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An introduction to measure theory

Terence Tao

Department of Mathematics, UCLA, Los Angeles, CA 90095

E-mail address: [email protected]

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

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 Lp^ 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 Rd^ (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

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 xn ∈ [0, +∞] increases to x ∈ [0, +∞], and yn ∈ [0, +∞] increases to y ∈ [0, +∞], then xnyn increases to xy) but not downward continuous (e.g. 1 /n → 0 but 1/n · +∞ 6 → 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, +∞],

Notation xiii

we can always form the sum

∑^ ∞

n=

xn ∈ [0, +∞]

as the limit of the partial sums

∑N

n=1 xn, which may be either finite or infinite. An equivalent definition of this infinite sum is as the supremum of all finite subsums:

∑^ ∞

n=

xn = sup F ⊂N,F finite

n∈F

xn.

Motivated by this, given any collection (xα)α∈A of numbers xα ∈ [0, +∞] indexed by an arbitrary set A (finite or infinite, countable or uncountable), we can define the sum

α∈A xα^ by the formula

(0.1)

α∈A

xα = sup F ⊂A,F finite

α∈F

xα.

Note from this definition that one can relabel the collection in an arbitrary fashion without affecting the sum; more precisely, given any bijection φ : B → A, one has the change of variables formula

(0.2)

α∈A

xα =

β∈B

xφ(β).

Note that when dealing with signed sums, the above rearrangement identity can fail when the series is not absolutely convergent (cf. the Riemann rearrangement theorem).

Exercise 0.0.1. If (xα)α∈A is a collection of numbers xα ∈ [0, +∞] such that

α∈A xα^ <^ ∞, show that^ xα^ = 0 for all but at most countably many α ∈ A, even if A itself is uncountable.

We will rely frequently on the following basic fact (a special case of the Fubini-Tonelli theorem, Corollary 1.7.23):

Theorem 0.0.2 (Tonelli’s theorem for series). Let (xn,m)n,m∈N be a doubly infinite sequence of extended non-negative reals xn,m ∈ [0, +∞]. Then ∑

(n,m)∈N^2

xn,m =

∑^ ∞

n=

∑^ ∞

m=

xn,m =

∑^ ∞

m=

∑^ ∞

n=

xn,m.

Notation xv

suffices to show that

∑^ N

n=

∑^ M

m=

xn,m ≤

(n,m)∈N^2

xn,m

for each finite M. But the left-hand side is

(n,m)∈{ 1 ,...,N }×{ 1 ,...,M } xn,m, and the claim follows. 

Remark 0.0.3. Note how important it was that the xn,m were non- negative in the above argument. In the signed case, one needs an additional assumption of absolute summability of xn,m on N^2 before one is permitted to interchange sums; this is Fubini’s theorem for series, which we will encounter later in this text. Without absolute summability or non-negativity hypotheses, the theorem can fail (con- sider for instance the case when xn,m equals +1 when n = m, − 1 when n = m + 1, and 0 otherwise).

Exercise 0.0.2 (Tonelli’s theorem for series over arbitrary sets). Let A, B be sets (possibly infinite or uncountable), and (xn,m)n∈A,m∈B be a doubly infinite sequence of extended non-negative reals xn,m ∈ [0, +∞] indexed by A and B. Show that ∑

(n,m)∈A×B

xn,m =

n∈A

m∈B

xn,m =

m∈B

n∈A

xn,m.

(Hint: although not strictly necessary, you may find it convenient to first establish the fact that if

n∈A xn^ is finite, then^ xn^ is non-zero for at most countably many n.)

Next, we recall the axiom of choice, which we shall be assuming throughout the text:

Axiom 0.0.4 (Axiom of choice). Let (Eα)α∈A be a family of non- empty sets Eα, indexed by an index set A. Then we can find a family (xα)α∈A of elements xα of Eα, indexed by the same set A.

This axiom is trivial when A is a singleton set, and from math- ematical induction one can also prove it without difficulty when A is finite. However, when A is infinite, one cannot deduce this axiom from the other axioms of set theory, but must explicitly add it to the list of axioms. We isolate the countable case as a particularly useful

xvi Preface

corollary (though one which is strictly weaker than the full axiom of choice):

Corollary 0.0.5 (Axiom of countable choice). Let E 1 , E 2 , E 3 ,... be a sequence of non-empty sets. Then one can find a sequence x 1 , x 2 ,... such that xn ∈ En for all n = 1, 2 , 3 ,.. ..

Remark 0.0.6. The question of how much of real analysis still sur- vives when one is not permitted to use the axiom of choice is a delicate one, involving a fair amount of logic and descriptive set theory to an- swer. We will not discuss these matters in this text. We will however note a theorem of G¨odel[Go1938] that states that any statement that can be phrased in the first-order language of Peano arithmetic, and which is proven with the axiom of choice, can also be proven without the axiom of choice. So, roughly speaking, G¨odel’s theorem tells us that for any “finitary” application of real analysis (which includes most of the “practical” applications of the subject), it is safe to use the axiom of choice; it is only when asking questions about “infini- tary” objects that are beyond the scope of Peano arithmetic that one can encounter statements that are provable using the axiom of choice, but are not provable without it.

Acknowledgments

This text was strongly influenced by the real analysis text of Stein and Shakarchi[StSk2005], which was used as a secondary text when teaching the course on which these notes were based. In particular, the strategy of focusing first on Lebesgue measure and Lebesgue inte- gration, before moving onwards to abstract measure and integration theory, was directly inspired by the treatment in [StSk2005], and the material on differentiation theorems also closely follows that in [StSk2005]. On the other hand, our discussion here differs from that in [StSk2005] in other respects; for instance, a far greater emphasis is placed on Jordan measure and the Riemann integral as being an elementary precursor to Lebesgue measure and the Lebesgue integral.

I am greatly indebted to my students of the course on which this text was based, as well as many further commenters on my blog, including Marco Angulo, J. Balachandran, Farzin Barekat, Marek

Chapter 1

Measure theory

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 := [0, 2] are in one-to-one correspondence (using the bijection x 7 → 2 x from A to B), but of course B is twice as long as A. So one can disassemble A into an uncountable number of points and reassemble them to form a set of twice the length.

Of course, one can point to the infinite (and uncountable) number of components in this disassembly as being the cause of this break- down of intuition, and restrict attention to just finite partitions. But one still runs into trouble here for a number of reasons, the most striking of which is the Banach-Tarski paradox, which shows that the unit ball B := {(x, y, z) ∈ R^3 : x^2 + y^2 + z^2 ≤ 1 } in three dimensions^2 can be disassembled into a finite number of pieces (in fact, just five pieces suffice), which can then be reassembled (after translating and rotating each of the pieces) to form two disjoint copies of the ball B.

Here, the problem is that the pieces used in this decomposition are highly pathological in nature; among other things, their construction requires use of the axiom of choice. (This is in fact necessary; there are models of set theory without the axiom of choice in which the Banach-Tarski paradox does not occur, thanks to a famous theorem of Solovay[So1970].) Such pathological sets almost never come up in practical applications of mathematics. Because of this, the standard solution to the problem of measure has been to abandon the goal of measuring every subset E of Rd, and instead to settle for only measuring a certain subclass of “non-pathological” subsets of Rd, which are then referred to as the measurable sets. The problem of measure then divides into several subproblems:

(i) What does it mean for a subset E of Rd^ to be measurable? (ii) If a set E is measurable, how does one define its measure? (iii) What nice properties or axioms does measure (or the con- cept of measurability) obey?

(^2) The paradox only works in three dimensions and higher, for reasons having to do with the group-theoretic property of amenability; see §2.2 of An epsilon of room, Vol. I for further discussion.

4 1. Measure theory

(iv) Are “ordinary” sets such as cubes, balls, polyhedra, etc. measurable?

(v) Does the measure of an “ordinary” set equal the “naive geo- metric measure” of such sets? (e.g. is the measure of an a × b rectangle equal to ab?)

These questions are somewhat open-ended in formulation, and there is no unique answer to them; in particular, one can expand the class of measurable sets at the expense of losing one or more nice properties of measure in the process (e.g. finite or countable addi- tivity, translation invariance, or rotation invariance). However, there are two basic answers which, between them, suffice for most applica- tions. The first is the concept of Jordan measure (or Jordan content) of a Jordan measurable set, which is a concept closely related to that of the Riemann integral (or Darboux integral ). This concept is el- ementary enough to be systematically studied in an undergraduate analysis course, and suffices for measuring most of the “ordinary” sets (e.g. the area under the graph of a continuous function) in many branches of mathematics. However, when one turns to the type of sets that arise in analysis, and in particular those sets that arise as limits (in various senses) of other sets, it turns out that the Jordan concept of measurability is not quite adequate, and must be extended to the more general notion of Lebesgue measurability, with the corre- sponding notion of Lebesgue measure that extends Jordan measure. With the Lebesgue theory (which can be viewed as a completion of the Jordan-Darboux-Riemann theory), one keeps almost all of the de- sirable properties of Jordan measure, but with the crucial additional property that many features of the Lebesgue theory are preserved un- der limits (as exemplified in the fundamental convergence theorems of the Lebesgue theory, such as the monotone convergence theorem (Theorem 1.4.44) and the dominated convergence theorem (Theorem 1.4.49), which do not hold in the Jordan-Darboux-Riemann setting).