An introduction to measure theory Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: tao@math.ucla.edu To Garth Gaudry, who set me on the road; To my family, for their constant support; And to the readers of my blog, for their feedback and contributions. Contents Preface ix Notation x Acknowledgments xvi Chapter 1. Measure theory 1 §1.1. Prologue: The problem of measure 2 §1.2. Lebesgue measure 17 §1.3. The Lebesgue integral 46 §1.4. Abstract measure spaces 79 §1.5. Modes of convergence 114 §1.6. Differentiation theorems 131 §1.7. Outer measures, pre-measures, and product measures 179 Chapter 2. Related articles 209 §2.1. Problem solving strategies 210 §2.2. The Radamacher differentiation theorem 226 §2.3. Probability spaces 232 §2.4. Infinite product spaces and the Kolmogorov extension theorem 235 Bibliography 243 vii viii Contents Index 245 Preface In the fall of 2010, I taught an introductory one-quarter course on graduate real analysis, focusing in particular on the basics of mea- sure and integration theory, both in Euclidean spaces and in abstract measure spaces. This text is based on my lecture notes of that course, which are also available online on my blog terrytao.wordpress.com, together with some supplementary material, such as a section on prob- lem solving strategies in real analysis (Section 2.1) which evolved from discussions with my students. This text is intended to form a prequel to my graduate text [Ta2010] (henceforth referred to as An epsilon of room, Vol. I ), which is an introduction to the analysis of Hilbert and Banach spaces (such as L p and Sobolev spaces), point-set topology, and related top- ics such as Fourier analysis and the theory of distributions; together, they serve as a text for a complete first-year graduate course in real analysis. The approach to measure theory here is inspired by the text [StSk2005], which was used as a secondary text in my course. In particular, the first half of the course is devoted almost exclusively to measure theory on Euclidean spaces R d (starting with the more elementary Jordan-Riemann-Darboux theory, and only then moving on to the more sophisticated Lebesgue theory), deferring the abstract aspects of measure theory to the second half of the course. I found ix x Preface that this approach strengthened the student’s intuition in the early stages of the course, and helped provide motivation for more abstract constructions, such as Carath´eodory’s general construction of a mea- sure from an outer measure. Most of the material here is self-contained, assuming only an undergraduate knowledge in real analysis (and in particular, on the Heine-Borel theorem, which we will use as the foundation for our construction of Lebesgue measure); a secondary real analysis text can be used in conjunction with this one, but it is not strictly necessary. A small number of exercises however will require some knowledge of point-set topology or of set-theoretic concepts such as cardinals and ordinals. A large number of exercises are interspersed throughout the text, and it is intended that the reader perform a significant fraction of these exercises while going through the text. Indeed, many of the key results and examples in the subject will in fact be presented through the exercises. In my own course, I used the exercises as the basis for the examination questions, and signalled this well in advance, to encourage the students to attempt as many of the exercises as they could as preparation for the exams. The core material is contained in Chapter 1, and already com- prises a full quarter’s worth of material. Section 2.1 is a much more informal section than the rest of the book, focusing on describing problem solving strategies, either specific to real analysis exercises, or more generally applicable to a wider set of mathematical problems; this section evolved from various discussions with students through- out the course. The remaining three sections in Chapter 2 are op- tional topics, which require understanding of most of the material in Chapter 1 as a prerequisite (although Section 2.3 can be read after completing Section 1.4. Notation For reasons of space, we will not be able to define every single math- ematical term that we use in this book. If a term is italicised for reasons other than emphasis or for definition, then it denotes a stan- dard mathematical object, result, or concept, which can be easily Notation xi looked up in any number of references. (In the blog version of the book, many of these terms were linked to their Wikipedia pages, or other on-line reference pages.) Given a subset E of a space X, the indicator function 1 E : X → R is defined by setting 1 E (x) equal to 1 for x ∈ E and equal to 0 for x ∈ E. For any natural number d, we refer to the vector space R d := {(x 1 , . , x d ) : x 1 , . . . , x d ∈ R} as (d-dimensional) Euclidean space. A vector (x 1 , . . . , x d ) in R d has length |(x 1 , . . . , x d )| := (x 2 1 + . . . + x 2 d ) 1/2 and two vectors (x 1 , . . . , x d ), (y 1 , . . . , y d ) have dot product (x 1 , . . . , x d ) ·(y 1 , . . . , y d ) := x 1 y 1 + . . . + x d y d . The extended non-negative real axis [0, +∞] is the non-negative real axis [0, +∞) := {x ∈ R : x ≥ 0} with an additional element adjointed to it, which we label +∞; we will need to work with this system because many sets (e.g. R d ) will have infinite measure. Of course, +∞is not a real number, but we think of it as an extended real number. We extend the addition, multiplication, and order structures on [0, +∞) to [0, +∞] by declaring +∞ + x = x + +∞ = +∞ for all x ∈ [0, +∞], +∞ ·x = x · +∞ = +∞ for all non-zero x ∈ (0, +∞], +∞ ·0 = 0 ·+∞ = 0, and x < +∞ for all x ∈ [0, +∞). Most of the laws of algebra for addition, multiplication, and order continue to hold in this extended number system; for instance ad- dition and multiplication are commutative and associative, with the latter distributing over the former, and an order relation x ≤ y is preserved under addition or multiplication of both sides of that re- lation by the same quantity. However, we caution that the laws of xii Preface cancellation do not apply once some of the variables are allowed to be infinite; for instance, we cannot deduce x = y from +∞+x = +∞+y or from +∞ · x = +∞ · y. This is related to the fact that the forms +∞ − +∞ and +∞/ + ∞ are indeterminate (one cannot assign a value to them without breaking a lot of the rules of algebra). A gen- eral rule of thumb is that if one wishes to use cancellation (or proxies for cancellation, such as subtraction or division), this is only safe if one can guarantee that all quantities involved are finite (and in the case of multiplicative cancellation, the quantity being cancelled also needs to be non-zero, of course). However, as long as one avoids us- ing cancellation and works exclusively with non-negative quantities, there is little danger in working in the extended real number system. We note also that once one adopts the convention +∞ · 0 = 0 · +∞ = 0, then multiplication becomes upward continuous (in the sense that whenever x n ∈ [0, +∞] increases to x ∈ [0, +∞], and y n ∈ [0, +∞] increases to y ∈ [0, +∞], then x n y n increases to xy) but not downward continuous (e.g. 1/n → 0 but 1/n · +∞ → 0 · +∞). This asymmetry will ultimately cause us to define integration from below rather than from above, which leads to other asymmetries (e.g. the monotone convergence theorem (Theorem 1.4.44) applies for monotone increasing functions, but not necessarily for monotone decreasing ones). Remark 0.0.1. Note that there is a tradeoff here: if one wants to keep as many useful laws of algebra as one can, then one can add in infinity, or have negative numbers, but it is difficult to have both at the same time. Because of this tradeoff, we will see two overlapping types of measure and integration theory: the non-negative theory, which involves quantities taking values in [0, +∞], and the absolutely integrable theory, which involves quantities taking values in (−∞, +∞) or C. For instance, the fundamental convergence theorem for the former theory is the monotone convergence theorem (Theorem 1.4.44), while the fundamental convergence theorem for the latter is the dominated convergence theorem (Theorem 1.4.49). Both branches of the theory are important, and both will be covered in later notes. One important feature of the extended nonnegative real axis is that all sums are convergent: given any sequence x 1 , x 2 , . . . ∈ [0, +∞], [...]... problem of measure on other domains than Euclidean space, such as a Riemannian manifold, but we will focus on the Euclidean case here for simplicity, and refer to any text on Riemannian geometry for a treatment of integration on manifolds 1.1 Prologue: The problem of measure 3 the same number of points, need not have the same measure For instance, in one dimension, the intervals A := [0, 1] and B :=... also elementary If x ∈ Rd , show that the translate E + x := {y + x : y ∈ E} is also an elementary set 3There are other ways to extend Jordan measure and the Riemann integral, see for instance Exercise 1.6.53 or Section 1.7.3, but the Lebesgue approach handles limits and rearrangement better than the other alternatives, and so has become the standard approach in analysis; it is also particularly well suited... case when L is an elementary transformation, using Gaussian elimination Alternatively, work with the cases when L is a diagonal transformation or an orthogonal transformation, using the unit ball in the latter case, and use the polar decomposition.) Exercise 1.1.12 Define a Jordan null set to be a Jordan measurable set of Jordan measure zero Show that any subset of a Jordan null set is a Jordan null set... set E ⊂ Rd If those measures match, we say that E is Jordan measurable, 18 1 Measure theory and call m(E) = m∗,(J) (E) = m∗,(J) (E) the Jordan measure of E As long as one is lucky enough to only have to deal with Jordan measurable sets, the theory of Jordan measure works well enough However, as noted previously, not all sets are Jordan measurable, even if one restricts attention to bounded sets In fact,... × E2 ⊂ Rd1 +d2 is Jordan measurable, and md1 +d2 (E1 × E2 ) = md1 (E1 ) × md2 (E2 ) Exercise 1.1.17 Let P, Q be two polytopes in Rd Suppose that P can be partitioned into finitely many sub-polytopes which, after being rotated and translated, form a cover of Q, with any two of the sub-polytopes in Q intersecting only at their boundaries Conclude that P and Q have the same Jordan measure The converse... Jordan inner measure zero and Jordan outer measure one In particular, both sets are not Jordan measurable Informally, any set with a lot of “holes”, or a very “fractal” boundary, is unlikely to be Jordan measurable In order to measure such sets we will need to develop Lebesgue measure, which is done in the next set of notes Exercise 1.1.19 (Carath´odory type property) Let E ⊂ Rd be e a bounded set, and... which case the Riemann integral and Darboux integrals are equal Exercise 1.1.23 Show that any continuous function f : [a, b] → R is Riemann integrable More generally, show that any bounded, piecewise continuous8 function f : [a, b] → R is Riemann integrable Now we connect the Riemann integral to Jordan measure in two ways First, we connect the Riemann integral to one-dimensional Jordan measure: Exercise... and several anonymous commenters, for providing corrections and useful commentary on the material here These comments can be viewed online at terrytao.wordpress.com/category/teaching/245a-real-analysis The author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award Chapter 1 Measure theory 1 2 1 Measure theory 1.1 Prologue: The problem of measure. .. Measure theory As we shall see in Section 1.7, Lebesgue outer measure (also known as Lebesgue exterior measure) is a special case of a more general concept known as an outer measure In analogy with the Jordan theory, we would also like to define a concept of “Lebesgue inner measure to complement that of outer measure Here, there is an asymmetry (which ultimately arises from the fact that elementary measure. .. are not Jordan measurable: Exercise 1.1.18 Let E ⊂ Rd be a bounded set (1) Show that E and the closure E of E have the same Jordan outer measure (2) Show that E and the interior E ◦ of E have the same Jordan inner measure (3) Show that E is Jordan measurable if and only if the topological boundary ∂E of E has Jordan outer measure zero (4) Show that the bullet-riddled square [0, 1]2 \Q2 , and set of . problem of measure on other domains than Euclidean space, such as a Riemannian manifold, but we will focus on the Euclidean case here for simplicity, and refer to any text on Riemannian geometry. intended to form a prequel to my graduate text [Ta2010] (henceforth referred to as An epsilon of room, Vol. I ), which is an introduction to the analysis of Hilbert and Banach spaces (such as L p and. at terrytao.wordpress.com/category/teaching/245a-real-analysis The author is supported by a grant from the MacArthur Founda- tion, by NSF grant DMS-0649473, and by the NSF Waterman award. Chapter 1 Measure theory 1 2 1. Measure theory 1.1.