16

CHAPTER 2 Interlude

In this chapter we describe some useful classes of functions, sets of measure zero, and various ways (known as modes of convergence) in which a sequence of functions can approximate another function. We then describe situations when we are allowed to interchange limiting operations. Finally we state several density results that appear in different guises throughout the book. A leitmotif in analysis, in particular in harmonic analysis, is that often it is easier to prove results for a class of simpler functions that are dense in a larger class, and then to use limiting processes to pass from the approximating simpler functions to the more complicated limiting function.

This chapter should be read once to get a glimpse of the variety of function spaces and modes of convergence that can be considered. It should be revisited as often as necessary as the reader gets acquainted with the theory of Fourier series and integrals, and with the other types of time-frequency decompositions described in this book.

2.1. Nested classes of functions on bounded intervals

Here we introduce some classes of functions on bounded intervals I (the intervals can be open, closed or neither) and on the unit circle 𝕋 that will appear throughout the book. Later we will introduce analogous function classes on ℝ, and we will go beyond functions and introduce distributions (generalized functions) in Section 7.4.2.

2.1.1. Riemann-integrable functions. We review the concept of the Riemann integral, following the exposition in the book by Terence Tao^{1} [Tao1, Chapter 11]. We highly recommend Tao’s book to our readers.

The development of the Riemann integral is in terms of approximation by step functions. We begin with some definitions.

Definition 2.1. Given a bounded interval I, a partition P of I is a finite collection of disjoint intervals {J_{k}}_{k=1,...,n} such that I = ⋃_{k=1}^{n} J_{i}. The intervals J_{i} are contained in I and are necessarily bounded.

Definition 2.2. The characteristic function of J, χ_{J}(x), is defined to be the function that takes the value one if x lies in J, and zero otherwise:

χ_{J}(x) = \begin{cases}1, & if x ∈ J; \\ 0, & if x ∉ J. \end{cases}

1Terence Tao (1975–) is an Australian mathematician. He was awarded the Fields Medal in 2006.

Figure 2.1. Graph of the step function h = 3χ_{[−1,2)} − χ_{[2,4]} +
2χ_{(5,6)} in Example 2.4.

The building blocks in the Riemann integration theory are finite linear combinations of characteristic functions of disjoint intervals, called step functions.

Definition 2.3. A function h defined on I is a step function if there is a partition P of the interval I, and real numbers {a_{J}}_{J∈P}, such that

h(x) = ∑_{J∈P} a_{J}χ_{J}(x),

where χ_{J}(x) is the characteristic function of the interval J.

For step functions there is a very natural definition of integral. Let us first consider an example.

Example 2.4. The function h = 3χ_{[−1,2)} − χ_{[2,4]} + 2χ_{(5,6)} is a step function defined on the interval [−1, 6). Its graph is shown in Figure 2.1.

If we ask the reader to compute the integral of this function over the interval [−1, 6), she will add (with appropriate signs) the areas of the rectangles in the picture to get (3 × 3) + (−1 × 2) + (2 × 1) = 9. More generally, we can define the integral of a step function h associated to the partition P over the interval I by

(2.1)  ∫_{I} h(x) dx := ∑_{J∈P} a_{J}|J|,

where |J| denotes the length of the interval J. A step function could be associated to more than one partition, and one can verify that this definition of the integral is independent of the partition chosen. Notice that the integral of the characteristic function of a subinterval J of I is its length |J|.

Exercise 2.5. Given an interval J ⊂ I, show that

∫_{I} χ_{J}(x) dx = |J|.

Definition 2.6. A function f is bounded on I if there is a constant M > 0 such that |f(x)| ≤ M for all x ∈ I. If so, we say that f is bounded by M.

Exercise 2.7. Verify that step functions are always bounded functions.

Definition 2.8. A function f : I → ℝ is Riemann integrable if f is bounded on I, and if the supremum of the integrals of all step functions below f is equal to the infimum of the integrals of all step functions above f. In mathematical notation,

sup_{h_{1}≤f} ∫_{I} h_{1}(x) dx = inf_{h_{2}≥f} ∫_{I} h_{2}(x) dx < ∞,

Figure 2.2. This figure illustrates the approximation of a function from below and from above by step functions, as in the definition of the Riemann integral.

where the supremum and infimum are taken over step functions h_{1}(x) ≤ f(x) ≤ h_{2}(x) for all x ∈ I. We denote the common value by

∫_{I} f(x) dx, or ∫_{a}^{b} f(x) dx,

where a ≤ b are the endpoints of the interval I. We denote the class of Riemannintegrable functions on I by R(I). See Figure 2.2 illustrating the approximation from below and from above by step functions.

Riemann integrability is a property preserved under linear transformations, taking maximums and minimums, absolute values, and products. It also preserves order, meaning that if f and g are Riemann-integrable functions over I and if f < g then ∫_{I} f < ∫_{I} g, and satisfies the triangle inequality, | ∫_{I} f| ≤ ∫_{I} |f|. See [Tao1, Section 11.4].

Exercise 2.9. Given two step functions h_{1} and h_{2} defined on I, verify that the linear combination ah_{1} + bh_{2}, for a, b ∈ ℝ, is a step function. Also check that

∫_{I} (ah_{1}(x) + bh_{2}(x)) dx = a ∫_{I} h_{1}(x) dx + b ∫_{I} h_{2}(x) dx.

Warning: The partitions P_{1} and P_{2} associated to the step functions h_{1} and h_{2} might be different. Try to find a partition that works for both (a common refinement).

This linearity property is inherited by the Riemann-integrable functions.

One can deduce from Definition 2.8 the following very useful proposition.

Proposition 2.10. A bounded function f is Riemann integrable over I if and only if for each ε > 0 there are step functions h_{1} and h_{2} defined on I below and above f, that is h_{1}(x) ≤ f(x) ≤ h_{2}(x) for all x ∈ I, such that the integral of their difference is bounded by ε. More precisely,

0 ≤ ∫_{I} (h_{2}(x) − h_{1}(x)) dx < ε.

In particular, if f is Riemann integrable then there exists a step function h such that

∫_{I} |f(x) − h(x)| dx < ε.

See Figure 2.3 illustrating the integral of the difference of the two step functions in Figure 2.2.

Figure 2.3. The graph illustrates the integral of the difference of the two step functions in Figure 2.2.

Aside 2.11. Readers familiar with the definition of the Riemann integral via Riemann sums may like to observe that the integral of a step function that is below (respectively above) a given function f coincides with a lower (respectively upper) Riemann sum for f.

Our first observation is that if f is Riemann integrable on [−π, π), then the zeroth Fourier coefficient of f is well defined:

a_{0} = f̂(0) = \frac{1}{2π} ∫_{−π}^{π} f(θ) dθ.

We say that a complex-valued function f : I → ℂ is Riemann integrable if its real and imaginary parts are Riemann integrable. If f is Riemann integrable, then the complex-valued functions f(θ)e^{−inθ} are Riemann integrable for each n ∈ ℤ, and all the Fourier coefficients a_{n} = f̂(n), n ≠ 0, are also well defined.

Exercise 2.12. Verify that if f : [−π, π) → ℂ is Riemann integrable then the function g : [−π, π) → ℂ defined by g(θ) = f(θ)e^{−inθ} is Riemann integrable.

Here are some examples of familiar functions that are Riemann integrable and for which we can define Fourier coefficients. See Theorem 2.28 for a complete characterization of Riemann-integrable functions.

Example 2.13. Uniformly continuous functions on intervals are Riemann integrable. In particular continuous functions on closed intervals are Riemann integrable; see [Tao1, Section 11.5]. Monotone^{2} and bounded functions on intervals are Riemann integrable; see [Tao1, Section 11.6].

Here is an example of a function that is not Riemann integrable.

Example 2.14. (Dirichlet’s Example) The function

f(x) = \begin{cases}1 & if x ∈ [a, b] ⋂ ℚ \\ 0 & if x ∈ [a, b] \ ℚ \end{cases} ,

is bounded by one, but it is not Riemann integrable on [0, 1]. Notice also that it is discontinuous everywhere. ♢

Exercise 2.15. Show that the Dirichlet function is not Riemann integrable.

2A function f : I → ℝ is monotone if it is increasing on I (respectively decreasing), that is for all x, y ∈ I, x ≤ y then f(x) ≤ f(y) (respectively f(x) ≥ f(y)).

2.1.2. Lebesgue integrable functions and L^{p}-spaces. There are other more general notions of integral. We will consider the Lebesgue integral. The reader could consult the book by Robert G. Bartle [Bart2] for other notions.

The Lebesgue integral is studied in depth in courses on measure theory. For a brief introduction see [Tao2, Chapters 18, 19]. For more in-depth presentations at a level intermediate between advanced undergraduate and graduate students see [SS2] or [Bart1]. Classical graduate textbooks include the books by Gerald B. Folland [Fol], or Halsey L. Royden [Roy].

In this section we simply aim to give some ideas about what the Lebesgue integral is and what the L^{p}-spaces are.

Every Riemann-integrable function on a bounded interval I is also Lebesgueintegrable on I, and the values of its Riemann and Lebesgue integrals are the same. The Dirichlet function in Example 2.14 is Lebesgue integrable but not Riemann integrable, with Lebesgue integral equal to zero; see the last paragraph in Section 2.1.4. In practice when you see an integral, think Riemann even if sometimes we really mean Lebesgue.

In parallel to the definition of the Riemann integral just presented, we sketch the definition of the Lebesgue integral over the bounded interval I. We use as building blocks the so-called simple functions instead of step functions.

Simple functions are finite linear combinations of characteristic functions of disjoint measurable sets. The characteristic^{3} function of a set A ⊂ ℝ is defined, in analogy to the characteristic function of an interval, to be

(2.2)  χ_{A}(x) = \{ \begin{matrix}1 & if x ∈ A, \\ 0 & otherwise. \end{matrix}

Intervals are measurable sets, and we can assign to them a measure, namely their length. However there are sets that are non-measurable, and we need to exclude such sets. Once we assign a measure to each measurable set, we can define the Lebesgue integral of the characteristic function of a measurable set to be the measure of the set. The Lebesgue integral of a simple function is then defined to be the corresponding finite linear combination of the measures of the underlying sets, exactly like in (2.1) where now the Js are disjoint measurable subsets of the measurable set I, and the length |J| is replaced by the measure m(J) of the set J. One can then define Lebesgue integrability of a function f : I → ℝ in analogy to Definition 2.8, replacing the boundedness condition by asking the function to be measurable on I, and replacing step functions by simple functions. A function f : I → ℝ is measurable if the pre-image of each measurable subset of ℝ is a measurable subset of I. It all boils down to understanding well measurable sets and their measures.

We will not define measurable sets, or their measure, except for sets of measure zero in ℝ in Section 2.1.4, and for intervals or countable unions of disjoint intervals, where their measure is given, quite naturally, by the sum of the lengths of the intervals.

A measurable function is said to belong to L^{1}(I) if ∫_{I} |f(x)| dx < ∞, where the integral is in the sense of Lebesgue. In fact, there is a continuum of Lebesgue spaces, the L^{p}-spaces (indeed the L in L^{p} stands for Lebesgue), defined for each real number p such that 1 ≤ p < ∞ by

3In statistics and probability the characteristic function of a set A is called the indicator function of the set, and denoted 𝕀_{A}. The term “characteristic function” refers to the Fourier transform of a probability density (so-called because it “characterizes” the probability distribution).

L^{p}(I) := {f : I → ℂ : ∫_{I} |f(x)|^{p} dx < ∞} ,

and for p = ∞ by

L^{∞}(I) := {f : I → ℂ : f is essentially bounded}.

All the functions involved are assumed to be measurable to begin with. In Definition 2.29 we will state precisely what essentially bounded means. For now, just think of these as bounded functions on I. Let us denote B(I) the space of bounded functions on I, namely,

B(I) := {f : I → ℂ : f is bounded}.

Notice that B(I) ⊂ L^{∞}(I), but L^{∞}(I) is larger, that is there are essentially bounded functions that are not bounded.

These spaces are normed spaces (see Appendix B for the definition) with the L^{p}-norm defined to be

(2.3)  ∥f∥_{L^{p}(I)} := (∫_{I} |f(x)|^{p} dx)^{1/p} , 1 ≤ p < ∞,
(2.4)  ∥f∥_{L^{∞}(I)} := ess sup_{x∈I} |f(x)|, ∥f∥_{B(I)} := sup_{x∈I} |f(x)|.

In Definition 2.30 we will state precisely what essential supremum of f means.

Given a norm we can define a metric, in this case, the L^{p}-metric or L^{p}-distance, d_{p}, induced by the L^{p}-norm, for all 1 ≤ p ≤ ∞. The L^{p}-distance between two functions f, g ∈ L^{p}(I) is given by the norm of their difference:

(2.5)  d_{p}(f, g) := ∥f − g∥_{L^{p}(I)}.

The property that we will use repeatedly about normed spaces is the triangle inequality. In this case it states that for all f, g ∈ L^{p}(I), 1 ≤ p ≤ ∞,

(2.6)  ∥f + g∥_{L^{p}(I)} ≤ ∥f∥_{L^{p}(I)} + ∥g∥_{L^{p}(I)}

The Riemann-integrable functions are not only contained in all L^{p}-spaces for 1 ≤ p ≤ ∞, they are dense in L^{p}(I). This means that given any function f ∈ L^{p}(I), we can find a Riemann-integrable function arbitrarily close to f in the L^{p}-metric. Furthermore, for each p the space L^{p}(I) is the completion of R(I) in the L^{p}-metric. Being the completion of R(I) automatically implies that R(I) is dense in L^{p}(I), but not vice versa. It also implies that any convergent sequence in L^{p}(I) must converge to a function in L^{p}(I), even more than that, any sequence that has the “right to be convergent”, a Cauchy^{4} sequence in L^{p}(I), will converge and to a function in L^{p}(I).

Aside 2.16. The reader should be familiar with the construction of the real numbers ℝ as the completion of the rational numbers ℚ in the Euclidean metric d(x, y) = |x − y|. You should think of R(I) as playing the role of ℚ and L^{p}(I) as playing the role of ℝ. There is a procedure that allows the completion of any metric space mirroring the procedure used in the construction of the real numbers. See for example [Tao2, Exercise 12.4.8].

4The sequence {f_{n}} ⊂ L^{p}(I) is a Cauchy sequence if for each ε > 0 there is an N > 0 such that for all n, m > N, ∥f_{n} − f_{m}∥_{L^{p}(I)} < ε. Every convergent sequence is Cauchy, but not vice versa, unless the space is complete.

In fact, we could have defined the Lebesgue L^{p}-spaces as the completion of R(I) with the L^{p}-metric, instead of attempting to define the Lebesgue integral via measure theory. The important thing is that both processes lead to the same spaces. This fact is so important that we state it (without proof) as a theorem, for future reference.

Theorem 2.17. L^{p}(I) is the completion of the Riemann-integrable functions R(I) in the L^{p}-metric. In particular, every Cauchy sequence in L^{p}(I) converges to a function in L^{p}(I).

Therefore the L^{p}-spaces with the L^{p}-norms are complete normed spaces. Such spaces are so common and important that they have a special name, Banach^{5} spaces. See Appendix B for a definition of a complete normed space, and see Section 2.4 for some density results. You learn about Banach spaces in a functional analysis course. Classical graduate textbooks include [Fol], and the book by Martin Schechter [Sch].

Among the L^{p}-spaces, both L^{1} and L^{∞} play special roles. The space

L^{2}(I) := {f : I → ℂ : ∫_{I} |f(x)|^{2} dx < ∞}

of square-integrable functions is also very important. The term square-integrable emphasizes that the integral of the square of (the absolute value of) f ∈ L^{2}(I) is finite. Functions in L^{2}(I) are also said to have finite energy.

The L^{2}-norm is induced by an inner product,

(2.7)  ⟨f, g⟩_{L^{2}(I)} := ∫_{I} f(x)\overline{g(x)} dx,

meaning that ∥f∥_{L^{2}(I)} = \sqrt{⟨f, f⟩_{L^{2}(I)}}.

The space L^{2}(I) of square-integrable functions over I is a complete innerproduct vector space (a Hilbert^{6} space), and so it has geometric properties very similar to those of n-dimensional Euclidean space ℝ^{n}. In particular the notion of orthogonality is very important. See Appendix B for a more detailed description of inner-product vector spaces and Hilbert spaces. Another reference is [SS2, Chapters 4-5].

In Chapter 5 we will be especially interested in the Hilbert space L^{2}(𝕋) of square-integrable functions on 𝕋 with norm

∥f∥_{L^{2}(𝕋)}^{2} := \frac{1}{2π} ∫_{−π}^{π} |f(θ)|^{2} dθ,

induced by the inner-product

⟨f, g⟩ := \frac{1}{2π} ∫_{−π}^{π} f(θ)\overline{g(θ)} dθ.

The factor \frac{1}{2π} is just a normalization constant that forces the trigonometric functions e_{n}(θ) := e^{inθ} to have norm one in L^{2}(𝕋).

5Named after the Polish mathematician Stefan Banach (1892–1945).

6Named after the German mathematician David Hilbert (1862–1943).

Exercise 2.18. Verify that

⟨e_{n}, e_{m}⟩ = δ_{n,m},

in particular, ∥e_{n}∥_{L^{2}(𝕋)} = 1. This shows, in language we will meet in Chapter 5, that the trigonometric system is an orthonormal system.

2.1.3. Ladder of function spaces on 𝕋. So far in this chapter, we have been concerned with function spaces defined by integrability properties. Earlier we encountered function spaces defined by differentiability properties. It turns out that for functions defined on a given bounded and closed interval, all these function spaces are nicely nested.

Continuous functions on a closed and bounded interval I = [a, b] are Riemann integrable, and hence Lebesgue integrable. The collection of continuous functions on I = [a, b] is denoted by C([a, b]). Next, C^{k}([a, b]) denotes the collection of ktimes continuously differentiable functions on [a, b], in other words functions whose k^{th} derivatives exist and are continuous. For convenience we sometimes use the expression ‘f is C^{k}’ to mean ‘f ∈ C^{k}’. Finally, C^{∞}([a, b]) denotes the collection of functions on [a, b] that are differentiable infinitely many times. (At the endpoints a and b, we mean continuous and differentiable from the right and from the left respectively.)

Recall that in Section 1.3.2 we discussed periodic functions on the unit circle 𝕋 = [−π, π), as well as functions that are continuously differentiable on 𝕋. In that context we meant that f : 𝕋 → ℂ is C^{k} if f is k-times continuously differentiable on 𝕋 and if when f is extended periodically, f^{(j)}(−π) = f^{(j)}(−π) for all 0 ≤ j ≤ k. This boundary condition implies that when we restrict f to the closed interval [−π, π] it is C^{k}, in particular it is Riemann integrable on [−π, π). Define L^{p}(𝕋) to be L^{p}([−π, π)).

For complex-valued functions defined on 𝕋, or on any closed and bounded interval I = [a, b], all these spaces are nicely nested. There is a ladder of function spaces in 𝕋, with C^{∞}(𝕋) being the smallest space, and L^{1}(𝕋) the largest,

\begin{aligned}C^{∞}(𝕋) ⊂ C^{k} & (𝕋) ⊂ C^{2}(𝕋) ⊂ C^{1}(𝕋) ⊂ C(𝕋) \\ & ⊂ R(𝕋) ⊂ B(𝕋) ⊂ L^{∞}(𝕋) ⊂ L^{p_{1}}(𝕋) ⊂ L^{p_{2}}(𝕋) ⊂ L^{1}(𝕋), \end{aligned} ; (2.8)

for integers k ≥ 2 and real numbers p_{1} and p_{2} such that p_{2} < p_{1}. In particular notice that

L^{2}(𝕋) ⊂ L^{1}(𝕋).

(Warning: For functions defined on the real line, the L^{p}-spaces are no longer nested, nor are the continuous functions necessarily bounded or integrable.)

We have already encountered a family of function spaces on 𝕋, namely, the collection of 2π-periodic trigonometric polynomials of degree N. We denote these spaces by P_{N}(𝕋). More precisely,

P_{N}(𝕋) = { ∑_{|n|≤N} a_{n}e^{inθ} : a_{n} ∈ ℂ, |n| ≤ N}.

These spaces are clearly nested as N increases, and they are certainly subsets of C^{∞}(𝕋), so we have a ladder starting at P_{0}(𝕋) (the constant functions) and climbing all the way to C^{∞}(𝕋),

P_{0}(𝕋) ⊂ P_{N_{1}}(𝕋) ⊂ P_{N_{2}}(𝕋) ⊂ C^{∞}(𝕋),

for all 0 ≤ N_{1} ≤ N_{2}.

The step functions defined in Section 2.1.1 are Riemann integrable, although most of them have discontinuities. This will add a branch to the ladder,

{ step functions on I} ⊂ R(I). The simple functions mentioned in Section 2.1.2 are bounded functions not

necessarily Riemann integrable, however they are Lebesgue integrable, and in fact simple functions are in L^{p}(I) for all 1 ≤ p ≤ ∞. This will add another branch to the ladder,

{ simple functions on I} ⊂ B(I).

Exercise 2.19. Compute the L^{p}-norm of a step function defined on an interval I. Compute the L^{p}-norm of a simple function defined on an interval I. Use the notation m(A) for the measure of a measurable set A ⊂ I.

We will encounter other classes of functions on 𝕋, [a, b], or ℝ, such as functions that are piecewise continuous or piecewise smooth, monotone functions, functions of bounded variation, Lipschitz^{7} functions, and Hölder^{8}-continuous functions. We define what we mean as they appear in the text. As you encounter these spaces, you are encouraged to try to fit them in the ladder of spaces in Figure 2.1.4, which will start looking more like a tree with branches and leaves.

The results of the following exercises justify some of the inclusions in the ladder of function spaces. Operate with the Lebesgue integrals as you would operate with Riemann integrals. Try to specify what properties of the integral you are using.

Exercise 2.20. Let I be a bounded interval. Show that if f : I → ℂ is bounded by M > 0 and f ∈ L^{p}(I), then f ∈ L^{q}(I) for all p ≤ q < ∞, furthermore,

∫_{I} |f(x)|^{q}dx ≤ M^{q−p} ∫_{I} |f(x)|^{p}dx < ∞.

In the language of norms this inequality can be written as

∥f∥_{L^{q}(I)}^{q} ≤ ∥f∥_{L^{∞}(I)}^{q−p}∥f∥_{L^{p}(I)}^{p}.

For the last exercise you may find useful Hölder’s inequality.

Lemma 2.21 (Hölder’s Inequality). If 1 ≤ s ≤ ∞, \frac{1}{s} + \frac{1}{t} = 1, f ∈ L^{s}(I), and

g ∈ L^{t}(I), then

(2.9)  |∫_{I} f(x)g(x) dx| ≤ ∥f∥_{L^{s}(I)}∥g∥_{L^{t}(I)}.

Hölder’s inequality will reappear in later chapters, and it will be discussed and proved, together with other very important inequalities, at the end of the book. When s = t = 2 this is the celebrated Cauchy-Schwarz^{9}inequality. Inequalities are inherent to the way of analysis, and we will see them over and over throughout the book.

7Named after the German mathematician Rudolf Otto Sigismund Lipschitz (1832–1903).

8Named after the German mathematician Otto Ludwig Hölder (1859–1937).

9This inequality is named after the French mathematician Augustin Louis Cauchy (1789– 1857), and the German mathematician Hermann Amandus Schwarz (1843–1921). The Cauchy– Schwarz inequality was discovered 25 years earlier by the Ukranian mathematician Viktor Yakovlevich Bunyakovskii (1804–1889), and sometimes it is called the Cauchy–Bunyakovskii–Schwarz inequality.

Exercise 2.22. Let I be a bounded interval. Show that if 1 ≤ q ≤ p < ∞ and h ∈ L^{p}(I) then h ∈ L^{q}(I), moreover show that if h ∈ L^{p}(I) then

∥h∥_{L^{q}(I)} ≤ |I|^{1/q−1/p}∥h∥_{L^{p}(I)}.

Thus for those exponents, L^{p}(I) ⊂ L^{q}(I). Hint: Apply Hölder’s inequality (2.9) on I, with s = p/q > 1, f = |h|^{q}, g = 1, h ∈ L^{p}(I).

2.1.4. Sets of measure zero. For many purposes we often only need that a function has a given property except on an exceptional set. Such exceptional sets are small, and the precise notion of smallness is the concept of a set of measure zero that we introduce now.

Definition 2.23. A set E ⊂ ℝ has measure zero if, for each ε > 0, there are open intervals I_{j} such that E is covered by the collection of intervals I_{j}, and the total length of the intervals is less than ε:

(i) E ⊂ ⋃_{j=1}^{∞} I_{j}, and (ii) ∑_{j=1}^{∞} |I_{j}| < ε.

We say that a given property holds almost everywhere if the set of points where the property does not hold has measure zero. Often we use the short-hand a.e. for almost everywhere.

In particular, every countable set has measure zero. There are uncountable sets that have measure zero. The most famous example is the Cantor^{10} set, discussed in Example 2.25.

Exercise 2.24. Show that finite subsets of ℝ have measure zero. Show that countable subsets of ℝ have measure zero. In particular the set ℚ ⋂ ℝ has measure zero.

Example 2.25. (The Cantor set) The Cantor set C is obtained from the closed interval [0, 1] by a sequence of successive deletions of open intervals called the middle thirds as follows: Remove the middle third interval (1/3, 2/3) of [0, 1]. We are left with two closed intervals: [0, 1/3] and [2/3, 1]. Remove the two middle third intervals (1/9, 2/9) and (7/9, 8/9) of the remaining closed intervals. We are left with four closed intervals: [0, 1/9], [2/9, 1/3], [2/3, 7/9], and [8/9, 1]. Delete the four middle thirds intervals (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), and (25/27, 26/27) of the remaining closed intervals.

At the n^{th} stage we are left with 2^{n} disjoint closed intervals each of length 3^{n−1}. Removing the 2^{n} middle third intervals, each of length 1/3^{n}, we are left with 2^{n+1} disjoint closed intervals of length 1/3^{n}. This process can be continued indefinitely, so that the set of all points removed from the closed interval [0, 1] is the union of a sequence of disjoint open intervals, hence is an open set A ⊂ [0, 1].

The Cantor set C is defined to be the remaining set, that is C := [0, 1]\A. The Cantor set can be seen to coincide with the set of numbers in [0, 1] whose ternary expansion (base three) uses only the digits 0 and 2. Once this identification is made, one can verify that C is in a one-to-one (some special points in a two-to-one) correspondence with the interval [0, 1]. Therefore C has the same cardinality as ℝ, hence it is an uncountable set. However C has measure zero because its complement A has measure one^{11} since the sum of the lengths of the disjoint intervals removed is

10Named after the German mathematician Georg Cantor (1845–1918).

m(A) := 1 × \frac{1}{3} + 2 × \frac{1}{9} + 4 × \frac{1}{27} + ⋯ = ∑_{n=1}^{∞} \frac{2^{n−1}}{3^{n}} = 1.

The last equality holds because we are dealing with a geometric sum with ratio r = 2/3 < 1. ♢

Exercise 2.26. Verify that the Cantor set C is closed. Use Definition 2.23 to verify that C has measure zero.

Aside 2.27. We assume that the reader is familiar with basic notions of point set topology on ℝ, and in particular with the concepts of open and closed sets in ℝ. To review these notions, see for example [Tao2, Chapter 12].

Here is an important characterization of Riemann-integrable functions, which appeared in Lebesgue’s doctoral dissertation. A proof can be found in Stephen Abbott’s book [Abb, Section 7.6].

Theorem 2.28 (Lebesgue’s Theorem, 1901). A bounded function f : I → ℂ is Riemann integrable if and only if f is continuous almost everywhere.

We can now define what the terms essentially bounded and essential supremum mean.

Definition 2.29. A function f : 𝕋 → ℂ is essentially bounded if it is bounded except on a set of measure zero. More precisely, there exists a constant M > 0 such that |f(x)| ≤ M a.e. in I.

Definition 2.30. Let f : I → ℂ be an essentially bounded function. The essential supremum of f is defined to be the smallest constant M such that |f(x)| ≤ M a.e. on the interval I. In other words,

ess sup_{x∈I} |f(x)| = inf{M ≥ 0 : |f(x)| ≤ M a.e. in I}.

Exercise 2.31. Verify that if f is bounded on I then f is essentially bounded. In other words, the space B(I) of bounded functions on I is a subset of L^{∞}(I). Show that it is a proper subset. Furthermore, verify that if f is bounded then ess sup_{x∈I} |f(x)| = sup_{x∈I} |f(x)|.

Remark 2.32. For the purposes of Lebesgue integration theory, functions that are equal almost everywhere are regarded as being the same. To be really precise, one should talk about equivalence classes of L^{p}-integrable functions: f is equivalent to g if and only if f = g a.e. In that case ∫_{I} |f − g|^{p} = 0, and if f ∈ L^{p}(I) then so is g. We say f = g in L^{p}(I) if f = g a.e. In that case, ∫_{I} |f|^{p} = ∫_{I} |g|^{p}. In particular the Dirichlet function g in Example 2.14 is equal a.e. to the zero function which is in L^{p}([0, 1]). Hence g ∈ L^{p}([0, 1]), its integral in the Lebesgue sense vanishes, and its L^{p}-norm is zero for all 1 ≤ p ≤ ∞.

11Recall that a natural way of defining the length or measure of a subset A of ℝ which is a countable union of disjoint intervals {I_{j}} is to declare the measure of A to be the sum of the lengths of the intervals I_{j}.

Now that we have cleared all definitions, we summarize in Figure 2.1.4 the ladder of function spaces discussed in the Section 2.1.3. In Figure 2.1.4, p_{1} and p_{2} are real numbers such that 1 < p_{1} < 2 < p_{2} < ∞, N_{1} and N_{2} are natural numbers such that N_{1} ≤ N_{2}. The blank box appears because there is no commonly used symbol for the class of differentiable functions. Notice that as we move down the ladder into the smaller function classes, the functions become better behaved. For instance, the L^{p} spaces on 𝕋 are nested and decreasing as the real number p increases from 1 to ∞. Two warnings are in order. First, when the underlying space is changed from the circle 𝕋 to another space X, the L^{p} spaces need not be nested: for real numbers p, q with 1 ≤ p ≤ q ≤ ∞, L^{p}(X) need not contain L^{q}(X). For example, the L^{p} spaces on the real line ℝ are not nested. (By contrast, the C^{k} classes are always nested: C^{k}(X) ⊃ C^{k+1}(X) for all spaces X and for all k ∈ ℕ.) Second, there are important function classes that do not fall into the nested chain of classes shown here.

2.2. Modes of convergence

Given a bounded interval I and functions g_{n}, g : I → ℂ, what does it mean to say ‘g_{n} → g’? There are many different ways in which the g_{n}’s could approximate g: pointwise, uniformly, almost everywhere, in the sense of convergence in the mean or in L^{1}, in the sense of mean-square convergence or in L^{2}, in the sense of convergence in L^{p}. Some of these modes of convergence are stronger than others, uniform convergence being the strongest. We will discuss all these modes of convergence and their interrelations in this section.

In Chapter 3, we will be concerned with pointwise and uniform convergence of Fourier series, and in Chapter 5 we will be concerned with mean-square convergence of Fourier series. At the end of the book, we come full circle and show that convergence in L^{p}(𝕋) of Fourier series is a consequence of the boundedness in L^{p}(𝕋) of the periodic Hilbert transform.

Definition 2.33. A sequence of functions g_{n} : I → ℂ converges pointwise to g : I → ℂ if for each x ∈ I,

lim_{n→∞} g_{n}(x) = g(x).

Equivalently, if given ε > 0 and given x ∈ I there exists N = N(ϵ, x) > 0 such that for all n > N

|g_{n}(x) − g(x)| < ε.

Definition 2.34. A sequence of functions g_{n} : I → ℂ converges almost everywhere to g : I → ℂ if it converges pointwise except on a set of measure zero, that is

lim_{n→∞} g_{n}(x) = g(x) a.e. in I.

Equivalently if there exists a set of measure zero E ⊂ I such that given x ∈ I \ E and ε > 0 there exists N = N(ϵ, x) > 0 such that for all n > N

|g_{n}(x) − g(x)| < ε.

Definition 2.35. A sequence of bounded functions g_{n} : I → ℂ converges uniformly to g : I → ℂ if given ε > 0 there exists N > 0 such that for all n > N

|g_{n}(x) − g(x)| < ε for all x ∈ I.

Equivalently, lim_{n→∞} sup_{x∈I} |g_{n}(x) − g(x)| = 0.

⋮⋮⋮⋮⋮ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ L^{1}(𝕋) L^{p_{1}}(𝕋) L^{2}(𝕋) L^{p_{2}}(𝕋) L^{∞}(𝕋) B(𝕋) R(𝕋) C(𝕋) C^{1}(𝕋) C^{2}(𝕋) ⋮ C^{k}(𝕋) ⋮ C^{∞}(𝕋) ⋮ P_{N_{2}}(𝕋) P_{N_{1}}(𝕋) ⋮ P_{0}(𝕋)  Lebesgue integrable: ∫_{𝕋} |f(x)| dx < ∞ ∫_{𝕋} |f(x)|^{p_{1}} dx < ∞ , 1 < p_{1} < 2 square integrable: ∫_{𝕋} |f(x)|^{2} dx < ∞ ∫_{𝕋} |f(x)|^{p_{2}} dx < ∞ , 2 < p_{1} < ∞ essentially bounded: ess sup_{x∈𝕋} |f(x)| < ∞ bounded functions: |f(x)| ≤ M for all x ∈ 𝕋 Riemann integrable continuous differentiable continuously differentiable: f′ exists and is continuous twice continuously differentiable: f, f′ exist and are continuous k times continuously differentiable: f, f′, …, f^{(k)} exist and are continuous infinitely differentiable f, f′, f′′,…, exist and are continuous trigonometric polynomials of degree N_{2} > N_{1} trigonometric polynomials of degree N_{1} > 0 constant functions

Figure 2.4. Ladder of nested classes of functions, for functions f : 𝕋 → ℂ.

Definition 2.36. A sequence of L^{∞}-functions g_{n} : I → ℂ converges uniformly outside a set of measure zero to g : I → ℂ if lim_{n→∞} ∥g_{n} − g∥_{L^{∞}(I)} = 0. We often write g_{n} → g in L^{∞}(I).

Remark 2.37. If the functions g_{n}, and g are continuous then ∥g_{n} − g∥_{L^{∞}(I)} = sup_{x∈I} |g_{n}(x) − g(x)|, and in this case, convergence in L^{∞}(I) and uniform convergence coincide. This is why the L^{∞}-norm for continuous functions on closed and bounded intervals is often called the uniform norm. The metric space of continuous functions over a closed and bounded interval with the metric induced by the uniform norm is a complete metric space, or a Banach space, see [Tao2, Section 4.4]. In particular this means that if the functions g_{n} in Definition 2.35 are continuous and converge uniformly, then the limiting function is itself continuous; see Theorem 2.52.

Definition 2.38. A sequence of integrable functions g_{n} : I → ℂ converges in the mean or in L^{1}(I) to g : I → ℂ if for each ε > 0 there exists N > 0 such that for all n > N

∫_{I} |g_{n}(x) − g(x)| dx < ε.

Equivalently, lim_{n→∞} ∥g_{n} − g∥_{L^{1}(I)} = 0. We often write g_{n} → g in L^{1}(I).

The integral in the preceding definition is the Lebesgue integral, and by integrable we mean Lebesgue integrable. Similarly for the next two definitions.

Definition 2.39. A sequence of square-integrable functions g_{n} : I → ℂ converges in mean-square or in L^{2}(I) to g : I → ℂ if for each ε > 0 there exists N > 0 such that for all n > N

∫_{I} |g_{n}(x) − g(x)|^{2} dx < ε.

Equivalently, lim_{n→∞} ∥g_{n} − g∥_{L^{2}(I)} = 0. We often write g_{n} → g in L^{2}(I).

Figure 2.5 illustrates the relations between the different modes of convergence. Convergence in L^{1}(I) and in L^{2}(I) are particular cases of convergence in L^{p}(I).

Definition 2.40. A sequence of L^{p}-integrable functions g_{n} : I → ℂ converges in L^{p}(I) for 1 ≤ p < ∞ to g : I → ℂ if given ε > 0 there exists N > 0 such that for all n > N

∫_{I} |g_{n}(x) − g(x)|^{p} dx < ε.

Equivalently, lim_{n→∞} ∥g_{n} − g∥_{L^{p}(I)} = 0. We often write g_{n} → g in L^{p}(I).

We already noted that the L^{p}-spaces are complete normed spaces or Banach spaces, this means that the limiting functions in Definition 2.40 have no other choice that to be in L^{p}(I) themselves. In particular, in Definitions 2.38 and 2.39 the limiting functions must be in L^{1}(I) and L^{2}(I) respectively.

Exercise 2.41. Verify that uniform convergence on a bounded interval I implies the other modes of convergence on I.

Exercise 2.42. Show that for 1 ≤ q < p ≤ ∞ convergence in L^{p}(I) implies convergence in L^{q}(I). Hint: You may find Hölder’s Inequality (2.9) on I useful.

Uniform convergence: |g_{n}(x) − g(x)| → 0 uniformly in x. For each ε > 0 there is an N that works for all x ∈ 𝕋.

Pointwise convergence:

|g_{n}(x) − g(x)| → 0 for all x ∈ 𝕋.

For each ε > 0 and each x ∈ 𝕋 there is an N = N(x, ε) that works.

L^{2} (mean-square) convergence: ∫_{𝕋} |g_{n}(x) − g(x)|^{2} dx → 0.

Pointwise a.e. convergence:

|g_{n}(x) − g(x)| → 0 for almost all x ∈ 𝕋.

For each ε > 0 and almost every x ∈ 𝕋 there is an N = N(x, ε) that works.

L^{1} (mean) convergence:

∫_{𝕋} |g_{n}(x) − g(x)| dx → 0.

⇓ ⇓

⟹

⟹

Figure 2.5. Relations between different modes of convergence, for functions g_{n}, g : 𝕋 → ℂ, showing which modes imply which other modes. All potential implications other than those shown here are false.

The following examples show that the implications go only in the directions indicated in Figure 2.5. In these examples we are working on the unit interval [0, 1].

Example 2.43. (Pointwise but not uniform) The sequence of functions defined by f_{n}(x) = x^{n} for x ∈ [0, 1] converges pointwise to the function f defined by f(x) = 0 for 0 ≤ x < 1 and f(1) = 1, but it does not converge uniformly to f or to any other function. See Figure 2.6. ♢ Figure 2.6. The functions f_{n}(x) = x^{n} converge pointwise, but not uniformly, on [0, 1] to the function f(x) that is 0 for x ∈ [0, 1) and 1 at x = 1.

Example 2.44. (Pointwise but not in the mean) The sequence of functions g_{n} : [0, 1] → ℝ defined by

g_{n}(x) = \begin{cases}n, & if 0 < x < 1/n; \\ 0, & otherwise, \end{cases}

converges pointwise everywhere to g(x) = 0, but it does not converge in the mean or in L^{2}, nor does it converge uniformly. See Figure 2.7. ♢

Example 2.45. (In the mean but not pointwise) Given a positive integer n, let m, k be the unique non-negative integers such that n = 2^{k} + m and 0 ≤ m < 2^{k}. The sequence of functions h_{n}(x) : [0, 1] → ℝ defined by

h_{n}(x) = \begin{cases}1, & if \frac{m}{2^{k}} < x < \frac{m + 1}{2^{k}}; \\ 0, & otherwise, \end{cases}

converges in the mean to the zero function, but it does not converge pointwise at any point x ∈ [0, 1]. See Figure 2.8. This sequence of functions is sometimes called the “walking function”. ♢

Despite these examples, there is a relation between pointwise convergence and convergence in L^{p}. If f_{n}, f ∈ L^{p}(𝕋) for 1 ≤ p < ∞, and if f_{n} → f in L^{p}, then there exists a subsequence that converges to f a.e. on 𝕋. See [SS2, Corollary 2.2].

Exercise 2.46. Verify that the sequences described in Examples 2.43, 2.44, and 2.45 do have the convergence properties described there. Adapt the sequences in the examples to have the same convergence properties on the interval [a, b].

Figure 2.7. The functions g_{n} defined in Example 2.44 converge pointwise, but not in the mean, in L^{2}, or uniformly.

For yet more modes of convergence, such as convergence in measure and almost uniform convergence (not to be confused with uniform convergence outside a set of measure zero), see [Bart1, pp.68–72].

2.3. Interchanging limit operations

Uniform convergence on bounded closed intervals, is a very strong form of convergence which is amenable to interchange with other limiting operations and properties defined by limits, like integration, differentiation and continuity. We quote here several results that will be used throughout the text, and that belong to a course in advanced calculus. Some results do involve uniform convergence as one of the limit operations, some others involve other limit operations (like interchanging two integrals).

Theorem 2.47 (Interchange of Limits and Integrals). If f_{n} converges uniformly to f in [a, b] then

lim_{n→∞} ∫_{a}^{b} f_{n} = ∫_{a}^{b} lim_{n→∞} f_{n} = ∫_{a}^{b} f.

In other words, The limit of the integrals of a uniformly convergent sequence of functions on [a, b] is the integral of the limit function on [a, b].

Notice that the functions in Example 2.44 do not converge uniformly, and they provide an example where this interchange fails. In fact, ∫_{0}^{1} g_{n} = 1 for all n and ∫_{0}^{1} g = 0, so

lim_{n→∞} ∫_{0}^{1} g_{n} = 1 ≠ 0 = ∫_{0}^{1} lim_{n→∞} g_{n} = ∫_{0}^{1} g.

34 2. INTERLUDE

Figure 2.8. The functions h_{n}(x) defined in Example 2.45 converge in mean, but do not converge pointwise at any x.

Exercise 2.48. Given the functions g_{n} : I → ℂ, n ∈ ℕ, suppose the partial

sums ∑_{n=0}^{N} g_{n}(x) converge uniformly on I as N → ∞. Show that

∫_{I} ∑_{n=0}^{∞} g_{n}(x) dx = ∑_{n=0}^{∞} ∫_{I} g_{n}(x) dx.

Can interchange series and integration provided the convergence of the partial sums is uniform.

Under certain conditions, involving uniform convergence of a sequence of functions and their derivatives, one can exchange differentiation and the limit.

Theorem 2.49 (Interchange of Limits and Derivatives). Let f_{n} : [a, b] → ℂ be a sequence of C^{1} functions converging uniformly on [a, b] to the function f. Assume that the derivatives (f_{n})′ also converge uniformly on [a, b] to some function g. Then f is C^{1}, and f′ = g, that is

lim_{n→∞}(f_{n})′ = ( lim_{n→∞} f_{n})′ .

In other words, Can interchange limits and differentiation provided the convergence of the functions and the derivatives is uniform.

Exercise 2.50. Show that the hypotheses in Theorem 2.49 are necessary.

Exercise 2.51. What are the necessary hypothesis on a series ∑_{n=0}^{∞} g_{n}(x) so that

it is legal to interchange sum and differentiation? That is when can we guarantee that the following holds,

(∑_{n=0}^{∞} g_{n}(x))′ = ∑_{n=0}^{∞}(g_{n})′(x).

Here is another classical and useful results in the same vein, already mentioned in Remark 2.37.

Theorem 2.52. The uniform limit of continuous functions on [a, b] is continuous on [a, b].

Notice that this is the same as saying that if g_{n} → g uniformly, and g_{n}’s are continuous then for all x_{0} ∈ [a, b],

lim_{x→x_{0}} lim_{n→∞} g_{n}(x_{0}) = lim_{n→∞} lim_{x→x_{0}} g_{n}(x).

Perhaps inserting a few more equalities will make above statement clear:

lim_{x→x_{0}} lim_{n→∞} g_{n}(x_{0}) = lim_{x→x_{0}} g(x)
= g(x_{0})
= lim_{n→∞} g_{n}(x_{0})
= lim_{n→∞} lim_{x→x_{0}} g_{n}(x).

Example 2.43 shows a sequence of continuous functions on [0, 1] that converge pointwise to a discontinuous function f, hence the convergence cannot be uniform (as you were asked to verify), nor can the limit function be continuous (f is discontinuous at x = 1).

There is a partial converse to this result that is called Dini’s Theorem.

Theorem 2.53 (Dini’s Theorem). Suppose f, f_{n} : I → ℝ are continuous functions. Assume (i) f_{n} converges pointwise to f as n → ∞; (ii) f_{n} is a decreasing sequence, namely f_{n}(x) ≥ f_{n+1}(x) for all x ∈ I. Then f_{n} converges uniformly to f.

Exercise 2.54. Prove Dini’s theorem, and show that condition (ii) cannot be removed. A continuous version of Example 2.44, where the “steps” are replaced by “tents” might help.

Interchanging integrals is another instance of these interchange of limit operations. The theorems that allow for such interchange go by the name Fubini’s Theorems^{12}. Here is the statement of one such theorem valid for continuous functions.

Theorem 2.55 (Fubini’s Theorem for Continuous Functions). Let I and J be closed intervals. Let F : I × J → ℂ be a continuous function. Assume

∫∫_{I×J} |F (x, y)| dA < ∞,

where dA denotes the differential of area. Then

∫_{I} ∫_{J} F (x, y) dx dy = ∫_{J} ∫_{I} F (x, y) dy dx = ∫∫_{I×J} F (x, y) dA.

Usually Fubini’s Theorem refers to an L^{1}-version of Theorem 2.55 that is stated and proved in every measure theory book. You can find a 2-dimensional version in [Tao2, Section 19.5].

Interchanging limit operations is a delicate maneuver that cannot always be accomplished. We will illustrate throughout the book settings where this interchange is allowed and settings where this interchange is illegal, like the ones just described involving uniform convergence. If we insist on carrying on the interchange, false conclusions could be deduced.

Unlike the Riemann integral, the Lebesgue theory allows for the interchange of a pointwise limit and an integral. There are several landmark theorems that you learn in a course on measure theory. We state one such result.

Theorem 2.56 (Lebesgue Dominated Convergence Theorem). Given a sequence of measurable functions f_{n} defined on the interval I, such that they converge pointwise to f. Suppose there exists a “dominating function” g ∈ L^{1}(I), meaning that |f_{n}(x)| ≤ g(x) a.e. for each n > 0. Then

lim_{n→∞} ∫_{I} f_{n}(x) dx = ∫_{I} f(x) dx.

Exercise 2.57. Verify that the functions in Example 2.44 cannot be dominated by an integrable function g.

12Named after the Italian mathematician Guido Fubini (1879-1943).

2.4. Density

Having different ways to measure convergence of functions to other functions provides ways of deciding when we can approximate functions in a given class by functions in another class. In particular we can decide when a subset is dense in a larger set.

Definition 2.58. Given a normed space X of functions over a bounded interval I, with norm denoted by ∥ · ∥_{X}, we say that a subset A ⊂ X is dense in X if given any f ∈ X there exists a sequence of functions f_{n} ∈ A such that

lim_{n→∞} ∥f_{n} − f∥_{X} = 0.

Equivalently, A is dense in X if given any ε > 0 there exists a function h ∈ A such that

∥h − f∥_{X} < ε.

One can deduce from the definition of the Riemann-integrable functions on a bounded interval I that the step functions (see Definition 2.3) are dense in R(I) with respect to the L^{1}-norm; see Proposition 2.10. We state this result as a theorem, for future reference.

Theorem 2.59. Given a Riemann-integrable function on a bounded interval I, f ∈ R(I), and given ε > 0, there exists a step function h such that

∥h − f∥_{L^{1}(I)} < ε.

Remark 2.60. We can choose the step functions that approximate a given Riemann-integrable function to be below the function (or above).

In Chapter 4 we will learn that

Trigonometric polynomials can approximate uniformly continu-

ous functions on 𝕋.

This is the celebrated Weierstrass theorem (see Theorem 3.4) that we will prove in this book.

In Chapter 8 we will learn that on bounded and closed intervals,

Step functions can approximate uniformly continuous functions, despite the fact that step functions are not continuous.

It is not hard to deduce directly from the fact that continuous functions can be approximated uniformly by step functions that they can also be approximated in the L^{p}-norm by step functions.

Step functions can approximate continuous functions in the L^{p}-

norm. What is perhaps surprising is that

Continuous functions can approximate step functions pointwise

and in the L^{p}-norm. We will verify this result in Chapter 4, when we introduce the concepts of

convolution and approximations of the identity. If you draw a picture of a step function and then round-off the corners, this result might seem very reasonable, see Figure 2.4

Assuming these results, that we will prove later in the book, we can show the density of the continuous functions in the class of Riemann-integrable functions and in L^{p}-spaces.

Figure 2.9. The graph shows a step function whose corners have been rounded.

Theorem 2.61. Continuous functions on a bounded and closed interval I are dense in R(I) with respect to the L^{p}-norm.

Proof. Start with f ∈ R(I). There is a step function h that is ε-close in the L^{p}-norm to f, that is ∥f −h∥_{L^{p}(I)} < ε. There is also a continuous function g ∈ C(I) that is ε-close in the L^{p}-norm to the step function h, that is ∥h − g∥_{L^{p}(I)} < ε. The continuous function g is 2ε-close in the L^{p}-norm to our initial function f, by the triangle inequality,

∥f − g∥_{L^{p}(I)} ≤ ∥f − h∥_{L^{p}(I)} + ∥h − g∥_{L^{p}(I)} ≤ 2ε.

Hence continuous functions can get arbitrarily close in L^{p}-norm to any Riemannintegrable function. □

Remark 2.62. We will use a very precise version of this result for p = 1 in Chapter 4, see Lemma 4.8, where we require the approximating continuous functions to be uniformly bounded by the same bound that controls the Riemannintegrable function. Here is the idea. Given f ∈ R(I) and ε > 0, there exists a continuous function g on I (by Theorem 2.61) such that ∥g − f∥_{L^{1}(I)} < ε. Let f be bounded by M > 0, that is |f(x)| ≤ M for all x ∈ I. If g is bounded by M there is nothing to be done. If g is not bounded by M, consider the new function defined on I by g_{0}(x) = g(x) if |g(x)| ≤ M, and g(x) = M otherwise. One can show that g_{0} is continuous, g_{0}(x) ≤ M for all x ∈ I, and g_{0} is “closer” to f than g is. In particular the L^{1}-norm of the difference is at most ε, and possibly even smaller.

Exercise 2.63. Show that g_{0} as defined in Remark 2.62 satisfies the properties claimed there.

Exercise 2.64. Let X be a normed space of functions over a bounded interval I. Suppose A is a dense subset of X, and A ⊂ B ⊂ X, then B is a dense subset of X (all with respect to the norm of the ambient space X).

Exercise 2.65. (Density is a Transitive Property.) Let X be a normed space of functions over a bounded interval I. Suppose A is a dense subset of B, and B is a dense subset of X, then A is a dense subset of X (all with respect to the norm of the ambient space X).

Notice that the notion of density is strongly dependent on the norm or metric of the ambient space. For example, when considering nested L^{p}-spaces on a closed and bounded interval, in Section 2.1 we learned that C(I) ⊂ L^{2}(I) ⊂ L^{1}(I). Suppose we knew that the continuous functions are dense in L^{2}(I) (with respect to the L^{2}- norm), and the square-integrable functions are dense in L^{1}(I) (with respect to the L^{1}-norm). Even so, we could not use transitivity to conclude that the continuous functions are dense in L^{1}(I). We need to know more about the relation between the L^{1}- and the L^{2}-norms. In fact, to complete the argument, it suffices to know that ∥f∥_{L^{1}(I)} ≤ C∥f∥_{L^{2}(I)} (see Exercise 2.22), because that would imply that if the continuous functions are dense in L^{2}(I) (with respect to the L^{2}-norm), then they are also dense with respect to the L^{1}-norm, and now we can use the transitivity. Such norm inequality is stronger than the statement that L^{2}(I) ⊂ L^{1}(I). It says that the identity map or embedding E : L^{2}(I) → L^{1}(I), defined by Ef = f for f ∈ L^{2}(I), is a continuous map or a continuous embedding.

The following statements about various subsets of functions being dense in L^{p}(I) will not be proven since they rely in Lebesgue integration theory. However we will use these results for example to show that the trigonometric system is a basis in L^{2}(𝕋).

We know that L^{p}(I) is the completion of R(I) with respect to the L^{p}-norm; see Theorem 2.17. In particular,

The Riemann-integrable functions are dense in L^{p}(I) with re-

spect to the L^{p}-norm. With this fact in mind, we see that any set of functions that is dense in R(I)

with respect to the L^{p}-norm is also dense in L^{p}(I) with respect to the L^{p}-norm, by transitivity. For future reference we state here the specific density of the continuous functions in L^{p}(I).

Theorem 2.66. The continuous functions on a bounded and closed interval I are dense in L^{p}(I), for 1 ≤ p ≤ ∞.

Exercise 2.67. Given f ∈ L^{p}(I), 1 ≤ p < ∞, and ε > 0, and assuming all the theorems and statements in this section, show that there exists a step function h on I such that

∥h(x) − f(x)∥_{L^{p}(I)} < ε.