Open access peer-reviewed chapter

Derivation and Integration on a Fractal Subset of the Real Line

Written By

Donatella Bongiorno

Submitted: 05 May 2023 Reviewed: 15 May 2023 Published: 26 June 2023

DOI: 10.5772/intechopen.1001895

From the Edited Volume

Fractal Analysis - Applications and Updates

Dr. Sid-Ali Ouadfeul

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Abstract

Ordinary calculus is usually inapplicable to fractal sets. In this chapter, we introduce and describe the various approaches made so far to define the theory of derivation and integration on fractal sets. In particular, we study some Riemann-type integrals (the s-Riemann integral, the sHK integral, the s-first-return integral) defined on a closed fractal subset of the real line with finite and positive s-dimensional Hausdorff measure (s-set) with particular attention to the Fundamental Theorem of Calculus. Moreover, we pay attention to the relation between the s-Riemann integral, the sHK integral, and the Lebesgue integral with respect to the Hausdorff measure Hs, respectively, and we give a characterization of the primitives of the sHK integral.

Keywords

  • Hausdorff measure
  • s-set
  • s-Riemann integral
  • sHK integral
  • s-derivatives

1. Introduction

For many years, it was thought that the structure of fractal sets had so many irregularities to render too difficult the definition of standard-type methods and techniques of ordinary calculus in such type of sets. In fact, for example, the usual derivative of the classical Lebesgue-Cantor staircase function is zero almost everywhere, and the usual Riemann integral of a function defined on a fractal set is undefined (see Refs. [1, 2, 3]). Therefore analysis of fractals has been studied by different methods such as harmonic analysis, stochastic processes, and fractional processes. Very recently, a non-Newtonian calculus on fractal sets of the real line that starts by elementary non-Diophantine arithmetic operations of a Burgin type was formulated by M. Czachor (see Ref. [4]).

In this chapter, we present a method of standard calculus for fractal subsets of the real line, that was independently formulated by various authors (see Refs. [5, 6, 7, 8, 9]). Such formulation is aimed at those self-similar fractal subsets of the real line with finite and positive s-dimensional Hausdorff measure, briefly called s-sets. Moreover, it differs from the classical one, for the use of the Hausdorff measure instead of the natural distance. The idea of replacing the usual distance with the Hausdorff measure was used for the first time in the definition of s-derivative given in 1991 by De Guzmann, Martin, and Reyes (see Ref. [5]), in order to study the problem of existence and uniqueness of the solutions of ordinary differential equations in which the independent variable takes value in a fractal set. Later, in 1998, this concept was taken up by Jung and Su in Ref. [6] to define an integral of the Riemann type called s-integral. Moreover, Parvate and Gangal in Ref. [7], independently by Jung and Su, introduced an integral of Riemann type on an s-set of the real line, called Fs-integral. Such integration processes were defined as the classical Riemann integral but with the Hausdorff measure and the mass function, respectively, taking over the role of the distance. Since the Hausdorff measure and the mass function are proportional (see Ref. [7], Section 4), it follows that Jiang and Su in Ref. [6] and Parvate and Gangal in Ref. [7] defined, independently, the same integral. In this chapter, we call it the s-Riemann integral.

Both authors proved a version of the Fundamental Theorem of Calculus. About this, we recall that, in the real line, such fundamental theorem states that: iff:abRis differentiable on [a, b], then the functionfxis integrable (in some sense) on [a, b] andabftdt=fbfa.

Unfortunately, as it happens in the real line, the s-Riemann integral is not the best integral for the formulation of the Fundamental Theorem of Calculus. This was the reason that motivated Bongiorno and Corrao in Ref. [8] to define an Henstock-Kurzweil integration process on an s-set of the real line and to formulate in Ref. [9] the best version of the Fundamental Theorem of Calculus on such s-sets. We recall that, in the real line, it is the Henstock-Kurzweil integral that, solving the problem of primitives, provides the best version of the Fundamental Theorem of Calculus (see Ref. [10]). Later and independently, also Golmankhaneh and Baleanu extended, in Ref. [11], the s-Riemann integral by introducing an integral of the Henstock–Kurzweil type. However, no version of the Fundamental Theorem of Calculus is proved in Ref. [11].

More precisely, the integral introduced in Ref. [8] is based on the use of the Hausdorff measure instead of the notion of the classical distance, as it was already done by Jung and Su in Ref. [6]. On the other hand, the mass function, instead of the classical distance, is used in Ref. [11], as it was already done by Parvate and Gangal in Ref. [7]. Precisely, Golmankhaneh and Baleanu have revised the notion of mass function given in Ref. [7] by defining a special mass function through the use of a gauge function previously introduced in Ref. [12]. However, since the Hausdorff measure and the mass function are proportional, without loss of generality, we infer that the two integrals coincide and we call the resulting integral the sHK integral. Moreover, following the literature, we prefer the use of the Hausdorff measure instead of the mass function to define the sHK integral. Finally, through the characterization of the sHK primitives, a descriptive definition of the sHK integral is given.

All sections of this chapter are related to each other; moreover, in order not to burden the reader, some proofs are reported in Appendix A, while Appendix B contains a brief history of the Henstock-Kurzweil integral. Particular attention is paid to the formulation of the best version of the Fundamental Theorem of Calculus. Afterward, the last section of this chapter is devoted to the formulation of a new integration process. Precisely an integral of the first return-type, called s-first-return integral, is defined. The idea of the first-return technique comes from the Poincaré first-return map of differentiable dynamics, and it was already used in differentiation and in integration theory (see Refs. [13, 14, 15]).

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2. The s-derivatives

Let L be the usual Lebesgue measure, and let E be a compact subset of the real line. Given s, with 0<s1, let

HsE=limδ0infi=1LAis:Ei=1AiLAiδE1

be the s-dimensional Hausdorff measure of E.

Recall that Hs is a Borel regular measure and that the unique number s for which HtE=0 if t>s and HtE= if t<s is called the Hausdorff dimension of E.

Definition 2.1. Let E be a compact subset of the real line and let 0<s<1. E is called an s-set if it is measurable with respect to the s-dimensional Hausdorff measure Hs (briefly Hs-measurable) and 0<HsE<.

Therefore Hs is a Radon measure on each s-set (see Ref. [16]).

From now on we will denote by Eab an s-set of the real line, by a=minE and by b=maxE.

Definition 2.2. For x,yE we set

dxy=HsxyE,ifx<y;HsyxE,ify<x.E2

Proposition 2.1. The function xydxy from E×ER+0 is a metric, and the space (E, d) is a complete metric space.

Proposition 2.2. The topology of the metric space (E, d) coincides with the topology induced on E by the usual topology of R.

Definition 2.3. Let F:ER and let x0E. The s-derivatives of F on the left and on the right at the point x0 are defined, respectively, as follows:

Fsx0=limxx0xEFx0Fxdxx0E3
Fs+x0=limxx0+xEFxFx0dxx0E4

when these limits exist.

We say that the s-derivative of F at x0 exists if Fsx0=Fs+x0 or if the s-derivative of F on the left (resp. right) at x0 exists, and for some, ε>0 we have dx0x0+ε=0 (resp. dx0εx0=0). The s-derivative of F at x0, when it exists, will be denoted by Fsx0.

It is trivial to observe that by the previous definition, it follows the linearity of the s-derivative. Moreover.

Remark 2.1. If F is s-derivable at the point x0, then F is continuous at x0 according to the topology induced on E by the usual topology of R.

Hereafter, for each interval Aab, we set A˜=AE.

Example 2.1. Let E01 be the ternary Cantor set. E is an s-set for s=log32 and HsE=1 (see Ref. [2], Theorem 1.14). Moreover Hs23n13n1˜=12n=Hs013n˜, and Hs23n73n+1˜=14n=Hs83n+113n1˜.

The function

Fx=2nnHs23nx˜,x23n73n+1˜;2nnHs83n+1x˜,x83n+113n1˜;0,x=0.E5

is s-derivable on E with

Fsx=2nn,x23n13n1˜;0,x=0.E6

Infact, for x023n13n1˜ it is:

Fsx0=limxx0xEFx0FxHsxx0˜=2nn,E7
Fs+x0=limxx0+xEFxFx0Hsx0x˜=2nn.E8

So Fsx0=2n/n.

Moreover, for x23n73n+1˜, it is

FxF0Hs0x˜=2nnHs23nx˜Hs0x˜2nn14n2n=1nE9

and, for x83n+113n1˜, it is

FxF0Hs0x˜=2nnHs83n+1x˜Hs0x˜2nn14n2n=1nE10

Thus

Fs0=limx0xEFxF0Hs0x˜=0.E11

Definition 2.4. Let F:RR be a function. A point xR is said to be a point of change of F if it is not a constant over any open interval (c, d) containing x. The set of all points of change of F is called the set of change of F, and it is denoted by Sch(F).

Theorem 2.1 Let F:abR be a continuous function such that SchFE. Let us suppose that F is s-derivable at each point xab and that Fa=Fb=0. Then there exists a point cE such that Fsc0 and a point dE such that Fsd0.

In Ref. [7], it is possible to find the proof of the previous theorem, and an example that shows as the “fragmented nature” of the fractal set does not allow us to make an analog of Rolle’s theorem. Furthermore, the law of mean and the Leibniz rule is also discussed in Ref. [7].

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3. The s-Riemann integral

Definition 3.1. Let E be a closed s-set of the real line, a=minE and b=maxE. Let Aab be an interval. We call interval of the set A˜=AE.

Definition 3.2.A partition of E is any collection P=A˜ixii=1p of pairwise disjoint intervals A˜i and points xiA˜i such that E=iA˜i.

Definition 3.3. Let f:ER be a function. It is said that f is s-Riemann integrable on E, if there exists a number I such that, for each ε>0 there is a δ>0 with

i=1pfxiHsA˜iI<εE12

for each partition P=A˜ixii=1p of E with HsA˜i<δ,i=1,2,,p. The number I is called the s-Riemann integral of f on E and we set

I=sEftdHst.E13

The collection of all functions that are s-Riemann integrable on E will be denoted by sR(E).

Theorem 3.1 If f:ER is continuous on E with respect to the induced topology, then fsRE.

Remark 3.1. The classical Riemann integral properties, such as linearity, additivity with respect to integrating domain and the mean-value theorem for integrals hold too for this new integral. See Refs. [6, 7] for more details.

Moreover, it is useful to remark that, as it happens in the real case, the Lebesgue integral of f with respect to the Hausdorff measure Hs, here denoted by LEftdHst, includes the s-Riemann integral.

Theorem 3.2 If fsRE then f is Lebesgue integrable on E with respect to the Hausdorff measure Hs, and

LEftdHst=sEftdHst.E14

The proof can be found in Appendix A.

3.1 The Fundamental Theorem of Calculus

Definition 3.4. Let F:ER be a function. We say that F is Hs-absolutely continuous on E if ε>0δ>0 such that

k=1nFbkFak<εE15

whenever k=1nHsakbk˜<δ, with ak,bkE,k=1,,n, and a1<b1a2<b2.an<bn.

Jiang and Su in Ref. [6] announced, without proof, the following version of the Fundamental Theorem of Calculus:

Theorem 3.2 Let f:ER be a continuous function. If F:ER is Hs-absolutely continuous on E with Fs'x=fxHs- a.e. in E, then

sEftdHst=FbFa.E16

On the other hand Parvate and Gangal in Ref. [7] proved the following version of the Fundamental Theorem of Calculus:

Theorem 3.3 If F:RR is s-derivable on E, and if Fs is continuous with SchFE, then

sEFsxdHsx=FbFa.E17

Now, we give an example of a very simple function, that is not Hs-absolutely continuous on a fractal set E and that satisfies the condition SchFE for which the Fundamental Theorem of Calculus fails.

Example 3.1. Let E01 be the classical Cantor set. Let F be a real function on [0, 1] defined by

Fx=0,0x133x1,13<x<231,23x1.E18

and let f be the restriction of F on the Cantor set E.

Since, the Hausdorff dimension of the Cantor set E is s=log32 and Hlog32E=1 it is trivial to observe that:

  • f is log32-derivable on E with flog32x=0,xE,

  • flog32 is log32-Riemann integrable on E,

log32Eflog32xdHlog32x=0f1f0=1.E19

In the next section, we will prove a more general formulation of the Fundamental Theorem of Calculus on an s-set, that enclose Theorem 3.2 and Theorem 3.3.

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4. The sHK integral

Definition 4.1. It is called gauge on E any positive real function δ defined on E.

Definition 4.2. It is called partition of E any collection P=A˜ixii=1p of pairwise disjoint intervals A˜i of E, and points xiA˜i,i=1,,p, such that E=iA˜i.

Definition 4.3. Let P=A˜ixii=1p be a partition of E. If δ is a gauge on E, then we say that P is a δ-fine partition of E whenever A˜i]xiδxi,xi+δxi[, for i=1,2,,p.

Lemma 4.1. If δ is a gauge on E, then there exists a δ-fine partition of E.

This lemma, known in the literature as the Cousin lemma, is fundamental for the definition of an Henstock-Kurzweil type integral because it addresses the existence of δ-fine partitions. For completeness, we report the proof of this lemma in Appendix A, even if it is possible to find it in Ref. [8].

Now, let P=A˜ixii=1p be a partition of E, let f:ER be a function and let

σfP=i=1pfxiHsA˜i,E20

be the s-Riemann sum of f with respect to P.

Definition 4.4. We say that f is sHK integrable on E if there exists IR such that, for all ε>0, there is a gauge δ on E with:

σfPI<ε,E21

for each δ-fine partition P=A˜ixii=1p of E.

The number I is called the sHK integral of f on E, and we write

I=sHKEfdHs.E22

The collection of all functions that are sHK integrable on E will be denoted by sHK(E).

Remark 4.1. Let us notice that the difference between the sHK integral (see Definition 4.4) and the s-Riemann integral (see Definition 3.3) is due to the fact that while in Definition 4.4 the gauge δ is a positive real function in Definition 3.3 it is a positive constant.

About this, remark that, if f is an sHK integrable function, but it is not s-Riemann integrable, then infxEδx=0, for the gauge δ involved in the definition of the sHK integral. In fact, the condition infxEδx=δ>0 would imply that the choice of points xi inside the intervals A˜i,i=1,2,,p, may be arbitrary; therefore f would be s-Riemann integrable.

Here we list some basic properties of the sHK integral:

  1. the number I from Definition 4.4 is unique,

  2. if f,gsHKE therefore f+gsHKE and

    sHKEf+gtdHst=sHKEftdHst+sHKEgtdHst,

  3. if fsHKE and kR, then kfsHKE and

    sHKEkftdHst=ksHKEftdHst,

  4. if f,gsHKE with fg,Hs-almost everywhere on E, therefore

    sHKEftdHstsHKEgtdHst,

  5. if fsRE, then fsHKE and

    sHKEftdHst=sEftdHst,

  6. if fsHKE and a=minE<x<b=maxE, then the function

Fx=sHKax˜fdHsE23

is continuous and

sHKEfdHs=sHKax˜fdHs+sHKxb˜fdHs.E24

Just as in the case of the Henstock-Kurzweil integral (see Ref. [10]), there is a Cauchy criterion for a function to be sHK integrable on E. This is the content of the following theorem.

Theorem 4.1 A function f:ER is sHK-integrable on E if and only if for each ε>0 there exists a gauge δ on E such that

σfP1σfP2<ε,E25

for each pair P1 and P2 of δ-fine partitions of E.

The proof can be found in Ref. [8].

4.1 Relation with the Lebesgue integral

In this section, we prove that the sHK integral includes the Lebesgue integral with respect to Hs. In order to do this, we recall the following Vitali-Carathéodory Theorem:

The Vitali-Carathéodory Theorem Let f be a real function defined on E. If f is Lebesgue integrable on E with respect to Hs and ε>0, then there exist functions u and v on E such that ufv, u is upper semicontinuous and bounded above, v is lower semicontinuous and bounded below, and

LEvudHs<ε.E26

Theorem 4.2 Let f:ER be a function. If f is Lebesgue integrable on E with respect to Hs, then f is sHK-integrable on E and

LEfdHs=sHKEfdHs.E27

The proof can be found in Appendix A.

Now we give a simple example of an sHK integrable function which is not Lebesgue integrable with respect to the Hausdorff measure Hs.

Example 4.1. Let E01 be the classical Cantor set, and let f:ER be the function defined as follows

fx=1n+12nn,forx23n13n1˜n=1,2,3,0,forx=0.E28

We show that fsHKE where s=log32.

In order to do that, fixed ε>0, we can find a gauge δ on E such that

  • if xE and x0, f is constant on xδx,x+˜δx;

  • δ0<13n+1 with 1n<ε.

Choose kN with k>n+1 and set c=13k.

Let us consider P=A1˜x1A2˜x2Am˜xm a δ-fine partition of E such that A1˜=0c˜ and Ai˜23p13p1˜ (i=2,,m) for some pN. Our choice of δ implies that x1=0 and i=2mAi˜=i=1k23i13i1˜.

sHK23k1˜fdHs=i=1ksHK23i13i1˜fdHs==i=1k1i+12iiHs23i,13i1˜==i=1k1i+12ii12i=i=1k1i+1i.E29

Therefore

sHK23k1˜fdHslog2.E30

Then

σfPlog2f0Hs(A1)˜+i=2mfxiHs(Ai)˜sHK23k1˜fdHs+sHK23k1˜fdHslog2<ε.E31

In conclusion, we prove that f is not Lebesgue integrable on E with respect to Hs, where s=log32. In fact, if f were Lebesgue integrable on E with respect to Hs,f would be Lebesgue integrable on E with respect to Hs. But we have

LEfdHs=n=11n=+,E32

hence f is not Lebesgue integrable on E with respect to Hs.

4.2 The Fundamental Theorem of Calculus

Definition 4.5. An interval αβ is said to be contiguous to a set E, if:

  • αE and βE

  • αβE=0

In Ref. [9] we proved the following version of the Fundamental Theorem of Calculus.

Theorem 4.3 Let E be a closed s-set and let ajbjjN be the contiguous intervals of E. If F:ER is s-derivable on E and if j=1FbjFaj<+, then Fs'sHKE and

sHKEFs'dHs=FbFaj=1FbjFaj.E33

Remark 4.2. By the previous theorem, it is possible to extend the versions of the Fundamental Theorem of Calculus given by Jung and Su (i.e., Theorem 3.2) and that given by Parvate and Gangal (i.e., Theorem 3.3).

Extension of Theorem 3.2.

Let E be a closed s-set and let F:ER be a function Hs-absolutely continuous on E such that Fs' exists Hs-almost everywhere in E. Then

sHKEFstdHst=FbFa.E34

The proof of the Extension of Theorem 3.2 can be found in Ref. [9].

Extension of Theorem 3.3 If F:RR is continuous and s-derivable on E and if SchFE, then

sHKEFstdHst=FbFa.E35

Proof:

Condition SchFE implies that F is constant on each contiguous interval ajbj of E, then Faj=Fbj for jN. Therefore j=1FbjFaj<+, and Theorem 4.3 can be applied.

Remark 4.3. If we assume, like in Theorem 3.3 that Fs is continuous on E, then FssRE (Ref. [7], Theorem 39) and by (e) and by Extension of Theorem 3.3, we have that:

sEFstdHst=FbFa.E36

Remark 4.4. The absolute convergence of the series j=1FbjFaj is a necessary condition

  • for the sHK integrability of Fs

  • for the validity of some formulation of the Fundamental Theorem of Calculus.

See Ref. [9] for more details.

4.3 The primitives

Bongiorno and Corrao in Ref. [17] introduced an Henstock-Kurzweil type integral defined on a complete metric measure space X endowed with a Radon measure μ and with a family F of μ-cells that enclose the sHK integral (see Ref. [17], Example 2.4). For such an integral, they proved an extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line in terms of ACG* functions (see Ref. [17] and Appendix B). Here we report such descriptive characterization in the particular case in which X=ab is endowed with the Euclidean distance of R,Eab is an s-set and the family F of μ-cells coincides with the family F of all closed subintervals of ab˜.

Definition 4.6. A finite collection A˜1A˜m of pairwise disjoint elements of ab˜ is called a division of ab˜ if i=1mA˜i=ab˜.

Definition 4.7. Let E be an s-set and let δ be a gauge on ab˜. A collection P=A˜ixii=1m of finite ordered pairs of intervals and points is said to be

  • a partial partition of ab˜ if A˜1A˜m is a subsystem of a division of ab˜ and xiA˜i for i=1,2,,m;

  • E-anchored if the points x1,,xm belong to E.

Definition 4.8. Let π:FR be a function. We say that π is an additive function of interval if for each A˜F an for each division A˜1A˜m of A˜ we have

πA˜=i=1mπA˜i.E37

Definition 4.9. Let E be an s-set. Let π be a fixed additive function of interval. We say that π is ACΔ on E if for ε>0 there exists a gauge δ on E and a positive constant η such that the condition i=1mHsA˜i<η, implies i=1mπA˜i<ε, for each δ-fine E-anchored partial partition P=A˜ixii=1m of ab˜.

We say that π is ACGΔ on ab˜ if there exists a countable sequence of s-sets Ekk such that kEk=ab˜ and π is ACΔ on Ek, for each kN.

Theorem 4.4 Let Eab be an s-set. A function f:ER is sHK integrable on E if and only if there exists an additive function of interval F that is ACGΔ on ab˜ and s-derivable on ab˜ such that Fsx=fxHs-almost everywhere on ab˜.

Remark 4.5. An interested reader can find the proof of Theorem 4.4 in a more general version (i.e. for an Henstock-Kurzweil type integral defined on a complete metric measure space X endowed with a Radon measure μ) in Ref. [17].

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5. The s-first-return integral

Darji and Evans in Ref. [15] have defined on the real line the first-return integral. The motivation that led the authors to define such a new integration process lies in the fact that the gauge function involved in the definition of the first-return integral is a constant like in the Riemann integral. Borrowing such an idea, in this section, we define a new integral on an s-set E, called the s-first-return integral. Such a new integral is different from the sHK integral because, in the definition of this new integral, the function δ:ER+ is a positive constant, but the choice of points xi is not arbitrary in A˜i, for i=1,2,,p.

Definition 5.1. We call trajectory on E any sequence Γ of distinct points of E dense in E. Given a trajectory Γ on E and given an interval, A˜ we denote by rΓA˜ the first element of Γ that belongs to A˜.

Definition 5.2. Let f:ER and let Γ be a trajectory on E. We say that f is s-first-return integrable on E with respect to Γ if there exists a number IR such that, for all ε>0, there is a constant δ>0 with:

i=1pfrΓA˜iHsA˜iI<εE38

for each division P=A˜ii=1p of E with HsA˜i<δ.

The number I is called the s-first-return integral of f on E with respect to Γ, and we write

I=sfrΓEfdHs.E39

The collection of all functions that are s-first-return integrable on E with respect to Γ will be denoted by sfrΓE.

Theorem 5.1 Let f:ER such that fsRE, therefore, there exists a trajectory Γ on E such that fsfrΓE and

sfrΓEftdHst=sEftdHst.E40

Theorem 5.2 There exists an s-derivative f:ER such that fsfrΓE, for each trajectory Γ on E.

Remark 5.1. To prove Theorem 5.2 it is enough to consider the function fx=Fsx of the Example 2.1 and to show that fsfrΓE, for a given trajectory Γ. This is equivalent to find, for each M>0 and for each δ>0, a finite system of pairwise disjoint intervals A˜i, with i=1,2,,p, such that HsA˜i<δ,iA˜i=E and

i=1pf(rΓA˜iHsA˜i>M.E41

See Ref. [18] for more details.

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6. Conclusions

In this chapter we have developed a method of calculus on a closed fractal subset of the real line with finite and positive s-dimensional Hausdorff measure. Much of the development of such new calculus on s-sets is carried in analogy with the ordinary calculus with some differences due to the “fragmented nature” of fractal sets, like in the formulation of Rolle’s theorem or in the formulation of the Fundamental Theorem of Calculus. Therefore, in order to give the best version of the Fundamental Theorem of Calculus (see Theorem 4.3), we have generalized the s-Riemann integral by defining the sHK integral and the s-first-return integral. Finally, by Theorem 5.2, we have noted that to obtain the best version of the Fundamental Theorem of Calculus, we need to consider an Henstock-Kurzweil type integral, i.e., an integral in which the gauge δ is not a constant.

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Appendix A

Proof of Theorem 3.1.

By standard techniques, it follows that f is bounded and Hs-measurable; then f is Lebesgue integrable with respect to the Hausdorff measure Hs (briefly, Hs-Lebesgue integrable).

Given ε>0, by Definition 3.3, there exists δ>0 such that

i=1nfxiHsA˜isEftdHst<εE42

holds for each partition P=A˜ixii=1p of E with HsA˜i<δ,i=1,2,,p. Hence

i=1psupA˜iftinfA˜iftHsA˜i<2ε.E43

Now remark that i=1pfxiHsA˜i is the Hs-Lebesgue integral of the Hs-simple function i=1pfxi1A˜it, where 1A˜it denotes the characteristic function of A˜i; i.e., 1A˜it=1 for tA˜i, and 1A˜it=0 for tA˜i.

So

LEinfA˜ift1A˜itdHstLA˜iftdHstLEsupA˜ift1A˜itdHstE44

and

LEi=1pinfA˜ift1A˜itdHstεsEftdHstLEi=1psupA˜ift1A˜itdHst+ε.E45

Thus we have

LEftdHstsEftdHstLEi=1psupA˜iftinfA˜ift1A˜itdHst+2ε<4ε.E46

By the arbitrariness of ε we end the proof.

Proof of Lemma 4.1.

Let c be the midpoint of [a, b] and let us observe that if P1 and P2 are δ-fine partitions of ac˜ and cb˜, respectively, then P=P1P2 is a δ-fine partition of E. Using this observation, we proceed by contradiction.

Let us suppose that E does not have a δ-fine partition, then at least one of the intervals ac˜ or cb˜ does not have a δ-fine partition, let us say ac˜. Therefore ac˜ is not empty. Let us relabel the interval [a, c] with a1b1 and let us repeat indefinitely this bisection method. So, we obtain a sequence of nested intervals: aba1b1anbn such that anbn˜ is not empty. Since the length of the interval anbn is ba/2n, therefore, for the Nested Intervals Property, there is a unique number ξab such that:

n=0anbn=ξ.E47

Let ξnanbn˜. Therefore ξnξ<bnan=ba/2n. So limnξn=ξ. Now since E is a closed set, ξE.

Since δξ>0, we can find kN such that akbk˜]ξδξ,ξ+δξ[. Therefore (akbk,ξ)} is a δ-fine partition of akbk˜, contrarily to our assumption.

Proof of Theorem 4.2.

By Vitali-Carathéodory Theorem, given ε>0 there exist functions u and v on E that are upper and lower semicontinuos respectively such that ufv+ and LEvudHs<ε. Define on E a gauge δ so that

utfx+εandvtfxε,E48

for each tE with xt<δx.

Let P=A˜1x1A˜2x2A˜pxp be a δ-fine partition of E. Then, for each i12p, we have

LA˜iudHsLA˜ifdHsLA˜ivdHs.E49

Moreover, by utfxi+ε for each tA˜i, it follows

LA˜iuεdHsLA˜ifxidHsE50

and therefore

LA˜iudHsεHsA˜ifxiHsA˜i.E51

Similarly, by vtfxi+ε for each tA˜i, it follows

fxiHsA˜iLA˜ivdHs+εHsA˜i.E52

So, for i=1,2,,p, we have

LA˜iudHsεHsA˜ifxiHsA˜iLA˜ivdHs+εHsA˜i.E53

Hence,

LEudHsεσfPLEvdHs+ε,E54

and, by (Eq. (49)),

LEudHsLEfdHsLEvdHs.E55

Thus

σfPLEfdHsLEvudHs+2ε<3ε,E56

and the theorem is proved.

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Appendix B

The problem of primitives is the problem of recovering a function from its derivative (i.e. the problem of whether every derivative is integrable).

In the real line the Riemann integral is inadequate to solve it, in fact the function

Fx=x2sin1/x2,x01;0,x=0;E57

is differentiable everywhere on [0, 1], but its derivative

Fx=2xsin1/x22xcos1/x2,x01;0,x=0;E58

it is not Riemann integrable since it is unbounded. Moreover, a more detailed exam reveals that F is neither Lebesgue integrable, since F is not absolutely continuous on [0, 1]. Therefore the Lebesgue integral does not solve the problem of primitives. So it was natural to find an integration process for which the following theorem holds:

The Fundamental Theorem of CalculusIfF:abRis differentiable on [a, b], then the functionFxis integrable on [a, b] andabFtdt=FbFa.

The first solution to this problem was given in 1912 by Denjoy, shortly followed by Perron. Both definitions are constructive. While Denjoy developed a new method of integration, called totalization, that includes the Lebesgue integral and that gets the value of the integral of a function through a transfinite process of Lebesgue integrations and limit operations, Perron used an approach that does not require the theory of measure, based on families of major and minor functions previously introduced by de la Vallée Poussin. Later Hake, Alexandroff, and Looman independently proved that the Denjoy integral and the Perron integral are equivalent (see Ref. [19]). Hence from now on, we will refer to the integral of Denjoy–Perron.

Subsequently, denoted by ωFcd the oscillation of F on a given interval [c, d] (i.e. ωFcd=supαβcdFβFα), Luzin introduced the following notions of functions AC* and ACG*.

Definition 6.1. A function F:abR is said to be AC* on a set Eab if, given ε>0, there is a constant η>0 such that

i=1pω(F,aibi<εE59

for each finite system aibi of nonoverlapping intervals such that aibiE=0.

Definition 6.2. A function F:abR is said to be ACG* on [a, b] if it is continuous and there is a decomposition ab=iEi with Ei closed sets, such that F is AC* on Ei, for each i.

Then Luzin proved the following characterization of the Denjoy–Perron integrable functions: A functionf:abRis Denjoy–Perron integrable on [a, b] if and only if there exists a functionF:abRthat is ACG* on [a, b] and differentiable almost everywhere on [a, b] such thatFx=fxalmost everywhere on [a, b].

Note that the “if” part of this characterization is often called “the descriptive” definition of the Denjoy–Perron integral.

Moreover, since the constructive definition of the Denjoy-Perron integral was not as immediate as that of the Riemann integral, only a few mathematicians were interested in working with it, so the descriptive definition became the most known definition of the Denjoy–Perron integral.

So, in spite of the general conviction that no modification of Riemann’s method could possibly give such powerful results as that of the Lebesgue integral, Kurzweil in Ref. [20] and Henstock in Ref. [21] introduced independently, a generalized version of the Riemann integral that is known as the Henstock-Kurzweil integral. Such an integral solves the problem of primitives and encloses the Lebesgue integral, in the sense that if a function f is Lebesgue integrable, therefore, it is also integrable in the sense of Henstock-Kurzweil and the two integrals coincide.

The method of Henstock-Kurzweil is based on the notions of gauge and partition. Precisely, given a subinterval [a, b] of the real line R, a gauge on [a, b] is, by definition, any positive function δ defined on it. A partition of [a, b] is, by definition, any collection P=Aixii=1p of pairwise disjoint intervals Ai and points xiAi such that ab=iAi. Moreover, P is said to be δ-fine whenever Aixiδixi+δi,i=1,2,,p.

Given a gage δ on [a, b], the existence of δ-fine partitions of [a, b] is ensured by the following lemma.

Cousin’s lemma.For any gageδon [a, b], there exists aδ-fine partition of [a, b].

Given a function f:abR and a partition P=Aixii=1p of [a, b] the Riemann sum of f with respect to P is defined as follows

Pf=i=1pfxiLAi,E60

Definition 6.3. Let f:abR. We say that f is Henstock–Kurzweil integrable on [a, b] if there exists a number IR satisfying the following condition: to each positive ε there is a gage δ on [a, b] such that

PfI<ε,E61

for each δ-fine partition P of [a, b].

The number I is called the Henstock–Kurzweil integral of f on [a, b], and it is denoted by HKabfxdx.

The proofs of the basic properties of the Henstock–Kurzweil integral, like the linearity and the Cauchy criterion, can be found in Ref. [10].

Moreover, the integral of Henstock-Kurzweil, even if its definition is completely different from that of Denjoy-Perron, it is equivalent to the Denjoy-Perron integral. This result was shown by several mathematicians, see Refs. [22, 23] for instance, by proving the following characterization of the Henstock-Kurzweil primitives: The family of all Henstock-Kurzweil primitives coincides with the class of all ACG*-functions.

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Written By

Donatella Bongiorno

Submitted: 05 May 2023 Reviewed: 15 May 2023 Published: 26 June 2023