Weights and Measures

Several times throughout this book we've hinted at a physical basis for some of our notation. It's time that we made this more precise; a simple calculus problem will help explain.

Consider a thin rod, or wire, positioned along the interval [a, b] on the x-axis and having a nonuniform distribution of mass. For example, the rod might vary slightly in thickness or in density (mass per unit length) as x varies. Our job is to compute the density (at a point) as a function f(x), if at all possible.

What we can measure effectively is the distribution of mass along the rod. That is, we can easily measure the mass of any segment of the rod, and so we know the mass of the segment lying along the interval [a, x] as a function F(x). Said in slightly different terms, we are able to measure small, discrete “chunks” of mass as dm = F(x + dx) − F(x) = dF, and so we're led to define the density f(x) = dm/dx = F′(x) as the derivative of the distribution F(x), provided that F is differentiate, of course.

But F is an arbitrary increasing function – is every such function differentiate? And, if not, can we say anything meaningful about this problem? Could we, for example, still find the center of mass (the line x = µ through which the rod balances) when F is not differentiable?

Discontinuous Functions

We have had a lot to say so far about continuous functions, but what about discontinuous functions? Is there anything meaningful we might say about them? In order that we might ask more precise questions, let's fix our notation. Throughout this section, we will be concerned with a function f : ℝ → ℝ, and we will write D(f) for the set of points at which f is discontinuous. The questions are: What can we say about D(f)? What kind of set is it? Can any set be realized as the set of discontinuities of a function, or does D(f) have some distinguishing characteristics? To get us started, let's recall a few examples.

Examples 9.1

1. (a) If f is monotone, then D(f) is countable. Conversely, any countable set is the set of discontinuities for some monotone f (see Exercise 2.34).

2. (b)There are examples of functions f, g with D(f) = ℚ and D(g) = ℝ. (What are they?)

In particular, we might ask whether D(f) can be a proper, uncountable subset of ℝ. For example, is there an f with D(f) = ℝ\ℚ? or with D(f) = Δ? The answer to the first question is: No, and to the second: Yes, but to understand this will require a bit of machinery.

Functions of Bounded Variation

Throughout this book we've encountered the theme that C(X) determines X. Said another way, to fully understand X we want to understand C(X) as well. Taking this one step further, though, raises a curious question: How are we to understand C(X) without knowing something about C(C(X))? If we want to be true to our principles, we will have to consider continuous real-valued functions on C(X). If that sounds too esoteric to bother with, fear not. As it happens, we need only to consider the continuous linear real-valued functions on C(X), and such functions have a simple and altogether user-friendly description: Definite integrals! But we're getting a little ahead of ourselves. We'll talk about integrals in the next chapter. For the present, we'll content ourselves with the study of a class of functions that turns out to be of paramount interest in this postponed discussion of integration.

To motivate the inevitable blur of definitions ahead of us, let's consider a simple example. Suppose that f(t) = (x(t), y(t)), for a ≤ t ≤ b, is a “nice” curve. What would we mean by the length of this curve?

This book is based on a course in real analysis offered to advanced undergraduates and first-year graduate students at Bowling Green State University. In many respects it is a perfectly ordinary first course in analysis, but there are some important differences. For one, the typical audience for the class includes many nonspecialists, students of statistics, economics, and education, as well as students of pure and applied mathematics at the undergraduate and graduate levels. What's more, the students come from a wide variety of backgrounds. This makes the course something of a challenge to teach. The material must be presented efficiently, but without sacrificing the less well-prepared student. The course must be essentially self-contained, but not so pedestrian that the more experienced student is bored. And the course should offer something of value to both the specialist and the nonspecialist. The following pages contain my personal answer to this challenge.

To begin, I make a few compromises: Extra details are given on metric and normed linear spaces in place of general topology, and a thorough attack on Riemann–Stieltjes and Lebesgue integration on the line in place of abstract measure and integration. On the other hand, I avoid euphemisms and specialized notation and, instead, attempt to remain faithful to the terminology and notation used in more advanced settings. Next, to make the course more meaningful to the nonspecialist (and more fun for me), I toss in a few historical tidbits along the way.

Compact Metric Spaces

A metric space (M, d) is said to be compact if it is both complete and totally bounded. As you might imagine, a compact space is the best of all possible worlds.

Examples 8.1

1. (a) A subset K of ℝ is compact if and only if K is closed and bounded. This fact is usually referred to as the Heine–Borel theorem. Hence, a closed bounded interval [a, b] is compact. Also, the Cantor set Δ is compact. The interval (0, 1), on the other hand, is not compact.

2. (b) A subset K of ℝn is compact if and only if K is closed and bounded. (Why?)

3. (c) It is important that we not confuse the first two examples with the general case. Recall that the set {en:n ≥ 1} is closed and bounded in ℓ∞ but not totally bounded – hence not compact. Taking this a step further, notice that the closed ball {x: ∥x∥∞ ≤ 1} in ℓ∞ is not compact, whereas any closed ball in ℝn is compact.

4. (d) A subset of a discrete space is compact if and only if it is finite. (Why?)

Just as with completeness and total boundedness, we will want to give several equivalent characterizations of compactness. In particular, since neither completeness nor total boundedness is preserved by homeomorphisms, our newest definition does not appear to be describing a topological property.

Algebras and Lattices

We continue with our study of B(X), the space of bounded real-valued functions on a set X. As we have seen, B(X) is a Banach space when supplied with the norm ∥f∥∞ = supx∈X |f(x)|. Moreover, convergence in B(X) is the same as uniform convergence. Of course, if X is a metric space, we will also be interested in C(X), the space of continuous real-valued functions on X, and its cousin Cb(X) = C(X) ∩ B(X), the closed subspace of bounded continuous functions in B(X). Finally, if X is a compact metric space, recall that Cb(X) = C(X).

But now we want to add a few more ingredients to the recipe: It's time we made use of the algebraic and lattice structures of B(X). In this chapter we will make formal our earlier informal discussions of algebras and lattices. In particular, we will see how this additional structure leads to a generalization of the Weierstrass approximation theorem in C(X), where X is a compact metric space.

To begin, an algebra is a vector space A on which there is defined a multiplication (f, g) ↦ fg (from A × A into A) satisfying

1. (i) (fg)h = f(gh), for all f, g, h ∈ A;

2. (ii) f(g + h) = fg + fh, (f + g)h = fh + gh, for all f, g, h ∈ A;

3. (iii) α(fg) = (αf)g = f(αg), for all scalars α and all f, g ∈ A.

4. […]

Our goal in this chapter is to provide a quick review of a handful of important ideas from advanced calculus (and to encourage a bit of practice on these fundamentals). We will make no attempt to be thorough. Our purpose is to set the stage for later generalizations and to collect together in one place some of the notation that should already be more or less familiar. There are sure to be missing details, unexplained terminology, and incomplete proofs. On the other hand, since much of this material will reappear in later chapters in a more general setting, you will get to see some of the details more than once. In fact, you may find it entertaining to refer to this chapter each time an old name is spoken in a new voice. If nothing else, there are plenty of keywords here to assist you in looking up any facts that you have forgotten.

The Real Numbers

First, let's agree to use a standard notation for the various familiar sets of numbers. ℝ denotes the set of all real numbers; ℂ denotes the set of all complex numbers (although our major concern here is ℝ, we will use complex numbers from time to time); ℤ stands for the integers (negative, zero, and positive); ℕ is the set of natural numbers (positive integers); and ℚ is the set of rational numbers.

In the beginning there were operations – hundreds of them – limits, derivatives, integrals, sums; all of the many operations on functions, sequences, sets, vectors, matrices, and whatever else you might have encountered in calculus. The hallmark of twentieth-century mathematics is that we now view these operations as functions defined on entire collections of “abstract” objects rather than as specific actions taken on individual objects, one at a time. Maurice Fréchet, in a short expository article from 1950, had this to say (the italics are his own):

Lebesgue's Differentiation Theorem

In the last several chapters, we have raised questions about differentiation and about the Fundamental Theorem of Calculus that have yet to be answered. For example:

• For which f does the formula hold? If f′ is to be integrable, then at the very least we will need f′ to exist almost everywhere in [a, b]. But this alone is not enough: Recall that the Cantor function f : [0, 1] → [0, 1] satisfies f′ = 0 a.e., but.

• Stated in slightly different terms: If g is integrable, is the function differentiable? And, if so, is f′ = g in this case? For which f is it true that for some integrable g?

In our initial discussion of the Stieltjes integral, we briefly considered the problem of finding the density of a thin metal rod with a known distribution of mass. That is, we were handed an increasing function F(x) that gave the mass of that portion of the rod lying on [a, x], and we asked for its density f(x) = F′(x). We side-stepped this question entirely at the time, defining a new integral in the process, but perhaps it merits posing again.

The Weierstrass Theorem

While we now know something about convergence in C(X), there are many more things that we would like to know about C(X). We will find the task unmanageable, however, unless we place some restrictions on the metric space X. If we focus our attention on the case when X is compact, for example, we will be afforded plenty of extra machinery: In this case, C(X) is not only a vector space, an algebra, and a lattice (where algebraic operations are defined pointwise), but also a complete normed space under the supnorm. With all of these tools to work with, we will be able to accomplish quite a bit. And at least a few of our results will apply equally well to the space Cb(X) of bounded continuous functions on a general metric space X. For the remainder of this chapter, then, unless otherwise specified, X will denote a compact metric space.

We will concentrate on two questions in particular, and each of these will lead to some interesting applications:

• Is C(X) separable? More importantly, are there any “useful” dense subspaces, or even dense subalgebras, or sublattices of C(X)?

• What are the compact subsets of C(X)? And are such sets “useful”?

Either question is tough to answer in full generality, but the first one has a very satisfactory and easy to understand answer for C[a, b].

Connected Sets

We have a few details to clean up before we move on to other things; these concern the special role of intervals in ℝ and their use in characterizing the open sets in ℝ given in Chapter Four (see Theorem 4.6 and Exercise 4.25). As we'll see in this section, a better understanding of the special nature of intervals in ℝ will allow us to generalize the intermediate value theorem of calculus. The intermediate value theorem is the formal statement of the informal notion that the graph of a continuous function is “unbroken.” The historical basis of the theorem is the concept of a function as measuring, over time, the relative position of an object moving along a straight line. Thus, if we track the position y = f(x) of a moving object between time x = a and some subsequent time x = b, we would expect the object to “visit” all of the positions y that are intermediate to f(a) and f(b). In short, the continuous image of the time interval [a, b] should contain (at least) the full interval of positions between f(a) and f(b).

The secret here is the intuitively obvious fact that no interval in ℝ can be split into two relatively open parts. Let's prove this by “brute force” for the interval [a, b] (we'll do the other cases shortly).

Historical Background

Unarguably, modern analysis was formed during the resolution of an important controversy (or, rather, controversies) concerning the representation of “arbitrary” functions. This controversy has unfolded slowly over the last two centuries and was put to its final rest only in our own time.

The story begins in 1746 with the famous vibrating string problem. Briefly, an elastic string of length L has each end fastened to one of the endpoints of the interval [0, L] on the x-axis and is set into motion (as you might pluck a guitar string, for example). The problem is to determine the position y = F(x, t) of the string at time t, given only its initial position y = f(x) = F(x, 0) at time t = 0 where, for simplicity, we assume that the initial velocity Ft(x, 0) = 0. The function F(x, t) is the solution to d'Alembert's wave equation: Ftt = a2Fxx, where a is a positive constant determined by certain physical properties of the string. The initial data for the problem is F(x, 0) = f(x), Ft(x, 0) = 0, and f(0) = 0 = f(L).

The controversy, initially between d'Alembert and Euler, centers around the nature of the functions f that may be permitted as initial positions.

We've set the stage for the Lebesgue integral in the previous two chapters; now it's time for the star to make her entrance. By way of a reminder, recall that we want our new integral to satisfy at least the following few, loosely stated properties:

• ∫ χE = m(E), whenever E is measurable.

• The integral should be linear: ∫(αf + βg) = α ∫ f + β ∫ g.

• The integral should be positive (or monotone): f ≥ 0 ⇒ ∫ f ≥ 0 (or f ≥ g ⇒ ∫ f ≥ ∫ g). In the presence of linearity, these are the same.

• The integral should be defined for a large class of functions, including at least the bounded Riemann integrable functions, and it should coincide with the Riemann integral whenever appropriate.

The first two properties tell us how to define the integral for simple functions. Once we know how to integrate simple functions, the third property suggests how to define the integral for nonnegative measurable functions: If f ≥ 0 is measurable, then we can find a sequence (ϕn) of simple functions that increase to f. Now set ∫ f = limn→∞ ∫ ϕn. Finally, linearity supplies the appropriate definition for the general case: If f is measurable, then f+ and f− are nonnegative, measurable, and f = f+ − f−.