Integral, Measure and Derivative: A Unified Approach

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INTRODUCTION

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One of the basic concepts of analysis is that of the integral. The classical theory of integration, perfected in the middle of the last century by Cauchy and Riemann, is entirely adequate for solving many mathematical problems, both pure and applied. However, it does not meet the needs of a number of important branches of mathematics and physics of comparatively recent vintage, being deficient in at least three respects:

1. As classically defined, the integral applies only to functions of one or several variables, whereas nowadays one must be able to integrate over sets which cannot be described by a finite number of real parameters. This necessity arises, for example, in investigations ranging from probability theory and partial differential equations to hydrodynamics and quantum mechanics.

2. Even in the case of finitely many variables, only relatively few functions (e.g., those that are continuous, piecewise continuous or satisfy other rather strong requirements) can be integrated by using Riemann’s classical definition of the integral. Some indication of the smallness of the class of Riemann-integrable functions is shown by the following fact: It is an easy matter to construct a sequence of functions {fn(x)} on the interval a x b, say, which satisfies the Cauchy convergence criterion in the mean, in the sense that

without the sequence having a limit function which is Riemann integrable. This lack of completeness of the class of Riemann-integrable functions is a grave drawback, since completeness is well-nigh indispensable in any branch of modern analysis.

3. In the classical theory, the domain of integration X (e.g., a line or a plane) is homogeneous in the sense that the values of integrals over X do not change if the integrands are shifted. However, there are many problems where X can no longer be regarded as homogeneous. One can often take account of this lack of homogeneity by introducing a variable density, as is done in problems involving the vibration of inhomogeneous strings. But this device entails certain difficulties. For example, how should one define the density of a string loaded with point masses?

The above remarks amply illustrate the inadequacy of the classical theory of integration. All these difficulties disappear in the modern theory of integration, developed by some of the leading mathematicians of our time, from Lebesgue to the present day. The new theory does not require the domain X to be either finite-dimensional or homogeneous, and leads to a sufficiently large class of integrable functions, in particular, a class which is complete relative to convergence in the mean.

In presenting the general theory of the integral, we have chosen the Daniell method as our basic approach. This method gets to the crux of the matter more quickly and directly than the original method of Lebesgue, since it is not based on preliminary construction of a theory of measure. Moreover, from the Daniell standpoint, measure theory itself is particularly simple and natural, appearing as an almost self-evident consequence of the theory of the integral. In this regard, it should be pointed out that the Lebesgue and Daniell constructions of the integral are equivalent if finite-valued (step) functions are chosen as the elementary functions. However, there are cases where functions other than step functions should be chosen as the elementary functions (e.g., in studying linear functional on the space of continuous functions defined on a compact metric space), and then the Daniell method is effective while the Lebesgue method is not.

Having made these preliminary observations, we now give a brief sketch of the contents of the book. Part 1 is devoted to the integral, and consists of three chapters. In the first, we define the Riemann integral for a continuous function of n variables as the limit of a sequence of lower Darboux sums, or, what amounts to the same thing, a nondecreasing sequence of step functions. This approach has the merit of pointing the way to further generalization, by axiomatization of certain special properties of integrals of step functions. The most basic of these properties is upper continuity, i.e., if a nonincreasing sequence of step functions converges to zero, then so do their integrals. This generalization is carried out in Chap. 2, starting from a family of elementary functions defined on an arbitrary set X and equipped with an elementary integral which satisfies the axioms suggested by corresponding properties of integrals of step functions. The family of elementary functions is then enlarged by taking monotonic passages to the limit and forming differences. The result is a space of summable functions, which is complete relative to the natural norm based on the new definition of the integral. Finally, in Chap. 3, we apply the general theory to functions of n real variables, thereby obtaining the classical Lebesgue integral.

In Part 2, we consider the Stieltjes integral, corresponding to the case where X is inhomogeneous. Chapter 4 is concerned with the Riemann-Stieltjes integral in n-space, constructed from a quasi-volume (i.e., an additive function of n-dimensional parallelepipeds, called blocks). Here we digress to indicate some applications of Stieltjes integration to classical analysis, based on the use of the Helly limit theorems. In Chap. 5 the Daniell scheme (described in Chap. 2) is used to construct the Lebesgue-Stieltjes integral in n-space, starting from continuous functions as elementary functions and the Riemann-Stieltjes integral as elementary integral. One can also start from step functions as elementary functions, as in the construction of the ordinary Lebesgue integral, but then an extra requirement of upper continuity must be imposed on the quasi-volume. However, this causes no trouble, since every quasi-volume σ of bounded variation is equivalent to an upper continuous quasi-volume , in the sense that Riemann-Stieltjes integrals of continuous functions have the same values with respect to both σ and .

In Part 3 the general Daniell scheme is used to develop a theory of measure. We start in Chap. 6 with a family of elementary functions defined on an arbitrary set X, equipped with an integral satisfying the conditions stipulated in Chap. 2. A function on X is said to be measurable if it is the limit of a sequence of elementary functions in the sense of convergence almost everywhere. In particular, every summable function is measurable. A subset E ⊂ X is said to be measurable if its characteristic function χE(x) is measurable and summable if χE(x) is summable. In the latter case, the measure of E is defined as the integral of χE(x). It follows at once from earlier considerations that measure is countably additive. Then we give an alternative definition of the integral of a summable function f(x), based on Lebesgue’s original approach, in terms of the measures of the sets on which f(x) takes values lying in given intervals. Chapters 7 and 8 are devoted to a deeper study of measure theory. The first of these chapters explores constructive measure theory, where general measurable sets are approximated by countable unions and intersections of particularly simple measurable sets (blocks in the case where X is n-space); the second deals with axiomatic measure theory, where a theory of the integral is constructed from a postulated elementary measure which is susceptible to various extensions. Here again, consistent use of the Daniell scheme leads to great simplifications, and the two approaches, axiomatization of the integral (Chap. 2) and axiomatization of measure (Chap. 8) finally blend into a single theory. We conclude Part 3 with an introduction to the theory of Lebesgue-Stieltjes integration in infinite-dimensional spaces, a topic of great current interest.

The last part of the book (Part 4) is devoted to the theory of the derivative. In Chap. 9 we consider two countably additive set functions defined on the same abstract set X, one of which is still called a measure since it is non-negative. For the other set function, which is in general signed (i.e., which takes values of either sign), we establish a canonical decomposition (relative to the measure) into a discrete component and a continuous component, afterwards decomposing the continuous component in turn into a singular component and an absolutely continuous component A(E). It turns out that A(E) is the integral over E of a summable function g(x), called the density of A(E) (this is the celebrated Radon-Nikodým theorem). Particularizing the theory to the case of functions of one variable, we obtain the classical Lebesgue decomposition of an arbitrary (point) function of bounded variation into the sum of three terms, i.e., a discrete component, a singular component and an absolutely continuous component. The problem of finding the density g(x) is examined in Chap. 10. This leads to the operation of differentiation, which we first study for the case where X is an interval a x b. We consider three different ways of defining the derivative, one based on special intervals (with binary rational end points), another on arbitrary intervals, and a third on arbitrary Borel sets. Each of these three definitions can be generalized to the case where X is an arbitrary set, equipped with a Borel measure. The first corresponds to differentiation with respect to a net, the second to differentiation with respect to a Vitali system, and the third to differentiation with respect to the class of all Borel subsets of X. In each case we prove that the derivative exists and equals the density g(x) almost everywhere. Finally, as special cases, we prove de Possel’s theorem on differentiation with respect to a net and Lebesgue’s theorem on differentiation of a function of bounded variation.

Part 1

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THE INTEGRAL

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1

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THE RIEMANN INTEGRAL

AND STEP FUNCTIONS

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1.1. The Riemann Integral

By an n-dimensional rectangular parallelepiped we mean a set of points x = (x1, …, xn) of the form

where, naturally, it is assumed that

For brevity, such parallelepipeds will henceforth be called blocks. The largest of the numbers b1 − a1, …, bn − an will be called the size of the block B, and the quantity

will be called the volume of the block. The function s(B) is an additive function of its argument, in the sense that if the block B is divided into subblocks B1,…, Bp with no interior points in common (such subblocks are said to form a partition of B), then

A block which is fixed during the course of a given discussion will be called the basic block, denoted by boldface B.

We now recall how Riemann integrals are constructed. Let f(x) be a bounded real function defined in a basic block B. Let be a partition of B into subblocks B1, …, Bp, and in each block Bk choose an arbitrary point ξk (k = 1, …, p). Then form the Riemann sum

Let d( ) denote the largest size of the blocks B1, …, Bp, and let 1, …, q, … be a sequence of partitions such that d( q) → 0. If the sequence has a limit as q → , which is independent of the choice of the sequence q [provided only that d( q) → 0] or of the points ξk Bk, then the limit is called the Riemann integral of the function f(x) [over the block B], and we write

One would now like to know the class of functions f(x) for which this limit exists. In his Cours d’Analyse (1821), Cauchy proved that the integral (1) exists if f(x) is continuous.¹ By 1837 Dirichlet had already observed that there are (discontinuous) functions which are not Riemann integrable.² Some time later, necessary and sufficient conditions for a function f(x) to have a Riemann integral were found by Riemann, du Bois-Reymond and Lebesgue. In every case, it turns out that a Riemann-integrable function cannot be too discontinuous. (Lebesgue’s criterion for Riemann integrability will be given in Sec. 1.7.) Subsequently, various requirements of the theory led to a search for more general definitions of the integral, applicable to a much wider class of functions. The most important such definition was given by Lebesgue in 1902 (for n = 1), and later by Radon and Fréchet in the period 1912–1915 (for the general case). The construction of the Lebesgue integral can be approached in a variety of ways. For the reasons given in the Introduction, we choose the approach due to Daniell (1918). But first we must say more about Riemann integrals.

1.2. Lower and Upper Integrals

Let be a partition of the block B into subblocks B1, …, Bp, and let

Then the expression

is called the lower Darboux sum of f(x), corresponding to the partition . Similarly,

is called the upper Darboux sum of f(x). Obviously, for any choice of the points ξk Bk (k = 1, …, p), we have

where

We now compare the values of the lower and upper sums for two different partitions and ′ of the same basic block B. First suppose ′ is obtained by further subdividing the blocks of the partition (in which case, ′ is called a refinement of ). Then every term mks(Bk) of the sum is replaced by a sum of the form

where

Since mk mk,

and hence

Thus, in going from the partition to the finer partition ′, the lower Darboux sum can only increase. Similarly, in going from to ′, the upper Darboux sum can only decrease.

Next let and ′ be arbitrary partitions. Then the set of all intersections of blocks of with blocks of ′ forms a new partition ″, which is a refinement of both and ′. But then, as just shown,

i.e., a lower Darboux sum can never exceed an upper Darboux sum. Suppose we write

where the supremum and infimum are taken with respect to all partitions of the block B. The first of these integrals is called the lower (Riemann) integral of f(x), and the second is called the upper integral of f(x), both over the block B. Then it follows from (2) that

THEOREM 1. If 1, …, q, … is a sequence of partitions of the block B such that d( q) → 0, then

and similarly,

Proof. Given any > 0, there is a partition such that

Consider the quantity . The blocks of the partition q fall into two groups: The first group, denoted by , consists of blocks which are entirely contained in blocks of , and the second group, denoted by , consists of blocks which intersect the boundaries of blocks B of . Correspondingly, we represent in the form

Let denote the intersections of the blocks with the blocks B . Adding the blocks to the blocks , we obtain a new partition ′q of the basic block B, which is a refinement of the partition . Consequently,

and hence

Now let G denote the total area of the boundaries of all the blocks of the partition . Since the blocks and intersect boundaries of the partition and have sizes no larger than d( q), each of the sums in the right-hand side of (5) is no larger than MG d( q) in absolute value. Choosing q such that

we clearly have

where we have used (5) and (6). This proves (3), and (4) is proved similarly.

If the Riemann integral of the function f(x) exists, then the upper and lower sums must have the same limit, and hence

for any sequence of partitions q such that d( q) → 0. Conversely, if there is at least one pair of sequences of partitions q, ′q (q = 1,2, …) with d( q) → 0, d( ′q) → 0, such that

then, given any sequence q with d( ″q) → 0,

and hence f(x) is Riemann integrable.

1.3. Step Functions

Let

be a partition of the basic block B into subblocks B1, …, Bp with no interior points in common. Then a function h(x) taking constant values in each of the blocks B1, …, Bp, i.e., such that

is called a step function. The function h(x) can be defined in various ways (or even left undefined) on the boundary planes of the subblocks Bk, which are planes of discontinuity for h(x); the values of h(x) on these planes will not matter in our subsequent considerations.

The family of all step functions defined on a block B will be denoted by H, or if necessary, by H(B). The set H is a linear space with the usual operations of addition and multiplication by real numbers. Thus, if h(x) and k(x) are step functions, so is the linear combination

with real coefficients and . In fact, if h(x) is constant in the subblocks B1, …, Bp, while k(x) is constant in the subblocks B′i, …, Bq, then l(x) is constant in each of the intersections

which together constitute a partition of B.³

The space H is closed under operations other than the forming of linear combinations. For example, if h(x) is a step function, so is its absolute value |h(x)|. Moreover, if h(x) and k(x) are step functions, then so are the functions

In particular, the positive part h+(x) of any step function h(x), defined by

is itself a step function, and so is the negative part h−(x), defined by

Next we introduce the concept of the integral of a step function h(x).⁴ By the integral of h(x) over the block B, we mean the quantity

The integral of a step function has the following two properties:

a) If h, k are any two step functions and , are any two real numbers, then

b) If h and k are two step functions such that h(x) k(x) for all x B, then Ih Ik. In particular, if h(x) 0, then Ih 0.

To prove Property a, suppose h(x) is constant in the blocks B1, …, Bp, while k(x) is constant in the blocks B′1, …, B′q. Then both functions are constant in the blocks B1B′1, …, BqB′q, and moreover

It follows that

and hence

as asserted. Property b is proved similarly.

1.4. Sets of Measure Zero and Sets of Full Measure

In what follows, an important role will be played by coverings of sets by collections of blocks. We say that a set E (in the basic block B) is covered by a collection of blocks {B } if every point of E is an interior point of at least one block B . If E is closed, we have the finite subcovering lemma (a variant of the familiar Heine-Borel theorem): From every collection of blocks {B } covering a closed set E B, we can select a finite subcollection covering E.

DEFINITION. A set Z B is called a set of measure zero if given any > 0, there exists a countable (i.e., a finite or countably infinite) subcollection of blocks B1, B2, … covering Z such that the sum of the volumes of B1, B2, … is less than . The empty set will also be regarded as a set of measure zero.

Thus a sheet, i.e., the intersection of B with