Preface
Chapter 1 Cotructio and exteio of measures
1.1 Measurement of length: introductory remarks
1.2 Algebras and σ-algebras
1.3 Additivity and countable additivity of measures
1.4 Compact classes and countable additivity
1.5 Outer measure and the Lebesgue exteion of measures
1.6 Infinite and a-finite measures
1.7 Lebesgue measure
1.8 Lebesgue-Stieltjes measures
1.9 Monotone and σ-additive classes of sets
1.10 Souslin sets and the A-operation
1.11 Caratheodory outer measures
1.12 Supplements and exercises
Set operatio 48 Compact classes 50 Metric Boolean algebra
53.Measurable envelope, measurable kernel and inner measure
56.Exteio of measures 58 Some interesting sets 61 Additive,
but not countably additive measures 67 Abstract inner measures
70.Measures on lattices of sets 75 Set-theoretic problems in
measure theory 77 Invariant exteio of Lebesgue measure 80
Whitney''s decomposition 82 Exercises 83
Chapter 2 The Lebesgue integral
2.1 Measurable functio
2.2 Convergence in measure and almost everywhere
2.3 The integral for simple functio
2.4 The general definition of the Lebesgue integral
2.5 Basic properties of the integral
2.6 Integration with respect to infinite measures
2.7 The completeness of the space L1
2.8 Convergence theorems
2.9 Criteria of integrability
2.10 Connectio with the Riemann integral
2.11 The HSlder and Minkowski inequalities
2.12 Supplements and exercises
The a-algebra generated by a class of functio 143 Borel
mappings on IRn 145 The functional monotone class theorem 146
Baire classes of functio 148 Mean value theorems 150 The
Lebesgue-Stieltjes integral 152 Integral inequalities 153
Exercises 156
Chapter 3 Operatio on measures and functio
3.1 Decomposition of signed measures
3.2 The Radon-Nikodym theorem
3.3 Products of measure spaces
3.4 Fubini''s theorem
3.5 Infinite products of measures
3.6 Images of measures under mappings
3.7 Change of variables in IRn
3.8 The Fourier traform
3.9 Convolution
3.10 Supplements and exercises
On Fubini''s theorem and products of σ-algebras 209 Steiner''s
symmetrization 212 Hausdorff measures 215 Decompositio of set
functio 218 Properties of positive definite functio 220.The
Brunn-Minkowski inequality and its generalizatio 222.Mixed
volumes 226 The Radon traform 227 Exercises 228
Chapter 4 The spaces Lp and spaces of measures
4.1 The spaces Lp
4.2 Approximatio in Lp
4.3 The Hilbert space L2
4.4 Duality of the spaces Lp
4.5 Uniform integrability
4.6 Convergence of measures
4.7 Supplements and exercises
The spaces Lp and the space of measures as structures 277
The weak topology in LP280 Uniform convexity of LP283 Uniform
integrability and weak compactness in L1 285 The topology of
setwise convergence of measures 291 Norm compactness and
approximatio in Lp 294.Certain conditio of convergence in Lp
298 Hellinger''s integral and ellinger''s distance 299 Additive
set functio 302 Exercises 303
Chapter 5 Connectio between the integral and derivative
5.1 Differentiability of functio on the real line
5.2 Functio of bounded variation
5.3 Absolutely continuous functio
5.4 The Newton-Leibniz formula
5.5 Covering theorems
5.6 The maximal function
5.7 The Hetock-Kurzweil integral
5.8 Supplements and exercises
Covering theorems 361 Deity points and Lebesgue points
366.Differentiation of measures on IRn 367 The approximate
continuity 369 Derivates and the approximate differentiability
370.The class BMO 373 Weighted inequalities 374 Measures
with the doubling property 375 Sobolev derivatives 376 The
area and coarea formulas and change of variables 379 Surface
measures 383.The Calder6n-Zygmund decomposition 385 Exercises
386
Bibliographical and Historical Comments
References
Author Index
Subject Index
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