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The integrals introduced by J. Kurzweil [152] and independently by R. Henstock [118] in the late 1950s are equivalent to the restricted Denjoy integral and the Perron integral for integrands taking values in (see [108], for instance). In particular, the definitions of the so-called ”Kurzweil–Henstock” integrals are based on Riemannian sums, and are therefore easy to deal with even by undergraduate students. Not only that, but the Kurzweil–Henstock–Denjoy–Perron integral encompasses the integrals of Newton, Riemann, and Lebesgue.

In 1969, E. J. McShane (see [173, 174]) showed that a small change in the subdivision process of the domain of integration within the Kurzweil–Henstock (or Perron) integral leads to the Lebesgue integral. This is a very nice finding, since now the Lebesgue integral can be taught by presenting its Riemannian definition straightforwardly and, then, obtaining immediately some very interesting properties such as the linearity of the Lebesgue integral which comes directly from the fact that the Riemann sum can be split into two sums. The monotone convergence theorem for the Lebesgue integral is another example of a result which is naturally obtained from its equivalent definition due to McShane.

The Kurzweil integral and the variational Henstock integral can be extended to Banach space-valued functions as well as to the evaluation of integrands over unbounded intervals. The extension of the McShane integral, proposed by R. A. Gordon (see [107]) to Banach space-valued functions, gives a more general integral than that of Bochner–Lebesgue. As a matter of fact, the idea of McShane into the definition due to Kurzweil enlarges the class of Bochner–Lebesgue integrals.

On the other hand, when the idea of McShane is employed in the variational Henstock integral, one gets precisely the Bochner–Lebesgue integral. This interesting fact was proved by W. Congxin and Y. Xiabo in [47] and, independently, by C. S. Hönig in [131]. Later, L. Di Piazza and K. Musal generalized this result (see [55]). We clarify here that unlike the proof by Congxin and Xiabo, based on the Fréchet differentiability of the Bochner–Lebesgue integral, Hönig's idea to prove the equivalence between the Bochner–Lebesgue integral and the integral we refer to as Henstock–McShane integral uses the fact that the indefinite integral of a Henstock–McShane integrable function is itself a function of bounded variation and the fact that absolute Henstock integrable functions are also functions of bounded variation. In this way, the proof provided in [131] becomes simpler. We reproduce it in the next lines, since reference [131] is not easily available. We also refer to [73] for some details.

Definition 1.89: We say that a function is Bochner–Lebesgue integrable (we write ), if there exists a sequence of simple functions, , , such that

1  almost everywhere (i.e. for almost every ), and

2 .

With the notation of Definition 1.89, we define

Then, the space of all equivalence classes of Bochner–Lebesgue integrable functions, equipped with the norm , is complete.

The next definition can be found in [239], for instance.

Definition 1.90: We say that a function is measurable, whenever there is a sequence of simple functions such that almost everywhere. When this is the case,

(1.A.1)

Again, we explicit the “name” of the integral we are dealing with, whenever we believe there is room for ambiguity.

As we mentioned earlier, when only real-valued functions are considered, the Lebesgue integral is equivalent to a modified version of the Kurzweil–Henstock (or Perron) integral called McShane integral. The idea of slightly modifying the definition of the Kurzweil–Henstock integral is due to E. J. McShane [173, 174]. Instead of taking tagged divisions of an interval , McShane considered what we call semitagged divisions, that is,

is a division of and, to each subinterval , with , we associate a point called “tag” of the subinterval . We denote such semitagged division by and, by , we mean the set of all semitagged divisions of the interval . But what is the difference between a semitagged division and a tagged division? Well, in a semitagged division , it is not required that a tag belongs to its associated subinterval . In fact, neither the subintervals need to contain their corresponding tags. Nevertheless, likewise for tagged divisions, given a gauge of , in order for a semitagged division to be -fine, we need to require that

This simple modification provides an elegant characterization of the Lebesgue integral through Riemann sums (see [174]).

Let us denote by the space of all real-valued Kurzweil–McShane integrable functions , that is, is integrable in the sense of Kurzweil with the modification of McShane. Formally, we have the next definition which can be extended straightforwardly to Banach space-valued functions.

Definition 1.91: We say that is Kurzweil–McShane integrable, and we write if and only if there exists such that for every , there is a gauge on such that

whenever is -fine. We denote the Kurzweil–McShane integral of a function by .

The following inclusions hold

Moreover, as one can note by the next classical example.

Example 1.92: Let be defined by if , and , and consider . Since is Riemann improper integrable, , because the Kurzweil–Henstock (or Perron) integral contains its improper integrals (see Theorem 2.9, [158], or [213]). However, , since is not absolutely integrable (see also [227]).

Example 1.92 tells us that the elements of are not absolutely integrable.

When McShane's idea is applied to Kurzweil and Henstock vector integrals, the story changes. In fact, the modification of McShane applied to the Kurzweil vector integral originates an integral which encompasses the Bochner–Lebesgue integral (see Example 1.74). On the other hand, when McShane's idea is used to modify the variational Henstock integral, we obtain exactly the Bochner–Lebesgue integral (see [[47] and [131]]). Thus, if denotes the space of Henstock–McShane integrable functions , that is, is integrable in the sense of Henstock with the modification of McShane, then

We will prove this equality in the sequel. Furthermore, , and , where we use the notation and to denote, respectively, the spaces of Kurzweil–McShane and Riemann–McShane integrable functions from to . For other interesting results, the reader may want to consult [55].

Our aim in the remaining of this chapter is to show that the integrals of Bochner–Lebesgue and Henstock–McShane coincide. See [132, Theorem 10.4]. The next results are due to C. S. Hönig. They belong to a brochure of a series of lectures Professor Hönig gave in Rio de Janeiro in 1993. We include the proofs here, once the brochure is in Portuguese.

Lemma 1.93: Let be a sequence in and be a function. Suppose there exists

Then, and

Proof. Given , take such that for ,

and take a gauge on such that for every -fine ,

(1.A.2)

The limit exists, since for ,

Hence, if , then

Thus, the first summand on the right-hand side of the last inequality is smaller than by (1.A.2), the third summand is smaller than by the definition of and, if we refine the gauge , we may suppose, by the definition of , that the second summand is smaller than , and the proof is complete.

We show next that Lemma 1.93 remains valid if we replace by , that is, if instead of the space of Kurzweil–McSchane integrable functions, we consider the space of Henstock–McSchane integrable functions.

Lemma 1.94: Let be a sequence in and be a function. If , then and

Proof. By Lemma 1.93, , and we have the convergence of the integrals. It remains to prove that , that is, for every , there exists a gauge on such that for every -fine ,

However,

Since as tends to infinity, there exists such that the first summand in the last inequality is smaller than for all . Choose an . Then, we can take such that the third summand is smaller than , because it approaches . In addition, once , we can refine so that the second summand becomes smaller than , and we finish the proof.

For a proof of the next lemma, it is enough to adapt the proof found in [107, Theorem 16] for the case of Banach space-valued functions.

Lemma 1.95: .

Now, we are able to prove the next inclusion.

Theorem 1.96: .

Proof. By Lemma 1.95, . Then, following the steps of the proof of Lemma 1.95 and using Lemma 1.94, we obtain the desired result.

For the next result, which says that the indefinite integral of any function of belongs to , we employ a trick based on the fact that if almost everywhere, then and , that is, the indefinite integrals of and coincide. This fact follows by a straightforward adaptation of [108, Theorem 9.10] for Banach space-valued functions (see also [70]). Thus, if we change a function on a set of Lebesgue measure zero, its indefinite integral does not change. Therefore, we consider, for instance, that vanishes at such points.

Lemma 1.97: If , then .

Proof. It is enough to show that every has a neighborhood where is of bounded variation. By hypothesis, given , there exists a gauge on such that for every -fine semitagged division of ,

(1.A.3)

Let be any division of . If we take for , then the point-interval pair is a -fine tagged partial division of and, therefore, from (1.A.3) and since we can assume, without loss of generality (see comments in the paragraph before the statement), that for , we have

and the proof is complete.

Lemma 1.98: Suppose . The following properties are equivalent:

1  is absolutely integrable;

2 .

Proof. (i) (ii). Suppose is absolutely integrable. Since the variation of , , is given by

we have

(ii) (i). Suppose . We prove that the integral exists and . Given , we need to find a gauge on such that

whenever is -fine. However,

(1.A.4)

By the definition of , we may take such that the last summand in (1.A.4) is smaller than . Because , we may take a gauge such that for every -fine , the first summand in (1.A.4) is also smaller than (and we may suppose that the points chosen for the second summand are the points of the -fine tagged division ).

The next result is a consequence of the fact that and Lemmas 1.97 and 1.98. A proof of it can be found in [132, Theorem 10.3].

Corollary 1.99: All functions of are absolutely.integrable

The reader can find a proof of the next lemma in [35, Theorem 9].

Lemma 1.100: All functions of are.measurable

Finally, we can prove the following inclusion.

Theorem 1.101: .

Proof. The result follows from the facts that all functions of and, hence, of are measurable (Lemma 1.100), and all functions of are absolutely integrable by Corollary 1.99.

As we mentioned before, the inclusion always holds. However, when is an infinite dimensional Banach space, then for sure , as shown by the next result due to C. S. Hönig (personal communication by him to his students in 1990 at the University of São Paulo) and presented in [73].

Proposition 1.102 (Hönig): If is an infinite dimensional Banach space, then there exists .

Proof. Let denote the dimension of . If , then the Theorem of Dvoretsky–Rogers (see [60] and also [57]) implies there exists a sequence in which is summable but not absolutely summable. Thus, if we define a function by , for , then whenever the integral exists. However, , since

and this completes the proof.

In the next example, borrowed from [73, Example 3.4], we exhibit a Banach space-valued function which is integrable in the variational Henstock sense and also in the sense of Kurzweil–McShane. Nevertheless, it is not absolutely integrable.

Example 1.103: Let be given by

Then,

which is summable in . Since the Henstock integral contains its improper integrals (and the same applies to the Kurzweil integral), we have . However, because the sequence is non-summable in . By the Monotone Convergence Theorem for the Kurzweil–McShane integral (which follows the ideas of [71] with obvious adaptations), . But , since is not bounded, where by we denote the space of Riemann–McShane integrable functions from to .