The Henstock–Kurzweil Integral

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Department of Mathematics, Linköping University David Manolis LiTH-MAT-EX--2020/0

5

--SE Credits: 16 Level: G2 Supervisor: Anders Björn,

Department of Mathematics, Linköping University Examiner: Jana Björn,

Department of Mathematics, Linköping University Linköping: June 2020

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2

Abstract

Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventually becomes the standard integral.

Unfortunately, in the transition from the Riemann integral to the Lebesgue inte-gral there are certain topics of pure theoretical interest which to a certain extent are neglected. This is particularly the case for topics regarding the inverse relation-ship between differential and integral calculus and the integration of exceedingly complicated functions which for example might be of a highly oscillatory nature. From an applied mathematician’s point of view, the partial neglection of these topics in the case of highly problematic functions might be justified in the sense that this theory is unnecessary for modeling most problems that appear in na-ture. From a theoretician’s point of view however this negligence is unacceptable. Consequently, there are alternative integrals which give rise to theories which one can use in an attempt to study these aforementioned topics. An example of such an integral is the Henstock–Kurzweil integral, which can be developed in a rather similar manner to that of the Riemann integral.

In this thesis we will develop the Henstock–Kurzweil integral in order to answer some of the questions which to a certain extent are beyond the scope of the Lebesgue integral while using rather basic proof techniques from complex anal-ysis and measure theory. In addition to that we extended various properties of the Lebesgue integral to the Henstock–Kurzweil integral, in particular when it comes to Lebesgue’s fundamental theorem of calculus and the basic convergence theorems of the Lebesgue integral.

Keywords:

Henstock–Kurzweil integral, generalized Riemann integral, gauge integral, Denjoy–Perron integral, narrow Denjoy integral, Perron integral, non-absolute integration, Lebesgue outer measure, absolute integrability, fundamental the-orem of calculus, convergence thethe-orems, single variable calculus.

URL for electronic version:

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Acknowledgements

I would like to thank my supervisor Anders Björn for his feedback throughout the process of writing this thesis. Without his contributions this thesis would exhibit several glaring flaws. For example, the formatting of various extended computations would have lacked proper structure, in particular when it comes to the alignment of operators and expressions. Further, he identified errors and flaws in certain proofs and examples, corrected certain historical inaccuracies and suggested alternative terminology for terms which defied scientific convention. I would also like to thank my examiner Jana Björn for the feedback she gave me. She pointed out several errors in my work and gave me advice on how to better formulate logical statements in running text. Although I did not have time to implement all the advice she gave me, I will certainly make use of it in my master’s thesis.

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Contents 4

1 Introduction

In this section we will begin by briefly going over the content of each section of this thesis. Subsequently, we will briefly go over the history of integration theory so that the reader has an idea of which developments have paved the way for the discovery of the Henstock–Kurzweil integral. With this in mind, we will subsequently present the purpose of this thesis.

1.1 Section Outline

• Section 1: In this section we briefly discuss various historical developments which have paved the way for the discovery of the Henstock–Kurzweil in-tegral. In particular, we highlight the drawbacks of the more prominent integrals of Riemann and Lebesgue. Subsequently, we discuss the integrals of Denjoy och Perron which are able to remedy the aforementioned draw-backs at the unfortunate cost of considerably higher complexity. Finally, we discuss what role the Henstock–Kurzweil integral plays in all of this and why we should be interested in it. With this in mind, we then present the purpose of this thesis.

• Section 2: We develop some preliminary theory, in particular when it comes to the properties of the Lebesgue outer measure. Furthermore, we prove the Vitali covering theorem which is used in the more technical proofs of the subsequent sections.

• Section 3: The elementary properties of the Henstock–Kurzweil integral are developed. Most of these properties assure us that the Henstock–Kurzweil integral behaves the way we expect upon reading its definition. In section 4 these elementary properties are used to prove more technical theorems. • Section 4: In this section we prove the most technical theorems of this

thesis. Most of the necessary tools will have been developed in the previous sections. In particular, we prove a result which is reminiscent of the funda-mental theorem of calculus for the Lebesgue integral and we prove various convergence theorems.

• Section 5: At this point we have proved a lot of results. We begin by discussing to which extend the Henstock–Kurzweil integral is able to remedy the drawbacks of the Lebesgue integral. Subsequently, we discuss what role the Henstock–Kurzweil integral plays in education and why it is not more popular despite its power.

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1.2 A Brief Historical Background 6

1.2 A Brief Historical Background

At the end of the seventeenth century Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) independently discovered differential and integral calculus. Differential calculus was used to define the slope of a curve at a particular point and integral calculus was used to compute the area under a curve. As it turns out, for a large class of functions there was an inverse relationship between differential and integral calculus in the sense that a differentiable function could be obtained up to a constant by integrating its derivative. In fact, a function F : [a, b] → R which is differentiable everywhere on [a, b] except on a loosely stated small set S ⊆ [a, b], could in certain cases be recovered by integrating a function f : [a, b] → R which satisfies F0 = f on [a, b] \ S. Exactly how small S had to be and how well-behaved F had to be on S was not known at the time. The mathematics of that time simply did not allow mathematicians to conduct research into descriptive characterizations of integrable functions and consequently many aspects of integration theory remained shrouded in mystery.

During the eighteenth century mathematicians began to realize that the very foun-dation of mathematical analysis was highly unstable and loosely defined. When geometric intuition no longer was sufficient, this served as a massive obstacle in research. Consequently, in the nineteenth century mathematicians began devel-oping a rigorous framework for mathematical analysis using various epsilon-delta type definitions and proofs. In particular, it was the formal approach to continuity which laid the groundwork for many significant breakthroughs in analysis. The modern treatment of continuity is typically attributed to Augustin–Louis Cauchy (1789–1857), but there are many more mathematicians who made important con-tributions in this direction as well, see [1]. During this time integration theorists began to acquire tools which would allow them to rigorously define and study various integrals. In 1854 Bernhard Riemann (1826–1866) introduced the Rie-mann integral which was one of the first formally defined integrals. In 1904 Henri Lebesgue (1875–1941) was able to show that a real-valued function defined on a compact interval is Riemann integrable if and only if it is bounded and discontinu-ous on a set of Lebesgue measure 0.1 _{However, there are bounded derivatives which}

are discontinuous on a set of positive Lebesgue measure, and thus such derivatives are not Riemann integrable. An example of such a function was given by Vito Volterra (1860–1940) in 1881, which is constructed in [6, Example 1.4.1]. This

1_{In certain literature such as [2, Chapter 8.6.3] it is claimed that this result goes further}

back to 1901. Unfortunately I have been unable to find a proper reference to justify such a claim. However, in Lebesgue’s doctoral dissertation from 1902, Lebesgue attributes the forward direction of the aforementioned integrability criterion to Riemann who had produced a similar result, see [3, p. 254]. The full theorem can be found in Lebesgue’s book [4, p. 29] from 1904.

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drawback was remedied by the Lebesgue integral which Lebesgue introduced in [5] 1901. As it turns out, the Lebesgue integral is a generalization of the Riemann integral and it is able to integrate all bounded derivatives restricted to compact intervals. Going back to our functions F and f where F0 = f on [a, b] \ S, if [a, b] is compact then by Lebesgue’s fundamental theorem of calculus we have that f is Lebesgue integrable on [a, b] and F can be reconstructed by the Lebesgue integral of f if and only if F is absolutely continuous and S has Lebesgue measure 0. Note that the absolute continuity of F is a necessity here, thus it is easy to show that there are differentiable functions which cannot be recovered from their respective derivatives via the Lebesgue integral. If F is not absolutely continuous and S has Lebesgue measure 0, then in certain cases f may still be Lebesgue integrable but

F can then not be recovered from the Lebesgue integral of f . For example, if

[a, b] = [0, 1], F is the Cantor–Lebesgue function studied in [7], and f = 0 every-where on [0, 1], then F0 = f on [0, 1] \ S, where S has Lebesgue measure 0, yet F cannot be reconstructed from f via the Lebesgue integral.

Unfortunately, the full power of the Lebesgue integral as it is known today was not understood by the mathematical community during this time and thus Lebesgue’s work did not garner the immediate attention one perhaps would expect. On a second thought this is not too surprising, since measure theory was still a very new topic and functional analysis had not yet been developed to a point where these two areas of mathematics could be researched in unity to produce a more complete theory. Perhaps this is a partial explanation to why Lebesgue was not appointed as professor until 1919, as mentioned in [8, Chapter VI.72]. However, later on in the twentieth century mathematicians incrementally developed the necessary tools to study the Lebesgue integral more thoroughly and eventually the Lebesgue integral became the standard integral in advanced mathematical analysis.

Despite the power of the Lebesgue integral, there were still finite and unbounded derivatives that could not be integrated. In 1912 Arnaud Denjoy (1884–1974) presented a powerful integral which was able to integrate all finite derivatives and recover their primitive functions. As described in [9, Section 1.1], loosely stated, Denjoy constructed a transfinite sequence (Ik)k≤Ωof increasingly general integrals,

where Ω is the first uncountable ordinal, I0 is the Lebesgue integral and IΩis the so

called narrow Denjoy integral. Two years later in 1914 Oskar Perron (1880–1975) introduced an integral, which after some modification in 1915 by Hans Bauer2 _was

2_{I have been unable to confirm the dates for Bauer. In [10, p. 378] a certain Hans Adolf Bauer}

(1891–1953) is mentioned, who at some point in the early twentieth century was the private tutor of the famous theoretical physicist Wolfgang Ernst Pauli (1900–1958) in mathematics and physics. This may be the same Bauer as the one who worked on the Perron integral, but I have not been able to confirm this.

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1.3 Purpose of the Thesis 8

also able to integrate all finite derivatives and recover their primitive functions. It took quite some time, but by 1925 mathematicians had realized that the narrow Denjoy integral and the Perron integral are in fact equivalent, this is called the Hake–Aleksandrov–Looman theorem. Consequently, the aforementioned integral of Denjoy and Perron is today called the Denjoy–Perron integral.

Many years later in 1957, Jaroslav Kurzweil (born 1926) published a paper on differential equations in which he introduced a new integral. Four years later in 1961, while unaware of the work of Kurzweil, Ralph Henstock (1923–2007) published a paper on integration theory in which he introduced the same integral as Kurzweil. Throughout a series of papers in the sixties Henstock developed a substantial amount of properties of this integral. The definition of this integral as defined by Kurzweil in [11] and Henstock in [12] is quite elegant since it is highly reminiscent of the Riemann integral and since a substantial amount of its properties can be developed using Riemann sums and basic epsilon-delta proofs. Today the integral of Henstock and Kurzweil is called the Henstock–Kurzweil integral. As mathematicians later discovered, the Henstock–Kurzweil integral is in fact equivalent to the Denjoy–Perron integral. By the late nineties a lot of integration theorists had researched the Henstock–Kurzweil integral extensively and consequently the theory of this integral had been highly refined. Some of the most prominent results will be studied in this thesis.

For a more detailed historical background regarding the aforementioned integrals the interested reader may consult [13].

1.3 Purpose of the Thesis

Nowadays there is a rather substantial amount of literature in which the Henstock– Kurzweil integral is studied. However, seldom is integration on unbounded inter-vals considered. The theory of integration on unbounded interinter-vals can often be developed in a similar manner to that of integration on compact intervals, but there are certain subtle exceptions which are rarely discussed, let alone formally justified. Further, in the aforementioned literature real-valued functions are pre-dominantly studied, therefore one is left wondering if perhaps the corresponding theory for complex-valued functions can be developed in an analogous manner. In this thesis we will develop the Henstock–Kurzweil integral for complex-valued functions defined on closed intervals in the extended real numbers. In particular, we will investigate to what extent the Henstock–Kurzweil integral puts an end to the discussion of recovering a differentiable function from its derivative, and to which extent the convergence theorems from the theory of the Lebesgue integral can be extended to the Henstock–Kurzweil integral.

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2 Preliminaries

In order to study the Henstock–Kurzweil integral thoroughly, we need some theory from other areas of mathematics. In particular, we will need some basic results from measure theory related to sets of Lebesgue outer measure 0. This will ul-timately allow us to discover a deep connection between differential calculus and the Henstock–Kurzweil integral and it will allow us to prove stronger convergence theorems.

2.1 The Complex and Extended Real Numbers

In order to define the Henstock–Kurzweil integral for complex-valued functions defined on closed intervals which may be unbounded we need to define the extended real numbers.

Definition 2.1.1. We define the extended real numbers by

R := R ∪ {−∞, ∞}.

We partially extend the standard arithmetic operations of R to R in the following way:

• Subtraction: ∞ − x := ∞ and x − (−∞) := ∞ if x ∈ R. Further, ∞ − ∞ := 0, (−∞) − (−∞) := 0 and ∞ − (−∞) := ∞.

• Multiplication: 0 · ∞ := 0 and x · ∞ := ∞ if x ∈ R is such that x > 0. • Exponentiation: ∞−x

:= 0 if x ∈ R is such that x > 0.

We extend the standard order relations and the absolute value function from R to R in the following way:

• Partial order relations: −∞ < ∞ and −∞ < x < ∞ if x ∈ R. • Absolute value: | ± ∞| := ∞.

Remark 2.1.2. One has to be very careful with how algebraic expressions are

handled with the above definitions. The way we deal with infinite points in this thesis will not result in any contradictions and thus we may proceed with these definitions. Their only purpose is to simplify notation. For example, for any [u, v] ⊆ R we have that 0 · (v − u) = 0. As we will see later, this will make Riemann sums more convenient to deal with. Further, in this thesis an ε > 0 will simply refer to some ε ∈ R such that ε > 0 and not ε = ∞.

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2.1 The Complex and Extended Real Numbers 10

In order to prove the monotone convergence theorem and the dominated conver-gence theorem for the Henstock–Kurzweil integral for complex-valued functions we need to extend the order relations on R to partial orders on C.

Definition 2.1.3. For any x, y ∈ C we write

x ≤ y if Re x ≤ Re y and Im x ≤ Im y,

x ≥ y if Re x ≥ Re y and Im x ≥ Im y.

Definition 2.1.4. Consider a function f : [a, b] → C. We say that f is

increasing if f (x) ≤ f (y) whenever x, y ∈ [a, b] are such that x ≤ y,

decreasing if f (x) ≥ f (y) whenever x, y ∈ [a, b] are such that x ≤ y, and in both cases we say that f is monotone.

Definition 2.1.5. Consider a collection of functions f1, . . . , fn : [a, b] → C. We

define

min{f1, . . . , fn} := min{Re f1, . . . , Re fn} + i min{Im f1, . . . , Im fn},

max{f1, . . . , fn} := max{Re f1, . . . , Re fn} + i max{Im f1, . . . , Im fn}.

Definition 2.1.6. In this thesis N will denote the set of positive integers strictly

greater than 0.

Definition 2.1.7. For each k ∈ N consider a function fk : [a, b] → C. If there is

a function g : [a, b] → C such that g ≤ fk for each k ∈ N, then we define

inf{fk : k ∈ N} := inf{Re fk : k ∈ N} + i inf{Im fk: k ∈ N},

and if there is a function h : [a, b] → C such that fk ≤ h for each k ∈ N, then we

define

sup{fk : k ∈ N} := sup{Re fk: k ∈ N} + i sup{Im fk: k ∈ N}.

Definition 2.1.8. For any function f : [a, b] → C we define X

k∈∅

f (k) := 0.

Remark 2.1.9. Sometimes we will be computing sums and series over sets which

might be empty. Instead of splitting up certain proofs into several cases, the above definition allows us to compute sums and series over the empty set without any problems. As a result several proofs become more concise.

Definition 2.1.10. For each set E ⊆ R we let O(E) denote the set of all intervals

contained in E which are open with respect to E, and we let C(E) denote the set of all intervals [a, b] contained in E.

Example 2.1.11. For instance,

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2.2 Lebesgue Outer Measure

In this section we will define the Lebesgue outer measure and prove some of its properties. These results will be of great importance when we study the Henstock– Kurzweil integral.

Definition 2.2.1. For each interval ` ⊆ R we define the length of ` by

L(`) :=    sup ` − inf ` _{if ` 6= ∅,} 0 _{if ` = ∅.}

We define the Lebesgue outer measure of a set Ω ⊂ R by

µ∗(Ω) := inf ( _∞ X k=1 L(`k) : Ω ⊆ ∞ [ k=1

`k and `k∈ O(R) for each k ∈ N

)

.

Remark 2.2.2. We could go even further and define Lebesgue measure, but since

we will predominantly study sets of Lebesgue outer measure 0 this would be quite unnecessary since these are exactly the sets which have Lebesgue measure 0. Thus we shall content ourselves with the Lebesgue outer measure.

Definition 2.2.3. Consider a set E ⊆ R. We say that an assertion A regarding

the elements of E is true almost everywhere on E, if A is true everywhere on E except on a set of Lebesgue outer measure 0.

We will now prove that for any interval ` ⊆ R we have that L(`) = µ∗(`). This of course is to be expected. However, the proof of this assertion is not entirely trivial and thus we will prove it here. But first we need the following lemma, the proof of which was inspired by the one given in [14, Lemma 1.2].

Lemma 2.2.4. For any interval ` ∈ C(R) we have that

µ∗(`) = L(`).

Proof. Pick any ` ∈ C(R). The assertion is trivially true for the case ` = ∅, so

assume that ` 6= ∅. Let {`i : i ∈ N} ⊂ O(R) be any fixed but arbitrary collection

of intervals such that ` ⊆ S∞

i=1`i. By the Heine–Borel theorem there is a finite

collection of intervals {ì1, . . . , ìn} ⊆ {ì : i ∈ N} such that

` ⊆ n [ j=1 ìj. (2.1) Note that n X j=1 L(ìj) ≤ ∞ X i=1 L(ì). (2.2)

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2.2 Lebesgue Outer Measure 12

Without a loss of generality we may assume that ∅ /∈ {`i1, . . . , `in} and for any

j ∈ N such that j < n we have that

inf ìj < inf ìj+1 < sup ìj < sup ìj+1. (2.3)

With these assumptions (2.1) and (2.2) are still true. Define the interval

I :=

n

[

j=1

`ij, (2.4)

and note that

L(I)2.2.1= sup I − inf I

(2.3) (2.4) = sup ìn − inf ì1 (2.3) ≤ n X j=1 (sup ìj − inf ìj) 2.2.1 = n X j=1 L(ìj) (2.2) ≤ ∞ X i=1 L(ì). (2.5)

We have that (2.1), (2.4) and (2.5) imply that

µ∗(`) = inf{L(`0) : ` ⊂ `0 _{∈ O(R)} = L(`).}

Theorem 2.2.5. For any interval ` ⊆ R we have that

L(`) = µ∗(`).

Proof. Pick any interval ` ⊆ R. If ` ∈ C(R) then the assertion is true by Lemma

2.2.4, so assume that ` /_{∈ C(R). We begin by considering the case where ` is} bounded. Define the interval

` `6=∅:= ` ∪ {inf `, sup `}, and note that ` ∈ C(R). We have that

L(`)2.2.1= L(`) 2.2.4= µ∗(`)`⊇`≥ µ∗(`). (2.6) For the sake of a contradiction assume that L(`) > µ∗(`) and pick any ε > 0 such that

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Define the interval `0 := inf ` + ε 2, sup ` − ε 2 . (2.8) Note that L(`) − µ∗(`0)2.2.4= L(`) − L(`0) 2.2.1

= sup ` − inf ` − (sup `0− inf `0)

(2.8)

= sup ` − inf ` − (sup ` − inf ` − ε) = ε

(2.7)

< L(`) − µ∗(`), and therefore

µ∗(`0) > µ∗(`). (2.9) By Definition 2.2.1 we clearly have that µ∗(`0) ≤ µ∗(`) since `0 ⊂ ` and therefore we have reached a contradiction in (2.9). Thus we conclude that µ∗(`) = L(`).

Finally, we consider the case where ` is unbounded. For the sake of a contradiction we assume that there is a finite number N ≥ 0 such that µ∗(`) = N . Since ` is unbounded we can find a bounded interval I ⊂ ` such that

µ∗(I) = L(I) > N = µ∗(`). (2.10)

Again by Definition 2.2.1 we have that µ∗(I) ≤ µ∗(`) since I ⊂ ` and therefore we have reached a contradiction in (2.10). Thus µ∗(`) = ∞. Further, L(`) = ∞ by

Definition 2.2.1 and therefore µ∗(`) = L(`).

Theorem 2.2.6. Consider a set Ω ⊆ R and let {Ωi : i ∈ N} be a collection of sets

such that Ω ⊆S∞

i=1Ωi. We have that

µ∗(Ω) ≤ ∞ X i=1 µ∗(Ωi). Proof. If P∞

i=1µ∗(Ωi) = ∞ then the assertion is trivially true, so assume instead

that P∞

i=1µ

∗_(Ω

i) < ∞. Pick any ε > 0 and let {`i,j : i, j ∈ N} ⊂ O(R) be such

that for each i ∈ N we have that Ωi ⊆ ∞ [ j=1 `i,j and ∞ X j=1 µ∗(`i,j) < µ∗(Ωi) + ε 2i. (2.11)

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2.2 Lebesgue Outer Measure 14 Thus µ∗(Ω) ≤ µ∗ ∞ [ i=1 Ωi ! ≤ ∞ X i=1 ∞ X j=1 µ∗(`i,j) (2.11) < ∞ X i=1 µ∗(Ωi) + ε 2i = ∞ X i=1 µ∗(Ωi) + ε ∞ X i=1 1 2i = ∞ X i=1 µ∗(Ωi) + ε. (2.12)

Since ε > 0 was arbitrary, (2.12) implies that

µ∗(Ω) ≤

∞

X

i=1

µ∗(Ωi).

Corollary 2.2.7. Every countable subset of R has Lebesgue outer measure zero.

Proof. Pick any countable set Ω ∈ R and let σ : N → Ω be a surjection. µ∗(Ω)2.2.6≤

∞

X

i=1

µ∗({σ(i)})2.2.5= 0.

Remark 2.2.8. Intuitively it seems likely that if we added to the supposition of

Theorem 2.2.6 that the sets in {Ωi : i ∈ N} are pairwise disjoint and Ω =S∞i=1Ωi,

then we would instead have µ∗(Ω) = P∞

i=1µ

∗_(Ω

i). However, in general this is not

the case, so we need to impose further conditions on {Ωi : i ∈ N} in order for this

to be true.

Definition 2.2.9. Let S be a collection of intervals contained in R and define

S0 := {` \ {inf `, sup `} : ` ∈ S and ` 6= ∅}.

The intervals of S are pairwise internally disjoint if

`1∩ `2 = ∅ whenever `1, `2 ∈ S0 are such that `1 6= `2.

Lemma 2.2.10. Consider an interval ` ⊆ R and let {`i : i ∈ N} be a collection of

pairwise internally disjoint intervals contained in `. We have that

µ∗(`) ≥

∞

X

i=1

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Proof. By Definition 2.2.1 and Theorem 2.2.5 the assertion is trivially true for the

case where ` is empty or unbounded, so assume that ` is non-empty and bounded. For any j ∈ N let {I1,j, . . . , In(j),j} be such that {`1, . . . , `j} ∪ {I1,j, . . . , In(j),j} is a

collection of pairwise internally disjoint intervals for which

` = j [ i=1 ì∪ n(j) [ i=1 Ii,j. (2.13) Thus ∞ X i=1 µ∗(ì) = lim j→∞ j X i=1 µ∗(ì) 2.2.5 ≤ lim j→∞   j X i=1 L(ì) + n(j) X i=1 L(Ii,j)   2.2.1 (2.13) = lim j→∞L(`) 2.2.5 = µ∗(`).

Corollary 2.2.11. Consider an interval ` ⊆ R and let {`i : i ∈ N} be a collection

of pairwise internally disjoint intervals contained in ` such that ` = S∞

i=1`i. We have that µ∗(`) = ∞ X i=1 µ∗(`i).

Proof. The assertion is an immediate consequence of Theorem 2.2.6 and Lemma

2.2.10.

Definition 2.2.12. Let Ω ⊆ R be arbitrary. A set V ⊆ C(R) is a Vitali cover of

Ω if there for every ε > 0 and for every x ∈ Ω exists an interval ` ∈ V such that

x ∈ ` and 0 < µ∗(`) < ε.

The following theorem is of great importance in measure theory and will serve as a crucial lemma in several proofs in this thesis. The proof closely resembles the one given in [15], although we will prove a slightly stronger version which covers unbounded sets.

Theorem 2.2.13 (The Vitali covering theorem). Consider a set Ω ⊆ R and let V

be a Vitali cover of Ω. There exists a sequence (`k)∞k=1 of pairwise disjoint intervals

contained in V ∪ {∅} such that

µ∗ Ω \ ∞ [ k=1 `k ! = 0.

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Proof. We begin by proving the assertion for the case when Ω is bounded. Pick

any bounded interval U ⊂ O(R) for which Ω ⊂ U. If there is a finite collection of pairwise disjoint intervals in V that cover all of Ω then the assertion is trivially true, so assume that there is no such finite collection.

Our sequence (`k)∞k=1 will be constructed recursively. Fix an arbitrary non-empty

interval `1 ∈ V such that `1 ⊂ U . Assume that we have selected a sequence (`k)nk=1

of pairwise disjoint intervals contained in V. For each n ∈ N define

Un:= U \ n

[

k=1

`k. (2.14)

Pick an arbitrary x ∈ Ω∩Un. SinceSnk=1`kis closed, (2.14) implies that Unis open.

Further, by our assumption we have that Ω \Sn

k=1`k6= ∅ and since Ω ⊂ U we have

that Un 6= ∅. Therefore, since V is a Vitali cover of Ω there is a number ηn > 0

and an interval In ∈ V such that µ∗(In) > 0 and x ∈ In ⊂ (x − ηn, x + ηn) ⊂ Un.

Define

δn:= sup{µ∗(`) : ` ∈ V and ` ⊂ Un},

and note that by the existence of the aforementioned interval In we conclude that

δn > 0. Further, since U is bounded we have by Theorem 2.2.5 and Lemma

2.2.10 that δn ≤ µ∗(U ) = L(U ) < ∞. So there is an interval `n+1 ∈ V such that

`n+1 ⊂ Un and µ∗(`n+1) > δn/2. We take this interval and add it to our sequence

to get (`k)n+1k=1.

Thus we have our sequence (`k)∞k=1 of pairwise disjoint intervals contained in V,

certain desired properties of which we have yet to prove. We have that

∞ X k=1 µ∗(`k) 2.2.10 ≤ µ∗(U )2.2.5= L(U ) U is bounded < ∞, and thus lim M →∞ ∞ X k=M µ∗(`k) = 0. (2.15) Let M ∈ N be arbitrary. If x ∈ Ω\S∞ k=1`k, then x ∈ Ω\SMk=1`k ⊆ UM. By previous

arguments, there is a number η > 0 and an interval I ∈ V such that µ∗(I) > 0 and x ∈ I ⊂ (x − η, x + η) ⊂ UM. Since µ∗(`k+1) > δ₂k for each k ∈ N and since

P∞

k=1µ

∗_(`

k) < ∞ we have that limk→∞δk = 0. Thus there is a smallest positive

integer N for which I 6⊂ UN. Note that M < N , since if we assume otherwise we

have that I ⊂ UM ⊆ UN, which is a contradiction. Since I 6⊂ UN and I ⊂ UN −1

we have that I ∩ `N 6= ∅. Further, by the definition of δN −1 and `N we see that

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Let y ∈ I, xk ∈ `kand r, rk ≥ 0 be such that [y−r, y+r] := I and [xk−rk, xk+rk] :=

`k, for each k ∈ N. Since I ∩ `N 6= ∅ we have that

y + r ≤ xN + rN + 2r 2.2.5 (2.16) ≤ xN + rN + 2µ∗(`N) 2.2.5 = xN + rN + 4rN = xN + 5rN, and y − r ≥ xN − rN − 2r 2.2.5 (2.16) ≥ xN − rN − 2µ∗(`N) 2.2.5 = xN − rN − 4rN = xN − 5rN. Therefore x ∈ I ⊆ [xN − 5rN, xN + 5rN] ⊆ ∞ [ k=M [xk− 5rk, xk+ 5rk]. Since x ∈ Ω \S∞

k=1`k was arbitrary, we have that

Ω \ ∞ [ k=1 `k ⊆ ∞ [ k=M [xk− 5rk, xk+ 5rk]. (2.17) Note that µ∗ ∞ [ k=M [xk− 5rk, xk+ 5rk] !2.2.5 2.2.6 ≤ 5 ∞ X k=M µ∗(`k). (2.18)

Since M ∈ N was arbitrary, we have by (2.15) that lim M →∞5 ∞ X k=M µ∗(`k) = 0. (2.19)

Thus by (2.17), (2.18), (2.19) and Theorem 2.2.6 we acquire the desired result

µ∗ Ω \ ∞ [ k=1 `k ! = 0,

which completes the proof for the case when Ω is bounded. We now assume that Ω is unbounded. Let σ : N → Z be a bijection and define for each i ∈ N the set

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For each i ∈ N let (`σ(i),j)∞j=1 be a sequence of disjoint intervals in V ∪ {∅} such

that µ∗  Ω_i\ ∞ [ j=1 `σ(i),j  = 0. (2.20)

By the proof of the bounded case we may without a loss of generality assume that for each i ∈ N we have that

∞

[

j=1

`σ(i),j ⊆ (σ(i), σ(i) + 1). (2.21)

Therefore µ∗  Ω \ ∞ [ i=1 ∞ [ j=1 `σ(i),j   2.2.6 (2.21) ≤ ∞ X i=1 µ∗  Ω_i\ ∞ [ j=1 `σ(i),j  + µ ∗ (Z) 2.2.7 (2.20) = 0,

which implies that

µ∗  Ω \ ∞ [ i=1 ∞ [ j=1 `σ(i),j  = 0.

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3 The Henstock–Kurzweil Integral

In this section we will construct the Henstock–Kurzweil integral for functions with domains in C(R) and ranges in C. Our objective is to develop the most basic properties of the integral and save the more advanced results for Section 4. In Section 3.1 we will state various definitions which ultimately will allow us to define the integral. In Section 3.2 we will prove the most basic properties of the integral and assure ourselves that it behaves as one immediately would expect upon reading its definition. In Section 3.3 we will use the basic measure theoretic tools we developed in Section 2.2 to develop further properties of the integral. These results are interesting by themselves but they also serve as crucial lemmas for the more advanced results of Section 4. Finally, in Section 3.4 we will briefly prove some results regarding a class of functions which are called absolutely integrable. These functions are in some sense quite well-behaved and possess a lot of nice properties, some of which will be used in Section 4.3 to prove Fatou’s lemma, from which the dominated convergence theorem follows almost immediately.

3.1 Definition of the Integral

In order to define the Henstock–Kurzweil integral and subsequently develop its properties we need the following definitions.

Definition 3.1.1. A gauge on an interval [a, b] is a function δ : [a, b] → O([a, b])

which is such that for each x ∈ [a, b] we have that x ∈ δ(x).

Definition 3.1.2. Consider an interval [a, b] and let S := {δ1, . . . , δn} be a finite

set of gauges on [a, b]. LetTn

k=1δk denote the gauge δ on [a, b] which is defined by

δ(x) :=

n

\

k=1

δk(x).

Definition 3.1.3. Consider an interval [a, b] and let Q be a finite set of pairwise

internally disjoint intervals contained in C([a, b])\{∅}. We say that Q is a partition of [a, b] if

[

[u,v]∈Q

[u, v] = [a, b], and a subpartition of [a, b] if

[

[u,v]∈Q

[u, v] ⊆ [a, b].

Definition 3.1.4. Consider an interval [a, b] and a set P which consists of pairs

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3.1 Definition of the Integral 20

Define the set Q := {[u, v] : (t, [u, v]) ∈ P }. P is a tagged partition of [a, b] if Q is a partition [a, b], and a tagged subpartition of [a, b] if Q is a subpartition of [a, b]. If there is a gauge δ on [a, b] such that for any (t, [u, v]) ∈ P we have that [u, v] ⊆

δ(t) and v − u < ∞ whenever |t| < ∞, then we say that P is δ-fine or subordinate

to δ.

Definition 3.1.5. Let P be a tagged subpartition of an interval [a, b]. We define

the refinement of P by

Ref(P ) := {(t, [u, t]), (t, [t, v]) : (t, [u, v]) ∈ P }.

Definition 3.1.6. A function f : [a, b] → C is Henstock–Kurzweil integrable if

there is a number L ∈ C such that for any ε > 0 there is a gauge δ on [a, b] such

that X (t,[u,v])∈P : |t|<∞ f (t)(v − u) − L < ε,

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

Remark 3.1.7. By the above definition we may assume that for any function

f : [a, b] → C we have that f (x) = 0 if x ∈ [a, b] ∩ {−∞, ∞}. Consequently,

we may abolish the condition |t| < ∞ from the Riemann sum. This will make notation more convenient. Note that now we may get terms of the type 0 · ∞ and 0 · (∞ − ∞), but by the arithmetic operations of Definition 2.1.1 this will not be a problem.

Definition 3.1.8. We let HK([a, b]) denote the set of all functions f : [a, b] → C

which are Henstock–Kurzweil integrable.

The following theorem is of great importance and says that for any interval [a, b] and for any gauge δ on [a, b] there exists a δ-fine tagged partition of [a, b]. This implies that there is no function which can satisfy Definition 3.1.6 vacuously. The proof was inspired by the one given in [16, Theorem 5.6].

Theorem 3.1.9 (Cousin’s theorem). Consider an interval [a, b] and let δ be a

gauge on [a, b]. There exists a δ-fine tagged partition of [a, b].

Proof. In the case where a = b the assertion is trivially true, so assume that a < b.

We begin by considering the case where [a, b] is bounded. Define the set Ω := {x ∈ [a, b] : a < x ≤ b and there is a δ-fine tagged partition of [a, x]}.

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Since δ(a) 6= ∅ implies that Ω 6= ∅ and since b < ∞ we have that y := sup Ω exists and is finite. Define ya := inf δ(y) and yb := sup δ(y). Note that a ≤ ya< y.

Fix any x1 ∈ (ya, y] ∩ Ω and let Px1 be a δ-fine tagged partition of [a, x1]. Thus Px1 ∪ {(y, [x1, y])} is a δ-fine tagged partition of [a, y]. This implies that y ∈ Ω.

3

If y = b, then we are done, so assume that y < b. Thus y < yb ≤ b. Fix any

x2 ∈ (y, yb). We have that Px1∪ {(y, [x1, x2])} is a δ-fine tagged partition of [a, x2].

This implies that x2 ∈ Ω and y < x2, which is a contradiction to y being the

supremum of Ω. Thus we conclude that there must be a δ-fine tagged partition of [a, b].

Now we consider the case where [a, b] is unbounded. Let u, v ∈ (a, b) be such that u < v, [a, u] ⊂ δ(a) and [v, b] ⊂ δ(b). Define the gauge γ on [u, v] by

γ(x) := δ(x) ∩ [u, v], for each x ∈ [u, v]. By our previous result we may fix a

γ-fine tagged partition P of [u, v]. Thus P ∪ {(a, [a, u]), (b, [v, b])} is a δ-γ-fine tagged

partition of [a, b].

Remark 3.1.10. The classical definition of the Henstock–Kurzweil integral as

defined by Jaroslav Kurzweil in [11] and Ralph Henstock in [12] is defined for functions f : [a, b] → R where [a, b] ⊆ R, and is based on an alternative definition of a gauge which is defined as a function δ : [a, b] → (0, ∞). A δ-fine tagged partition P of [a, b] then satisfies (t, [u, v]) ∈ P ⇒ [u, v] ⊆ (t − δ(t), t + δ(t)) ∩ [a, b]. Clearly this gives rise to a theory which is just a special case of the one we are considering. The reader should also note that there are alternative definitions of the Henstock–Kurzweil integral which make use of upper and lower sums in a similar manner to that of the Darboux integral, see [17] and [18, Definition 12]. Note that the Darboux integral is equivalent to the Riemann integral and is commonly developed in elementary calculus courses.

3.2 Basic Properties

In this section we will develop various basic properties of the Henstock–Kurzweil integral. These properties are essential for further studies and assure us that the Henstock–Kurzweil integral possesses the basic properties we immediately expect upon reading its definition. Further, towards the end of this section we will prove some important theorems regarding improper integrals which will serve as the first collection of theorems that accentuate the strength of the Henstock–Kurzweil integral.

3_{To be more precise, it would have been sufficient to fix any x}

1 ∈ (ya, y) since (ya, y) ⊆ Ω

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3.2 Basic Properties 22

Theorem 3.2.1. Consider a function f ∈ HK([a, b]). There is a unique L ∈ C

such that for any ε > 0 there is a gauge δ on [a, b] which satisfies

X (t,[u,v])∈P f (t)(v − u) − L < ε,

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

Proof. Let ε > 0 be arbitrary. Assume that there are two numbers L1, L2 ∈ C and

two gauges δ1, δ2 on [a, b] such that

X (t,[u,v])∈P1 f (t)(v − u) − L1 < ε 2, (3.1) and X (t,[u,v])∈P2 f (t)(v − u) − L2 < ε 2, (3.2)

for any tagged partitions P1 and P2 of [a, b] which are subordinate to δ1 and δ2

respectively. Define a gauge δ on [a, b] by δ := δ1∩ δ2 and let P be a δ-fine tagged

partition of [a, b]. We have that |L1− L2| ≤ X (t,[u,v])∈P f (t)(v − u) − L1 + X (t,[u,v])∈P f (t)(v − u) − L2 (3.1) (3.2) < ε 2+ ε 2 = ε. (3.3)

Since ε > 0 was arbitrary, (3.3) implies that L1 = L2.

Definition 3.2.2. We will denote the unique number L ∈ C in Theorem 3.2.1 by Z

[a,b]

f := L.

Theorem 3.2.3. Consider n functions f1, . . . , fn∈ HK([a, b]). For any constants

c1, . . . , cn ∈ C we have that Pnk=1ckfk∈ HK([a, b]) and

Z [a,b] n X k=1 ckfk = n X k=1 ck Z [a,b] fk.

Proof. Let ε > 0 be arbitrary. For each k ∈ N such that k ≤ n there is a gauge δk

on [a, b] such that

X (t,[u,v])∈Pk fk(t)(v − u) − Z [a,b] fk < ε n(|ck| + 1) , (3.4)

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for any δk-fine tagged partition Pkof [a, b]. Define a gauge δ on [a, b] by δ :=Tnk=1δk

and let P be a δ-fine tagged partition of [a, b]. We have that

X (t,[u,v])∈P n X k=1 ckfk(t)(v − u) − n X k=1 ck Z [a,b] fk ≤ n X k=1 |ck| X (t,[u,v])∈P fk(t)(v − u) − Z [a,b] fk (3.4) ≤ n X k=1 |ck| ε n(|ck| + 1) < nε n = ε.

Sometimes we are interested in showing that a particular function f : [a, b] → C is integrable, while the value of R

[a,b]f is of no interest to us. Definition 3.1.6 is in

general quite weak for this purpose since typically the value ofR

[a,b]f is difficult to

find. To this end, the following theorem provides us with necessary and sufficient conditions for integrability without relying on the value of the integral of a function. The proof was inspired by the one given in [19, Theorem 14].

Theorem 3.2.4 (The Cauchy criterion). Consider a function f : [a, b] → C. We

have that f ∈ HK([a, b]) if and only if for any ε > 0 there is a gauge δ on [a, b] such that X (t,[u,v])∈P1 f (t)(v − u) − X (t,[u,v])∈P2 f (t)(v − u) < ε,

for any δ-fine tagged partitions P1 and P2 of [a, b].

Proof. We begin with the forward direction. Let ε > 0 be arbitrary. Since

f ∈ HK([a, b]) there is a gauge δ on [a, b] such that

X (t,[u,v])∈P1 f (t)(v − u) − Z [a,b] f < ε 2, (3.5) and X (t,[u,v])∈P2 f (t)(v − u) − Z [a,b] f < ε 2, (3.6)

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3.2 Basic Properties 24 X (t,[u,v])∈P1 f (t)(v − u) − X (t,[u,v])∈P2 f (t)(v − u) ≤ X (t,[u,v])∈P1 f (t)(v − u) − Z [a,b] f + X (t,[u,v])∈P2 f (t)(v − u) − Z [a,b] f (3.5) (3.6) < ε 2+ ε 2 = ε,

which completes the proof of the forward direction. We now prove the backwards direction. Let ε > 0 be arbitrary. For each k ∈ N there is a gauge γk on [a, b] such

that X (t,[u,v])∈Rk f (t)(v − u) − X (t,[u,v])∈Qk f (t)(v − u) < 1 k,

for any γk-fine tagged partitions Rk and Qk of [a, b]. Define for each n ∈ N the

gauge δn on [a, b] by δn :=Tnk=1γk. Note that for each n ∈ N and for each x ∈ [a, b]

we have that δn+1(x) ⊆ δn(x). Let (Pn)∞n=1 be a fixed sequence of tagged partitions

of [a, b] such that for each n ∈ N we have that Pn is subordinate to δn. By the

Archimedean property of R there is a number N1 ∈ N such that _N1₁ < ε. So for all

m, n ∈ N such that m, n ≥ N1 we have that

X (t,[u,v])∈Pm f (t)(v − u) − X (t,[u,v])∈Pn f (t)(v − u) < 1 N1 < ε 2, (3.7) since Pmand Pnare subordinate to δN1. This implies that the sequence of Riemann

sumsP

(t,[u,v])∈Pnf (t)(v − u)

∞

n=1is a Cauchy sequence in C, and thus a convergent

sequence in C. So there is a number L ∈ C and a number N2 ∈ N such that for

any n ∈ N such that n ≥ N2 we have that

X (t,[u,v])∈Pn f (t)(v − u) − L < ε 2. (3.8)

Let N := max{N1, N2}, define a gauge δ on [a, b] by δ := δN and let P be a δ-fine

tagged partition of [a, b]. Note that P and PN are both δN2-fine. We have that

X (t,[u,v])∈P f (t)(v − u) − L ≤ X (t,[u,v])∈P f (t)(v − u) − X (t,[u,v])∈PN f (t)(v − u) + X (t,[u,v])∈PN f (t)(v − u) − L

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(3.7) (3.8) < ε 2 + ε 2 = ε.

With that the proof is complete.

Corollary 3.2.5. Consider a function f ∈ HK([a, b]). For any [α, β] ⊆ [a, b] we

have that f ∈ HK([α, β]).4

Proof. Let ε > 0 be arbitrary. Since f ∈ HK([a, b]) we have by Theorem 3.2.4 that

there is a gauge δ on [a, b] such that

X (t,[u,v])∈P f (t)(v − u) − X (t,[u,v])∈Q f (t)(v − u) < ε, (3.9)

for any δ-fine tagged partitions P and Q of [a, b]. Let P[α,β] and Q[α,β] be δ-fine

tagged partitions of [α, β] and let R[a,α] and R[β,b] respectively be δ-fine tagged

partitions of [a, α] and [β, b]. Define P := R[a,α]∪ P[α,β]∪ R[β,b] and Q := R[a,α]∪

Q[α,β]∪ R[β,b]. We have that X (t,[u,v])∈P[α,β] f (t)(v − u) − X (t,[u,v])∈Q[α,β] f (t)(v − u) = X (t,[u,v])∈P f (t)(v − u) − X (t,[u,v])∈Q f (t)(v − u) (3.9) < ε.

Thus by Theorem 3.2.4 we have that f ∈ HK([α, β]). The proof of the following corollary closely resembles the one given in [20, Theorem 1.3.5].

Corollary 3.2.6. If the function f : [a, b] → C is continuous and [a, b] is bounded,

then f ∈ HK([a, b]).

Proof. The assertion is trivially true for the case a = b, so assume that a < b. Pick

any ε > 0. Since f is continuous there is a gauge δ on [a, b] which is such that for each x ∈ [a, b] we have that

|f (y) − f (x)| < ε

2(b − a), (3.10)

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whenever y ∈ δ(x). Let P = {(t1,1, I1,1), . . . ,(t1,p, I1,p)} and Q = {(t2,1, I2,1), . . . ,

(t2,q, I2,q)} be two fixed but arbitrary δ-fine tagged partitions of [a, b]. For each

i, j ∈ N such that i ≤ p and j ≤ q, if I1,i∩ I2,j 6= ∅ let zi,j be a fixed but arbitrary

point in I1,i∩ I2,j, otherwise let zi,j be any fixed but arbitrary point in [a, b]. We

have that p X i=1 f (t1,i)µ∗(I1,i) − q X j=1 f (t2,j)µ∗(I2,j) 2.2.11 = p X i=1 q X j=1 f (t1,i)µ∗(I1,i∩ I2,j) − q X j=1 p X i=1 f (t2,j)µ∗(I1,i∩ I2,j) ≤ p X i=1 q X j=1

|f (t1,i) − f (zi,j)|µ∗(I1,i∩ I2,j)

+ q X j=1 p X i=1 |f (t2,j) − f (zi,j)|µ∗(I1,i∩ I2,j) 2.2.11 (3.10) ≤ ε 2 + ε 2 = ε. (3.11)

Thus by (3.11) and Theorems 2.2.5 and 3.2.4 we have that f ∈ HK([a, b]).

Theorem 3.2.7. Consider a function f : [a, b] → C and let u := Ref and v :=

Imf . We have that f ∈ HK([a, b]) if and only if u, v ∈ HK([a, b]), in which case

Z [a,b] f = Z [a,b] u + i Z [a,b] v.

Proof. The assertion is an immediate consequence of Theorems 3.2.3 and 3.2.4.

Lemma 3.2.8. Consider a function f : [a, b] → C. If {[a1, b1], . . . , [an, bn]} is a

partition of [a, b] and for each k ∈ N such that k ≤ n we have that f ∈ HK([ak, bk]),

then f ∈ HK([a, b]) and

Z [a,b] f = n X k=1 Z [ak,bk] f.

Proof. For the cases where a = b or n = 1 the assertion is trivially true, so assume

that a < b and n > 1. Further, we may without a loss of generality assume that none of the n intervals are singletons and that they are indexed in ascending order. Let ε > 0 be arbitrary. For each k ∈ N such that k ≤ n there is a gauge δk on

[ak, bk] such that X (t,[u,v])∈Pk f (t)(v − u) − Z [ak,bk] f < ε n, (3.12)

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for any δk-fine tagged partition Pk of [ak, bk]. Define a gauge δ on [a, b] by δ(x) :=             

δk(x) ∪ δk+1(x) if k ∈ N is such that k < n and x = bk = ak+1,

δk(x) if k ∈ N is such that ak < x < bk,

δ1(x) if x = a1 = a,

δn(x) if x = bn= b,

and let P be an arbitrary δ-fine tagged partition of [a, b]. We have that

X (t,[u,v])∈P f (t)(v − u) − n X k=1 Z [ak,bk] f δ ≤ n X k=1 X (t,[u,v])∈Ref(P ): [u,v]⊆[ak,bk] f (t)(v − u) − Z [ak,bk] f (3.12) < n X k=1 ε n = n ε n = ε.

Theorem 3.2.9. Consider a function f : [a, b] → C and let {[a1, b1], . . . , [an, bn]}

be a partition of [a, b]. We have that f ∈ HK([a, b]) if and only if for each k ∈ N such that k ≤ n we have that f ∈ HK([ak, bk]), in which case

Z [a,b] f = n X k=1 Z [ak,bk] f.

Proof. The assertion follows from Corollary 3.2.5 and Lemma 3.2.8.

The following lemma is of great importance and can be used to prove many deep results related to the Henstock–Kurzweil integral. The most important implication of this lemma will be stated in the theorem following it. The proof closely resembles the one given in [20, Lemma 2.4.3].

Lemma 3.2.10. Consider a function f ∈ HK([a, b]). For any ε > 0 and for any

gauge δ on [a, b] which satisfies

X (t,[u,v])∈Q f (t)(v − u) − Z [a,b] f < ε, (3.13)

for any δ-fine tagged partition Q of [a, b]. Then

X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f ! ≤ ε (3.14)

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3.2 Basic Properties 28 and X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f ≤ 4ε, (3.15)

for any δ-fine tagged subpartition P of [a, b].

Proof. Let P be a fixed δ-fine tagged subpartition of [a, b]. Let {[u1, v1], . . . , [uq, vq]}

be a collection intervals such that P ∪ {[u1, v1], . . . , [uq, vq]} is a partition of [a, b].

Since f ∈ HK([a, b]) we have by Corollary 3.2.5 that for any η > 0 and for each

k ∈ N such that k ≤ q there is a gauge δk on [uk, vk] such that

X (t,[u,v])∈Pk f (t)(v − u) − Z [uk,vk] f < η q, (3.16)

for every δk-fine tagged partition Pk of [uk, vk]. For each k ∈ N such that k ≤ q

define the gauge γk on [uk, vk] by γk(x) := δ(x) ∩ δk(x), for each x ∈ [uk, vk],

and fix a tagged partition Qk of [uk, vk] which is subordinate to γk. Note that

Q := P ∪Sq

k=1Qk is a δ-fine tagged partition of [a, b]. We have that

X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f ! 3.2.9 =   X (t,[u,v])∈Q f (t)(v − u) − q X k=1 X (t,[u,v])∈Qk f (t)(v − u)   − Z [a,b] f − q X k=1 Z [uk,vk] f ! ≤ X (t,[u,v])∈Q f (t)(v − u) − Z [a,b] f + q X k=1 X (t,[u,v])∈Qk f (t)(v − u) − Z [uk,vk] f (3.13) (3.16) < ε + qη q = ε + η.

Since η was arbitrary and since P is independent of η, we have that (3.14) is true. Let x := Ref , y := Imf and note that x, y ∈ HK([a, b]) by Theorem 3.2.7. Thus by Corollary 3.2.5 we may for each function z ∈ {x, y} define the sets

Pz,+ := ( (t, [u, v]) ∈ P : z(t)(v − u) − Z [u,v] z > 0 )

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and Pz,− := ( (t, [u, v]) ∈ P : z(t)(v − u) − Z [u,v] z < 0 ) .

Since we have already deduced that (3.14) is true we have that

X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f 3.2.3 ≤ X (t,[u,v])∈Px,+ x(t)(v − u) − Z [u,v] x ! − X (t,[u,v])∈Px,− x(t)(v − u) − Z [u,v] x ! + X (t,[u,v])∈Py,+ y(t)(v − u) − Z [u,v] y ! − X (t,[u,v])∈Py,− y(t)(v − u) − Z [u,v] y ! 3.2.3 ≤ 4 X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f (3.14) ≤ 4ε,

which completes the proof of (3.15). Thus we are done.

Theorem 3.2.11 (The Saks–Henstock theorem). Consider a function f ∈ HK([a, b]).

For any ε > 0 there is a gauge δ on [a, b] such that

X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f < ε,

for any δ-fine tagged subpartition P of [a, b].

Proof. This is an immediate consequence of Lemma 3.2.10.

Definition 3.2.12. Consider a function f ∈ HK([a, b]). By Corollary 3.2.5 we can

define a function F : [a, b] → C by F (x) :=R

[a,x]f , for each x ∈ [a, b]. We will call

F the indefinite integral of f .

Theorem 3.2.13. Consider a function f ∈ HK([a, b]). We have that the indefinite

integral F of f is continuous.

Proof. The assertion is trivially true for the case a = b, so assume that a < b. Let ε > 0. By Theorem 3.2.11 there is a gauge γ1 on [a, b] such that

X (t,[u,v])∈P f (t)(v − u) − Z [u,v] f < ε 2, (3.17)

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for any γ1-fine tagged subpartition P of [a, b]. Define a gauge γ2 on [a, b] by

γ2(x) :=    γ1(x) ∩ x −_{2(1+|f (x)|)}ε , x + _{2(1+|f (x)|)}ε if |x| < ∞, γ1(x) if |x| = ∞. (3.18)

Let x ∈ [a, b] be arbitrary and pick any δ ∈ R such that

      

0 < δ < min({x − inf γ2(x), sup γ2(x) − x} \ {0}) if |x| < ∞,

δ < sup γ2(x) if x = −∞,

δ > inf γ2(x) if x = ∞.

Pick any y ∈ [a, b] such that

       |y − x| < δ if |x| < ∞, −∞ < y < δ if x = −∞, ∞ > y > δ if x = ∞.

Note that {(x, [min{x, y}, max{x, y}])} is a γ2-fine tagged subpartition of [a, b] and

by Remark 3.1.7 we may without a loss of generality assume that |x| = ∞ implies that f (x) = 0. Thus we have that

|F (y) − F (x)| 3.2.9 ≤

f (x)(max{x, y} − min{x, y}) −

Z [min{x,y},max{x,y}] f + |f (x)(y − x)| (3.17) (3.18) < ε 2+ ε 2 = ε.

Lemma 3.2.14. Consider a function f : [a, b] → C for which [a, x] ⊆ [a, b) implies

that f ∈ HK([a, x]). If limx→bR[a,x]f exists and is finite, then f ∈ HK([a, b]) and

Z [a,b] f = lim x→b Z [a,x] f.

Proof. The assertion is trivially true for the case where a = b, so assume that a < b. Let (ξk)∞k=1 be a strictly increasing sequence contained in [a, b) such that

ξ1 = a and limk→∞ξk = b. By Theorem 3.2.5 we can for each k ∈ N find a gauge

δk on [a, b] such that

X (t,[u,v])∈Pk f (t)(v − u) − Z [ξk,ξk+1] f < ε 3 · 2k+1, (3.19)

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for any δk-fine tagged partition Pk of [ξk, ξk+1]. By Theorem 3.2.13 there is a

number γ1 ∈ [a, b) such that γ ∈ (γ1, b) implies that

lim x→b Z [γ,x] f = Z [a,γ] f + lim x→b Z [γ,x] f ! − Z [a,γ] f 3.2.9 = lim x→b Z [a,x] f − Z [a,γ] f < ε 3. (3.20)

If b < ∞, then there is a γ2 ∈ [a, b) such that γ ∈ (γ2, b) implies that

b − γ < ε

3(1 + |f (b)|), (3.21)

otherwise let γ2 ∈ [a, b) be arbitrary (since the fact that b = ∞ implies that

f (∞) = 0 will be sufficient, see Remark 3.1.7). Define γ3 := max{γ1, γ2}, define a

gauge δ on [a, b] by δ(x) :=              δk(x) ∪ δk+1(x) if k ∈ N is such that x = ξk+1, δk(x) if k ∈ N is such that x ∈ (ξk, ξk+1), δ1(x) if x = ξ1 = a, (γ3, b] if x = b,

and let P be a δ-fine tagged partition of [a, b]. Note that there is a pair (b, [γ, b]) ∈

P . We have that X (t,[u,v])∈P f (t)(v − u) − lim x→b Z [a,x] f = X (t,[u,v])∈Ref(P ) f (t)(v − u) − lim x→b Z [a,x] f δ 3.2.9 ≤ X k∈N:ξk≤γ X (t,[u,v])∈Ref(P ): [u,v]⊆[ξk,min{γ,ξk+1]} f (t)(v − u) − Z [ξk,min{γ,ξk+1]} f + |f (b)|(b − γ) + lim x→b Z [γ,x] f (3.19) (3.20),(3.21) < ε 3+ ε 3+ ε 3 = ε.

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Theorem 3.2.15 (Cauchy extension part 1). Consider a function f : [a, b] → C.

We have that f ∈ HK([a, b]) if and only if [a, x] ⊆ [a, b) implies that f ∈ HK([a, x]) and limx→b

R

[a,x]f exists and is finite, in which case

Z [a,b] f = lim x→b Z [a,x] f.

Proof. The forward direction is an immediate consequence of Corollary 3.2.5 and

Theorem 3.2.13 and the backwards direction is equivalent to Lemma 3.2.14.

We have that f ∈ HK([a, b]) if and only if [x, b] ⊆ (a, b] implies that f ∈ HK([x, b]) and limx→a

R

[x,b]f exists and is finite, in which case

Z [a,b] f = lim x→a Z [x,b] f.

We will not prove this here since the proof is virtually identical to that of Theorem 3.2.15.

We have that f ∈ HK([a, b]) if and only if [α, β] ⊆ (a, b) implies that f ∈ HK([α, β]) and for any γ ∈ (a, b) we have that

lim x→a Z [x,γ] f and lim x→b Z [γ,x] f

both exist and are finite, in which case

Z [a,b] f = lim x→a Z [x,γ] f + lim x→b Z [γ,x] f.

Proof. For the case a = b the assertion is trivially true, so assume that a < b. Pick

any γ ∈ (a, b). We begin with the forward direction. Since f ∈ HK([a, b]) we have by Corollary 3.2.5 that [α, β] ⊆ (a, b) implies that f ∈ HK([α, β]). By Theorems 3.2.15 and 3.2.16 we have that

Z [a,γ] f = lim x→a Z [x,γ] f and Z [γ,b] f = lim x→b Z [γ,x] f. (3.22)

Thus by Theorem 3.2.9 we have that

Z [a,b] f = Z [a,γ] f + Z [γ,b] f (3.22)= lim x→a Z [x,γ] f + lim x→b Z [γ,x] f. (3.23)

Now we prove the backwards direction. By Theorems 3.2.15 and 3.2.16 we have that (3.22) is true and thus by Theorem 3.2.9 we have that (3.23) is true.

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Remark 3.2.18. Theorems 3.2.15, 3.2.16 and 3.2.17 show us that there is no

need to separately discuss the set of functions which are improperly integrable, since this is exactly the set of functions which are properly integrable. This is one of several drawbacks of the Riemann integral, and even the Lebesgue integral, which the Henstock–Kurzweil integral is able to remedy. Although depending on the context this might not necessarily be considered a flaw of the Lebesgue inte-gral since it inhibits bad functions from being Lebesgue integrable, which in turn makes the space of Lebesgue integrable function possess a lot of nice properties. It can however be shown that in L1_{-theory (which we will briefly cover in Section}

3.4), under quite general circumstances the Henstock–Kurzweil integral and the Lebesgue integral are equivalent, see the introduction of [21]. Therefore a sub-stantial amount of the theory of the Lebesgue integral can be developed via the Henstock–Kurzweil integral.

3.3 Further Properties and Basic Measure Theory

In this section we will continue developing the properties of the Henstock–Kurzweil integral. Several basic theorems will be strengthened by allowing certain criteria in their respective suppositions to fail on a set of Lebesgue outer measure 0. As a consequence of the results developed in this section, we will in subsequent sections have tools at our disposal which will allow us to prove deeper theorems. For example, in Section 4.1 we will be able to quite elegantly characterize the set of indefinite integrals and in Section 4.3 we will be able to prove a stronger version of the monotone convergence theorem (where the condition of monotonicity is relaxed to hold almost everywhere) and a stronger version of the dominated convergence theorem (where the condition of dominance is relaxed to hold almost everywhere). The proof of the following lemma was inspired by the one given in [19, Theorem 10].

Lemma 3.3.1. Consider a function f : [a, b] → C. If f = 0 almost everywhere5

on (a, b), then [α, β] ⊆ [a, b] implies that f ∈ HK([α, β]) and

Z

[α,β]

f = 0.

Proof. The assertion is trivially true for the case where α = β, so assume that α < β. Let ε > 0 be arbitrary. Define the set

Ω := {x ∈ (α, β) : f (x) 6= 0},

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3.3 Further Properties and Basic Measure Theory 34

and for each i ∈ N define the set

Ωi := {x ∈ Ω : i − 1 < |f (x)| ≤ i}. (3.24)

Since µ∗(Ω) = 0 we have that µ∗(Ωi) = 0 for each i ∈ N. So for each i ∈ N there

is a collection of intervals {`i,j : j ∈ N} ⊂ O(R) such that

Ωi ⊆ ∞ [ j=1 `i,j and ∞ X j=1 L(`i,j) < ε i2i+1. (3.25)

Define recursively the sets

ωi,j :=        Ωi∩ `i,j if i ∈ N and j = 1, (Ωi∩ `i,j) \ j−1 [ k=1

ωi,k if i, j ∈ N are such that j ≥ 2,

(3.26)

and note that for each choice of i, j, k ∈ N we have that ωi,j ⊆ `i,j, ωi,j ∩ ωi,k 6=

∅ ⇔ j = k and Ωi =S∞n=1ωi,n. There is a gauge γ on [α, β] such that

[α, v∗] ⊆ γ(α) ∧ [u∗, β] ⊆ γ(β) ⇒ |f (α)|(v∗− α) + |f (β)|(β − u∗) < ε 2. 6 _(3.27) Define a gauge δ on [α, β] by δ(x) :=             

`i,j ∩ (α, β) if i, j ∈ N are such that x ∈ ωi,j,

(α, β) if x ∈ (α, β) \ Ω, [α, β) ∩ γ(x) if x = α,

(α, β] ∩ γ(x) if x = β,

and let P be a δ-fine tagged partition of [α, β]. Note that by the definition of δ there are two pairs (α, [α, v∗]), (β, [u∗, β]) ∈ P . We have that

X (t,[u,v])∈P f (t)(v − u) δ (3.24) ≤ |f (α)|(v∗− α) + |f (β)|(β − u∗) + ∞ X i=1 i X (t,[u,v])∈P : t∈Ωi (v − u) (3.26) (3.27) < ε 2 + ∞ X i=1 i ∞ X j=1 X (t,[u,v])∈P : t∈ωi,j (v − u)