[...]... and integration The problem of measure Chapter 1 Measure Theory xvi xvi xvii xviii xviii 1 1 2 3 4 Preliminaries The exterior measure Measurable sets and the Lebesgue measure Measurable functions 4.1 Definition and basic properties 4.2 Approximation by simple functions or step functions 4.3 Littlewood’s three principles 5* The Brunn-Minkowski inequality 6 Exercises 7 Problems Chapter 2 Integration Theory. .. Introduction 2 1 The Hilbert space L 2 Hilbert spaces 2.1 Orthogonality 2.2 Unitary mappings 2.3 Pre -Hilbert spaces 3 Fourier series and Fatou’s theorem 3.1 Fatou’s theorem 4 Closed subspaces and orthogonal projections 5 Linear transformations 5.1 Linear functionals and the Riesz representation theorem 5.2 Adjoints 5.3 Examples 6 Compact operators 7 Exercises 8 Problems Chapter 5 Hilbert Spaces: Several... measure spaces 1.1 Exterior measures and Carath´odory’s theorem e 1.2 Metric exterior measures 1.3 The extension theorem 2 Integration on a measure space 3 Examples 3.1 Product measures and a general Fubini theorem 3.2 Integration formula for polar coordinates 3.3 Borel measures on R and the Lebesgue-Stieltjes integral 4 Absolute continuity of measures 4.1 Signed measures 4.2 Absolute continuity 5*... theorem and key estimate 4* The Dirichlet principle 4.1 Harmonic functions 4.2 The boundary value problem and Dirichlet’s principle 5 Exercises 6 Problems 114 115 127 131 134 136 143 145 152 156 156 161 164 168 169 170 173 174 180 181 183 185 188 193 202 207 207 213 221 222 224 229 234 243 253 259 CONTENTS xiii Chapter 6 Abstract Measure and Integration Theory 262 1 Abstract measure spaces 1.1 Exterior measures... bounded variation by Jordan and later (1887) connection with rectifiability 1883 − Cantor’s ternary set 1890 − Construction of a space-filling curve by Peano 1898 − Borel’s measurable sets 1902 − Lebesgue’s theory of measure and integration 1905 − Construction of non-measurable sets by Vitali 1906 − Fatou’s application of Lebesgue theory to complex analysis 4 There is no such measure on the class of all... distinct rationals, and therefore I is countable, as desired ∞ Naturally, if O is open and O = j=1 Ij , where the Ij ’s are disjoint ∞ open intervals, the measure of O ought to be j=1 |Ij | Since this representation is unique, we could take this as a definition of measure; we would then note that whenever O1 and O2 are open and disjoint, the measure of their union is the sum of their measures Although... The exterior measure The notion of exterior measure is the first of two important concepts needed to develop a theory of measure We begin with the definition and basic properties of exterior measure Loosely speaking, the exterior measure m∗ assigns to any subset of Rd a first notion of size; various examples show that this notion coincides with our earlier intuition However, the exterior measure lacks... two theorems motivate the definition of exterior measure given later We shall use the following standard notation A point x ∈ Rd consists of a d-tuple of real numbers x = (x1 , x2 , , xd ), xi ∈ R, for i = 1, , d Addition of points is componentwise, and so is multiplication by a real scalar The norm of x is denoted by |x| and is defined to be the standard Euclidean norm given by |x| = x2 + · · ·... between two points x and y is then simply |x − y| The complement of a set E in Rd is denoted by E c and defined by E c = {x ∈ Rd : x ∈ E} / If E and F are two subsets of Rd , we denote the complement of F in E by E − F = {x ∈ Rd : x ∈ E and x ∈ F } / The distance between two sets E and F is defined by d(E, F ) = inf |x − y|, where the infimum is taken over all x ∈ E and y ∈ F Open, closed, and compact sets... sets, those which are “measurable.” This class of sets is closed under countable unions, intersections, and complements, and contains the open sets, the closed sets, and so forth.4 It is with the construction of this measure that we begin our study From it will flow the general theory of integration, and in particular the solutions of the problems discussed above A chronology We conclude this introduction . REAL ANALYSIS Ibookroot October 20, 2007 Princeton Lectures in Analysis I Fourier Analysis: An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces IV. Functional Analysis: Introduction to Further Topics in Analysis Princeton Lectures in Analysis III REAL ANALYSIS Measure Theory, Integration, and Hilbert Spaces Elias M. Stein & Rami Shakarchi PRINCETON. series and integrals. II. Complex analysis. III. Measure theory, Lebesgue integration, and Hilbert spaces. IV. A selection of further topics, including functional analysis, distri- butions, and