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This text is a self-contained treatment of material that is normally found in a first-year graduate course in real analysis. It covers topics such as measure and integration theory, functional analysis, and distributions. The book assumes that the student has had a solid course in Advanced Calculus and has been exposed to rigorous ε,δ arguments. It is intended to serve as a graduate text and a useful reference for the more experienced mathematician.
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Department of Mathematics, Indiana University, Bloomington, In- diana E-mail address: ziemer@indiana.edu
Department of Mathematics, Purdue University, West Lafayette, Indiana E-mail address: torres@math.purdue.edu
This text is an essentially self-contained treatment of material that is normally found in a first year graduate course in real analysis. Although the presentation is based on a modern treatment of measure and integration, it has not lost sight of the fact that the theory of functions of one real variable is the core of the subject. It is assumed that the student has had a solid course in Advanced Calculus and has been exposed to rigorous ε, δ arguments. Although the book’s primary purpose is to serve as a graduate text, we hope that it will also serve as useful reference for the more experienced mathematician. The book begins with a chapter on preliminaries and then proceeds with a chapter on the development of the real number system. This also includes an informal presentation of cardinal and ordinal numbers. The next chapter provides the basics of general topological and metric spaces. By the time this chapter has been concluded, the background of students in a typical course will have been equalized and they will be prepared to pursue the main thrust of the book. The text then proceeds to develop measure and integration theory in the next three chapters. Measure theory is introduced by first considering outer measures on an abstract space. The treatment here is abstract, yet short, simple, and basic. By focusing first on outer measures, the development underscores in a natural way the fundamental importance and significance of σ-algebras. Lebesgue measure, Lebesgue-Stieltjes measure, and Hausdorff measure are immediately developed as important, concrete examples of outer measures. Integration theory is presented by using countably simple functions, that is, functions that assume only a countable number of values. Conceptually they are no more difficult than simple functions, but their use leads to a more direct development. Important results such as the Radon-Nikodym theorem and Fubini’s theorem have received treatments that avoid some of the usual technical difficulties. A chapter on elementary functional analysis is followed by one on the Daniell integral and the Riesz Representation theorem. This introduces the student to a completely different approach to measure and integration theory. In order for the student to become more comfortable with this new framework, the linear functional
7
8 PREFACE
approach is further developed by including a short chapter on Schwartz Distribu- tions. Along with introducing new ideas, this reinforces the student’s previous encounter with measures as linear functionals. It also maintains connection with previous material by casting some old ideas in a new light. For example, BV functions and absolutely continuous functions are characterized as functions whose distributional derivatives are measures and functions, respectively. The introduction of Schwartz distributions invites a treatment of functions of several variables. Since absolutely continuous functions are so important in real analysis, it is natural to ask whether they have a counterpart among functions of several variables. In the last chapter, it is shown that this is the case by developing the class of functions whose partial derivatives (in the sense of distributions) are functions, thus providing a natural analog of absolutely continuous functions of a single variable. The analogy is strengthened by proving that these functions are absolutely continuous in each variable separately. These functions, called Sobolev functions, are of fundamental importance to many areas of research today. The chapter is concluded with a glimpse of both the power and the beauty of Dis- tribution theory by providing a treatment of the Dirichlet Problem for Laplace’s equation. This presentation is not difficult, but it does call upon many of the top- ics the student has learned throughout the text, thus providing a fitting end to the book. We will use the following notation throughout. The symbol denotes the end of a proof and a := b means a = b by definition. All theorems, lemmas, corollaries, definitions, and remarks are numbered as a.b where a denotes the chapter number. Equation numbers are numbered in a similar way and appear as (a.b). Sections marked with ∗^ are not essential to the main development of the material and may be omitted.
2 1. PRELIMINARIES
Sometimes a family of sets will be defined in terms of an indexing set I and then we write
(1.3) {x : x ∈ Aα for some α ∈ I} = ⋃ α∈I
Aα.
If the index set I is the set of positive integers, then we write (1.3) as
(2.1)
i= Ai.
The intersection of sets A and B is defined by {x : x ∈ A and x ∈ B} and is written as A ∩ B. Similar to (1.1) and (1.2) we have
{x : x ∈ A for all A ∈ A} = ⋂ A∈A
A family A of sets is said to be disjoint if A 1 ∩ A 2 = ∅ for every pair A 1 and A 2 of distinct members of A. If every element of the set A is also an element of B, then A is called a subset of B and this is written as A ⊂ B or B ⊃ A. With this terminology, the possibility that A = B is allowed. The set A is called a proper subset of B if A ⊂ B and A 6 = B. The difference of two sets is A \ B = {x : x ∈ A and x /∈ B}
while the symmetric difference is
A∆B = (A \ B) ∪ (B \ A). In most discussions, a set A will be a subset of some underlying space X and in this context, we will speak of the complement of A (relative to X) as the set {x : x ∈ X and x /∈ A}. This set is denoted by A˜ and this notation will be used if there is no doubt that complementation is taken with respect to X. In case of possible ambiguity, we write X \ A instead of A˜. The following identities, known as de Morgan’s laws, are very useful and easily verified:
α∈I
Aα
α∈I
A^ ˜α ( ⋂ α∈I
Aα
α∈I
A^ ˜α.
We shall denote the set of all subsets of X, called the power set of X, by P(X). Thus,
(2.3) P(X) = {A : A ⊂ X}.
1.2. FUNCTIONS 3 The notions of limit superior (lim sup) and lim inferior (lim inf ) are defined for sets as well as for sequences:
lim sup i→∞ Ei =
k=
i=k
Ei
lim inf i→∞ Ei =
k=
i=k
Ei
It is easily seen that
(3.1)
lim sup i→∞ Ei = {x : x ∈ Ei for infinitely many i }, lim inf i→∞ Ei = {x : x ∈ Ei for all but finitely many i }. We use the following notation throughout: ∅ = the empty set, N = the set of positive integers, (not including zero), Z = the set of integers, Q = the set of rational numbers, R = the set of real numbers.
We assume the reader has knowledge of the sets N, Z, and Q, while R will be carefully constructed in Section 2.1.
1.2. Functions In this section an informal discussion of relations and functions is given, a subject that is encountered in several forms in elementary analysis. In this development, we adopt the notion that a relation or function is indistinguishable from its graph.
If X and Y are sets, the Cartesian product of X and Y is
(3.2) X × Y = { all ordered pairs (x, y) : x ∈ X, y ∈ Y }.
The ordered pair (x, y) is thus to be distinguished from (y, x). We will discuss the Cartesian product of an arbitrary family of sets later in this section. A relation from X to Y is a subset of X × Y. If f is a relation, then the domain and range of f are
domf = X ∩ {x : (x, y) ∈ f for some y ∈ Y } rngf = Y ∩ {y : (x, y) ∈ f for some x ∈ X }.
1.2. FUNCTIONS 5 Given an equivalence relation ∼ on X, a subset A of X is called an equivalence class if and only if there is an element x ∈ A such that A consists precisely of those elements y such that x ∼ y. One can easily verify that dsitinct equivalence classes are disjoint and that X can be expressed as the union of equivalence classes. A sequence in a space X is a mapping f : N → X. It is convenient to denote a sequence f as a list. Thus, if f (k) = xk, we speak of the sequence {xk}∞ k=1 or simply {xk}. A subsequence is obtained by discarding some elements of the original sequence and ordering the elements that remain in the usual way. Formally, we say that xk 1 , xk 2 , xk 3 ,... , is a subsequence of x 1 , x 2 , x 3 ,... , if there is a mapping g : N → N such that for each i ∈ N, xki = xg(i) and if g(i) < g(j) whenever i < j. Our final topic in this section is the Cartesian product of a family of sets. Let X be a family of sets Xα indexed by a set I. The Cartesian product of X is denoted by (^) ∏
α∈I
Xα
and is defined as the set of all mappings
x : I → ⋃^ Xα
with the property that
(5.1) x(α) ∈ Xα
for each α ∈ I. Each mapping x is called a choice mapping for the family X. Also, we call x(α) the αth^ coordinate of x. This terminology is perhaps easier to understand if we consider the case where I = { 1 , 2 ,... , n}. As in the preceding paragraph, it is useful to denote the choice mapping x as a list {x(1), x(2),... , x(n)}, and even more useful if we write x(i) = xi. The mapping x is thus identified with the ordered n-tuple (x 1 , x 2 ,... , xn). Here, the word “ordered” is crucial because an n-tuple consisting of the same elements but in a different order produces a different mapping x. Consequently, the Cartesian product becomes the set of all ordered n-tuples:
(5.2)
∏^ n i=
Xi = {(x 1 , x 2 ,... , xn) : xi ∈ Xi, i = 1, 2 ,... , n}.
In the special case where Xi = R, i = 1, 2 ,... , n, an element of the Cartesian product is a mapping that can be identified with an ordered n-tuple of real numbers. We denote the set of all ordered n-tuples (also referred to as vectors) by
Rn^ = {(x 1 , x 2 ,... , xn) : xi ∈ R, i = 1, 2 ,... , n}
6 1. PRELIMINARIES
Rn^ is called Euclidean n-space. The norm of a vector x is defined as
(6.1) |x| =
x^21 + x^22 + · · · + x^2 n;
the distance between two vectors x and y is |x − y|. As we mentioned earlier in this section, the Cartesian product of two sets X 1 and X 2 is denoted by X 1 × X 2.
6.1. Remark. A fundamental issue that we have not addressed is whether the Cartesian product of an arbitrary family of sets is nonempty. This involves concepts from set theory and is the subject of the next section.
1.3. Set Theory The material discussed in the previous two sections is based on tools found in elementary set theory. However, in more advanced areas of mathematics this material is not sufficient to discuss or even formulate some of the concepts that are needed. An example of this occurred in the previous section during the discussion of the Cartesian product of an arbitrary family of sets. Indeed, the Cartesian product of families of sets requires the notion of a choice mapping whose existence is not obvious. Here, we give a brief review of the Axiom of Choice and some of its logical equivalences.
A fundamental question that arises in the definition of the Cartesian product of an arbitrary family of sets is the existence of choice mappings. This is an example of a question that cannot be answered within the context of elementary set theory. In the beginning of the 20th^ century, Ernst Zermelo formulated an axiom of set theory called the Axiom of Choice, which asserts that the Cartesian product of an arbitrary family of nonempty sets exists and is nonempty. The formal statement is as follows.
6.2. The Axiom of Choice. If Xα is a nonempty set for each element α of an index set I, then (^) ∏
α∈I
Xα
is nonempty.
6.3. Proposition. The following statement is equivalent to the Axiom of Choice: If {Xα}α∈A is a disjoint family of nonempty sets, then there is a set S ⊂ ∪α∈AXα such that S ∩ Xα consists of precisely one element for every α ∈ A.
Proof. The Axiom of Choice states that there exists f : A → ∪α∈AXα such that f (α) ∈ Xα for each α ∈ A. The set S := f (A) satisfies the conclusion of the statement. Conversely, if such a set S exists, then the mapping A −→ ∪f α∈AXα defined by assigning the point S ∩ Xα the value of f (α) implies the validity of the Axiom of Choice.
8 1. PRELIMINARIES
Cantor put forward the continuum hypothesis in 1878, conjecturing that every infinite subset of the continuum is either countable (i.e. can be put in 1-1 corre- spondence with the natural numbers) or has the cardinality of the continuum (i.e. can be put in 1-1 correspondence with the real numbers). The importance of this was seen by Hilbert who made the continuum hypothesis the first in the list of problems which he proposed in his Paris lecture of 1900. Hilbert saw this as one of the most fundamental questions which mathematicians should attack in the 1900s and he went further in proposing a method to attack the conjecture. He suggested that first one should try to prove another of Cantor’s conjectures, namely that any set can be well ordered. Zermelo began to work on the problems of set theory by pursuing, in particular, Hilbert’s idea of resolving the problem of the continuum hypothesis. In 1902 Zer- melo published his first work on set theory which was on the addition of transfinite cardinals. Two years later, in 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved that every set can be well ordered. This result brought fame to Zermelo and also earned him a quick promotion; in December 1905, he was appointed as professor in G¨ottingen. The axiom of choice is the basis for Zermelo’s proof that every set can be well ordered; in fact the axiom of choice is equivalent to the well ordering property so we now know that this axiom must be used. His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction. Although Zermelo certainly gained fame for his proof of the well ordering property, set theory at this time was in the rather unusual position that many mathematicians rejected the type of proofs that Zermelo had discovered. There were strong feelings as to whether such non-constructive parts of mathematics were legitimate areas for study and Zermelo’s ideas were certainly not accepted by quite a number of mathematicians. The fundamental discoveries of K. G¨odel [ 4 ] and P. J. Cohen [ 2 ], [ 3 ] shook the foundations of mathematics with results that placed the axiom of choice in a very interesting position. Their work shows that the Axiom of Choice, in fact, is a new principle in set theory because it can neither be proved nor disproved from the usual Zermelo-Fraenkel axioms of set theory. Indeed, G¨odel showed, in 1940, that the Axiom of Choice cannot be disproved using the other axioms of set theory and then in 1963, Paul Cohen proved that the Axiom of Choice is independent of the other axioms of set theory. The importance of the Axiom of Choice will readily be seen throughout the following development, as we appeal to it in a variety of contexts.
EXERCISES FOR CHAPTER 1 9 Exercises for Chapter 1 Section 1. 1.1 Two sets are identical if and only if they have the same members. That is, A = B if and only if for each element x, x ∈ A when and only when x ∈ B. Prove A = B if and only if A ⊂ B and B ⊂ A. that A ⊂ B if and only if B = A ∪ B. Prove de Morgan’s laws, 2.2. 1.2 Let Ei, i = 1, 2 ,... , be a family of sets. Use definitions (2.4) to prove
lim inf i→∞
Ei ⊂ lim sup i→∞
Ei
Section 1.
1.3 Prove that f ◦ (g ◦ h) = (f ◦ g) ◦ h for mappings f, g, and h. 1.4 Prove that (f ◦ g)−^1 (A) = g−^1 [f −^1 (A)] for mappings f and g and an arbitrary set A. 1.5 Prove: If f : X → Y is a mapping and A ⊂ B ⊂ X, then f (A) ⊂ f (B) ⊂ Y. Also, prove that if E ⊂ F ⊂ Y , then f −^1 (E) ⊂ f −^1 (F ) ⊂ X. 1.6 Prove: If A ⊂ P(X), then
f
A∈A
A∈A
f (A) and f
A∈A
A∈A
f (A).
and
f −^1
A∈A
A∈A f −^1 (A) and f −^1
A∈A
A∈A f −^1 (A).
Give an example that shows the above inclusion cannot be replaced by equality. 1.7 Consider a nonempty set X and its power set P(X). For each x ∈ X, let Bx = { 0 , 1 } and consider the Cartesian product ∏ x∈X Bx. Exhibit a natural one-to-one correspondence between P(X) and ∏ x∈X Bx. 1.8 Let X −→f Y be an arbitrary mapping and suppose there is a mapping Y −→g X such that f ◦ g(y) = y for all y ∈ Y and that g ◦ f (x) = x for all x ∈ X. Prove that f is one-to-one from X onto Y and that g = f −^1. 1.9 Show that A × (B ∪ C) = (A × B) ∪ (A × C). Also, show that in general A ∪ (B × C) 6 = (A ∪ B) × (A ∪ C). Section 1. 1.10 Use a one-to-one correspondence between Z and N to exhibit a linear ordering of N that is not a well-ordering. 1.11 Use the natural partial ordering of P({ 1 , 2 , 3 }) to exhibit a partial 1.12 For (a, b), (c, d) ∈ N × N, define (a, b) ≤ (c, d) if either a < c or a = c and b ≤ d. With this relation, prove that N × N is a well-ordered set.
2.1. The Real Numbers A brief development of the construction of the Real Numbers is given in terms of equivalence classes of Cauchy sequences of rational numbers. This construction is based on the assumption that properties of the rational numbers, including the integers, are known.
In our development of the real number system, we shall assume that properties of the natural numbers, integers, and rational numbers are known. In order to agree on what the properties are, we summarize some of the more basic ones. Recall that the natural numbers are designated as
N : = { 1 , 2 ,... , k,.. .}.
They form a well-ordered set when endowed with the usual ordering. The order- ing on N satisfies the following properties:
(i) x ≤ x for every x ∈ S. (ii) if x ≤ y and y ≤ x, then x = y. (iii) if x ≤ y and y ≤ z, then x ≤ z. (iv) for all x, y ∈ S, either x ≤ y or y ≤ x. The four conditions above define a linear ordering on S, a topic that was in- troduced in Section 1.3 and will be discussed in greater detail in Section 2.3. The linear order ≤ of N is compatible with the addition and multiplication operations in N. Furthermore, the following three conditions are satisfied:
(i) Every nonempty subset of N has a first element; i.e., if ∅ 6 = S ⊂ N, there is an element x ∈ S such that x ≤ y for any element y ∈ S. In particular, the set N itself has a first element that is unique, in view of (ii) above, and is denoted by the symbol 1, (ii) Every element of N, except the first, has an immediate predecessor. That is, if x ∈ N and x 6 = 1, then there exists y ∈ N with the property that y ≤ x and z ≤ y whenever z ≤ x. (iii) N has no greatest element; i.e., for every x ∈ N, there exists y ∈ N such that x 6 = y and x ≤ y. 11
12 2. REAL, CARDINAL AND ORDINAL NUMBERS
The reader can easily show that (i) and (iii) imply that each element of N has an immediate successor, i.e., that for each x ∈ N, there exists y ∈ N such that x < y and that if x < z for some z ∈ N where y 6 = z, then y < z. The immediate successor of x, y, will be denoted by x′. A nonempty set S ⊂ N is said to be finite if S has a greatest element. From the structure established above follows an extremely important result, the so-called principle of mathematical induction, which we now prove.
12.1. Theorem. Suppose S ⊂ N is a set with the property that 1 ∈ S and that x ∈ S implies x′^ ∈ S. Then S = N.
Proof. Suppose S is a proper subset of N that satisfies the hypotheses of the theorem. Then N \ S is a nonempty set and therefore by (i) above, has a first element x. Note that x 6 = 1 since 1 ∈ S. From (ii) we see that x has an immediate predecessor, y. As y ∈ S, we have y′^ ∈ S. Since x = y′, we have x ∈ S, contradicting the choice of x as the first element of N \ S. Also, we have x ∈ S since x = y′. By definition, x is the first element of N − S, thus producing a contradiction. Hence, S = N.
The rational numbers Q may be constructed in a formal way from the natural numbers. This is accomplished by first defining the integers, both negative and positive, so that subtraction can be performed. Then the rationals are defined using the properties of the integers. We will not go into this construction but instead leave it to the reader to consult another source for this development. We list below the basic properties of the rational numbers. The rational numbers are endowed with the operations of addition and multi- plication that satisfy the following conditions:
(i) For every r, s ∈ Q, r + s ∈ Q, and rs ∈ Q. (ii) Both operations are commutative and associative, i.e., r + s = s + r, rs = sr, (r + s) + t = r + (s + t), and (rs)t = r(st). (iii) The operations of addition and multiplication have identity elements 0 and 1 respectively, i.e., for each r ∈ Q, we have 0 + r = r and 1 · r = r. (iv) The distributive law is valid: r(s + t) = rs + rt whenever r, s, and t are elements of Q. (v) The equation r + x = s has a solution for every r, s ∈ Q. The solution is denoted by s − r.
