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 LP(280) Uniform convexity of LP(283) 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