
PART SIX APPENDICES 441(24)
Appendix A Continuous Functions, Topology, and Set Theory 443(11)
A.1 Limit Superior and Limit Inferior of Sequences of Sets 443(1)
A.2 Inverse Images 444(1)
A.3 Inequalities Between and Operations on Functions 444(1)
A.4 Elementary Topology 445(1)
A.5 Pointwise and Uniform Convergence 446(1)
A.6 Bounded and Continuous Functions 446(3)
A.7 Axiom of Choice 449(1)
A.8 Ordinal Numbers 450(1)
A.9 Equivalence of Sets and Cardinal Numbers 451(3)
Appendix B Functional Analysis 454(11)
B.1 Metric Spaces 454(1)
B.2 Baire's Theorem 455(3)
B.3 Norms 458(1)
B.4 Continuous Linear Functionals 459(2)
B.5 Inner Products 461(4)
Index of Notation 465(3)
Index 468


Ole A. Nielsen is the author of An Introduction to Integration and Measure Theory, published by Wiley.