Integral, Measure and Derivative: A Unified Approach

By G. E. Shilov and B. L. Gurevich

This graduate-level textbook and monograph defines the functions of a real variable through consistent use of the Daniell scheme, offering a rare and useful alternative to customary approaches. The treatment can be understood by any reader with a solid background in advanced calculus, and it features many problems with hints and answers. "The exposition is fresh and sophisticated," declared Sci-Tech Book News, "and will engage the interest of accomplished mathematicians."
Part one is devoted to the integral, moving from the Reimann integral and step functions to a general theory, and obtaining the "classical" Lebesgue integral in n space. Part two constructs the Lebesgue-Stieltjes integral through the Daniell scheme using the Reimann-Stieltjes integral as the elementary integral. Part three develops theory of measure with the general Daniell scheme, and the final part is devoted to the theory of the derivative.

• Language: English
• Publisher: Dover Publications
• Release date: May 13, 2013
• ISBN: 9780486165615

Book preview

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 Bbe 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, …, Bpq, … be a sequence of partitions such that dqhas a limit as q q [provided only that dq) → 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

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. 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

′ of the same basic block B′ is called a refinement ). Then every term mks(Bkis replaced by a sum of the form

where

Since mk mk,

and hence

′, the upper Darboux sum can only decrease.

′. 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 q, … is a sequence of partitions of the block B such that dq) → 0, then

and similarly,

Proofsuch that

q , consists of blocks which intersect the boundaries of blocks B in the form

with the blocks B ′q of the basic block B. Consequently,

and hence

Now let G and have sizes no larger than dq), each of the sums in the right-hand side of (5) is no larger than MG dq) 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

q such that dqq′q (q = 1,2, …) with dq) → 0, d′q) → 0, such that

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

. 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 real numbers, then

b) If h and k are two step functions such that h(xk(x) for all x B, then Ih Ik. In particular, if h(x0, 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 some hyperplane of dimension n > 0, there is a block B B (we need only choose B to have sufficiently small thickness). On the other hand, the whole block B is certainly not a set of measure zero. In fact, suppose B1, B2, … is a covering of B. By the finite subcovering lemma, we can select from B1, B2, … a finite subcollection which also covers B. But then the sum of the volumes of even this finite number of blocks must exceed the volume s(B< s(B).

It is easy to see that the union of a countable collection of sets of measure zero is itself a set of measure zero. In fact, if Z1, …, Zm> 0, we cover every set Zm /2m. As a result, the whole set Z = ZZ… is covered by a countable collection of blocks,. Therefore Z is a set of measure zero, as asserted.

A set E B is said to be a set of full measureE (relative to B) is a set of measure zero. The intersection of a countable collection of sets of full measure is itself a set of full measure. In fact, if E1, E2, … are sets of full measure and if ZE1, ZE2, … are the corresponding sets of measure zero, then, as just shown, the set

Em is a set of full measure, as asserted.

If a given property holds at every point of a set of full measure in the block B, we say that the property holds for almost all points of B (or almost everywhere in B). There are functions which are continuous almost everywhere, i.e., continuous except on a set of measure zero. Similarly, in the class of functions that are allowed to take infinite values, there are functions which are finite almost everywhere, i.e., finite except on a set of measure zero. The set of discontinuity points of a step function has measure zero, consisting as it does of a finite number of sheets. By the same token, the set of continuity points of a step function is a set of full measure.

The following theorem can be used to give another definition of a set of measure zero, in terms of integrals of step functions:

THEOREM 2. A set Z B is a set of measure zero if and only if given any > 0, there exists a nondecreasing sequence of nonnegative step functions

such that

and

Proof. If Z > 0, there exists a collection of blocks B1, …, Bm, … with total volume less than z which covers the set Zdenote the step function which equals 1 in the blocks B1, …, Bm obviously satisfies (7) and (8). Moreover, any point xZ belongs to some block Bm. But this implies (9), as required.

Conversely, suppose the properties (7), (8), (9) hold, and let B, with no interior points in common. Because of (9), the set Z is contained in the union of all the blocks Bj. We now calculate the sum of the volumes of the blocks Bj, and using (8), we have

If we take the limit as m , this gives

The blocks Bj may not cover the set Z, since points of Z need not be interior points of the blocks Bj. However, if we replace every block Bj by a concentric block B′j with twice the volume of Bj, we get a covering of Z by blocks B′j is arbitrary, Z is a set of measure zero, and the proof is complete.

COROLLARY. Given a set Z B, suppose that for every > 0, there exists a step function h)(x0 such that Ih)(xand h)(x1 on Z. Then Z is a set of measure zero.

Proof. We need merely write

1.5. Further Properties of Step Functions

We now prove two important lemmas:

LEMMA 1. If a sequence of nonnegative step functions h1(x), …, hp(x), … is nonincreasing,⁶ and if , then

almost everywhere.

Proof. The function

defined everywhere in the block B, is nonnegative, and the set

is the union of the sequence of sets

Therefore, to show that G is a set of measure zero, it is sufficient to show that every Gm is a set of measure zero. But on every Gm we have

and hence

The function mhp(x) is a nonnegative step function and

as p > 0, we can always find a p such that I(mhp. The fact that Gm is a set of measure zero now follows from the corollary at the end of Sec. 1.4.

LEMMA 2. If a sequence of nonnegative step functions h1(x), …, hp(x), … is nonincreasing, and if almost everywhere, then

Proof. First suppose hp(x) converges to zero everywhere, and let Z denote the set of discontinuity points of all the functions hp> 0, we cover Z with a collection of blocks B1, B. With each of the remaining points x we associate an integer m = m(x′) such that hm(x′and a block B′(x′) containing x′ such that hm has a constant value in B′(x′). Together, the blocks B1, B2, … and the blocks B′(x′) form a covering of the basic block B, from which we can select a finite subcovering, whose blocks will be denoted by B1, …, Br, B′1, …, B′q. Let p be the largest of the integers associated with the corresponding points x′1, …, x′q. Then the function hq(x) and all step functions with higher indices do not exceed e in the blocks B′1, …, B′q. Moreover, in the blocks B1, …, Brby construction, hp(x) does not exceed M1, the maximum of h1(x) on B. It can be assumed that no two of blocks B′1, …, B′q have interior points in common (this can always be achieved by going over to a finer collection of blocks and then excluding shared parts of blocks), and therefore the sum of the volumes of the blocks B′1, …, B′q can be regarded as no larger than the volume of the basic block B itself. Hence, for the integral of the function hp(x) over the block B and for the integral of any step function with a higher index, we have the estimate

can be chosen arbitrarily small, it follows that Ihp → 0, as asserted.

Now suppose hp(x) does not converge to zero everywhere, but only almost everywhere. Consider the set Z of measure zero on which the sequence hp(x) fails to approach zero. According to Theorem 2, > 0, there is a nonincreasing sequence of nonnegative step functions kp(x) such that

for every x Z and

for every p = 1, 2, …. Moreover, the limits

obviously exist, while the difference hp − M1kp is nonincreasing and has a nonpositive limit everywhere. Therefore, by the first part of the proof,

and hence

But then

is arbitrary,

and the lemma is proved.

1.6. Application to the Theory of the Riemann Integral

The lower Darboux sum (see Sec. 1.2)

, taking the value mk in the block Bk. Similarly, the upper Darboux sum

, taking the value Mk in Bkq, … of the block B qqis nondecreasing and the sequence of upper step functions is nonincreasing. Assuming that dq) → 0, we introduce the lower function

and the upper function

where obviously

THEOREM 3. The function f(x) is Riemann integrable if and only if