Integrals on R:

An Exposition

From Darboux to Henstock-Kurzweil

ZWEISTEIN

diagrams by lambda

Desu-Cartes

Contents

1 Introduction 3

1.1 Scope of the lecture . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Darboux-Riemann Integral 4

2.1 Prelude: Darboux Sums . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Waltz: The Darboux Integral . . . . . . . . . . . . . . . . . . . . 5

2.3 Duet: The Riemann Integral . . . . . . . . . . . . . . . . . . . . 6

2.4 Aria: Properties of the Darboux integral . . . . . . . . . . . . . . 7

3 The Riemann-Stieltjes Integrals 9

3.1 Intermezzo: The Riemann-Stieltjes Integral . . . . . . . . . . . . 9

3.2

Etude: Properties of the Riemann-Stieltjes integral . . . . . . . . 10

4 The Henstock-Kurzweil Integral 11

4.1 Cavatina: Gauge functions . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Sonata: The Gauge integral . . . . . . . . . . . . . . . . . . . . . 12

4.3 Concerto: Properties of the Gauge Integral . . . . . . . . . . . . 13

4.4 Finale: The Fundamental Theorem of Calculus . . . . . . . . . . 15

A Further Reading 16

1 Introduction

1.1 Scope of the lecture

In this lecture, we are going to discuss how to deﬁne integration on R. Any-

one that has taken a course in calculus knows that applications of integration

abound. In this lecture, we will not so much be interested in how one can apply

the deﬁnite integral, but rather, how one should go about deﬁning it in the ﬁrst

place. As is common in math, multiple deﬁnitions can capture the essence of

the object at hand, and the deﬁnite integral is no diﬀerent in that regard. From

Archimedes’ method of exhaustion, to Lebesgue’s measure theory, integration

has been of interest for thousands of years now, in some form or another. Be-

cause the topic is such a vast one, we will put two restrictions on this lecture.

First of all, this lecture should be understandable to anyone that has taken a

basic course in analysis at most. Eﬀectively, this means that we are not going to

talk about Lebesgue’s elegant deﬁnition of the integral. There are many great

books on the subject and we refer the reader to any such book of his liking. As

such, we will also restrict ourself to R. Integration can be deﬁned on many other

spaces (smooth manifolds for instance, which you can learn about in Killing-

form’s cotangent bundle lecture) but it would require us to develop too much

machinery.

Secondly, we will only discuss ”modern” integrals. Concretely, this means that

we will not talk about Archimedes’ method of exhaustion and will begin directly

with the deﬁnition of integral most people see in their ﬁrst course on analysis.

Our goal is to get to the beautiful theory of the Henstock-Kurzweil integral, and

to see a few of its most prominent results.

But what exactly are we trying to do? In order to understand our integral, we

need to make precise our goal: We wish to create a method that lets us ﬁnd

the area between the graph of a function f(x) and the x-axis. Since we want

to measure an area, it is only natural to ask that the function be bounded. We

will restrict integration to compact intervals on the reals. Once this has been

done, generalizations to other intervals or all of R are straightforward.

We also want our integral to respect a few basic properties. For instance, since

the integral represents an area, if f and g are integrable functions, then f + g

should be integrable, and its integral should be the sum of the integral of f and

g respectively. We now turn to the construction of such an integral, namely the

Darboux integral.

2 The Darboux-Riemann Integral

2.1 Prelude: Darboux Sums

The idea behind integration is to ”cut up” the function in a certain way, so that

we can approximate the area with measurable objects (such as rectangles) and

then take this process to inﬁnity. This idea will become clearer once we actually

get to the integral. To this purpose, we introduce some deﬁnitions.

Deﬁnition 2.1 (Partitions). Let I be the compact interval [a, b]. A partition

of I is a ﬁnite ordered set P := {a < x

< · · · < x

n−1

< b}.

We can then look at the widths x

− x

i−1

and multiply them by a certain

number to get a rectangle. This is what we are going to do to get a ﬁrst

approximation of our area: make a ﬁnite sum of rectangles, with height a certain

value of our function on this interval.

Deﬁnition 2.2. Let P be a partition of [a, b]. For each subinterval [x

i−1

, x

deﬁne m

:= inf{f (x) | x ∈ [x

i−1

, x

]}. Similarly, we deﬁne M

:= sup{f (x) |

x ∈ [x

i−1

, x

]}. m

(resp. M

) is simply the inﬁmum (resp. supremum) of the

function restricted to the interval [x

i−1

, x

We are now ready to deﬁne Darboux sums, which we will need in order to

deﬁne the Darboux integral.

Deﬁnition 2.3 (Darboux sums). Let P be a partition of [a, b]. Let M

and m

be deﬁned as above. The upper Darboux sum (resp. lower Darboux sum)

f,P

i=1

− x

i−1

) resp. L

f,P

i=1

− x

i−1

)

Intuitively, we understand that U

f,P

overestimates the area, whereas L

f,P

underestimates the area as one can see in ﬁgure 1. As such it is obvious that

f,P

≤ U

f,P

. Notice that these sums depend on the partition. The idea behind

the Darboux integral will be to ﬁnd the ”best” partition to make U

f,P

and L

f,P

closer and closer.

Figure 1: Comparison of Darboux sums

2.2 Waltz: The Darboux Integral

We have seen that the Darboux sums each miss the mark on the exact area

we’re looking for. U

f,P

is slightly more than desired, while L

f,P

is the opposite.

We do have one saving grace: We can vary the sums by choosing a particular

partition. One can picture the possible lower and upper Darboux sums as a

set indexed by their respective partition. By the axiom of least upper bounds,

we know such sets have suprema and minima. We can thus minimize (resp.

maximize) M

(resp. m

). This leads us to our next deﬁnition.

Deﬁnition 2.4 (Lower and Upper Integrals). The upper Darboux Integral

is deﬁned to be U

:= inf{U

f,P

| P is a partition of [a, b]}. Similarly, the

Lower Darboux Integral is L

:= sup{L

f,P

| P is a partition of [a, b]}. We

sometimes denote the upper integral as

f(x) dx and the lower integral as

f(x) dx.

Taking the supremum and the inﬁmum there allow us to minimize the error in

the area measurement. It is but a natural step to wonder what the integral itself

is. Well, since L

f,P

≤ U

f,P

for any P . In particular, L

≤ U

. Geometrically,

this is obvious because L

is just ”under” the area of f(x), whereas U

just ”above” the area of f(x). Equality thus means that they meet, which

geometrically implies that there is no error anymore. This is exactly what we

were looking for! One can see this more clearly with a geometric picture of the

process, which is given in ﬁgure 2.

Deﬁnition 2.5 (Darboux Integral). Let f : [a, b] → R be a bounded function.

If U

= L

, we say that f is Darboux integrable. We denote this common value

f(x) dx, the Darboux integral.

→

Figure 2: The Darboux Integral

2.3 Duet: The Riemann Integral

To deﬁne the Riemann integral, we will need to alter a bit our deﬁnition of a

partition. Instead of looking at suprema and inﬁma, we will tag every partition

with a point in the interval.

Deﬁnition 2.6 (Riemann Sums). Let P be a partition of a bounded f : [a, b].

For each interval [x

i−1

, x

] we take an arbitrary point c

. The Riemann sum

is the sum S(P, f) =

i=1

f(c

)(x

− x

i−1

). The value max(x

− x

i−1

) is called

the mesh of the partition.

Eﬀectively what we are doing here is simply summing rectangles of width

− x

i−1

and of height f(c

). As the partition gets thinner, the mesh tends to

zero, as can be seen in ﬁgure 3. This leads us to the proper deﬁnition of the

Riemann integral.

Deﬁnition 2.7 (Riemann Integral). Let f : [a, b] be a bounded function. We

say that A is the Riemann integral of f if for all  > 0, there exists δ > 0

such that for any tagged partition P

whose mesh is less than δ, we have

|S(P, f) − A| < .

One can see at a glance that Darboux’s deﬁnition is easier to digest, and

work with. Luckily for us, both integrals are equivalent! This is why we started

with the Darboux integral. It generalizes more easily than Riemann’s deﬁnition

while still being equivalent. This is contained in the following theorem.

Theorem 2.1 (Equivalence of the integrals). Let f be a bounded function de-

ﬁned on a compact interval. Then, f is Darboux-integrable if and only if it is

Riemann-integrable. Furthermore, if the integrals exist, they are the same.

Remark. Since the two integrals coincide, we will just call it the integral

f(x) dx.

→

Figure 3: The Riemann Integral

2.4 Aria: Properties of the Darboux integral

In this chapter, we list without proofs a few properties that our integral enjoys.

You can try to prove a few of these theorems or ﬁnd a proof in any standard

book on the topic. These properties all come from our rigorous deﬁnition of the

integral.

Theorem 2.2 (Linear properties). Let D [a, b] be the set of all integrable func-

tions. Then D [a, b] is a vector space, that is

1. f, g ∈ D [a, b] implies that f+g ∈ D [a, b]. Furthermore,

[f(x) + g(x)] dx =

f(x) dx +

g(x) dx

2. f ∈ D [a, b] implies that λf ∈ D [a, b], where λ ∈ R. Furthermore

λf(x) dx =

f(x) dx.

This theorem enables us to put a nice structure on the set of integrable func-

tions, namely that of a vector space. This makes sense geometrically. Another

interesting property is that of the middle point.

Lemma 2.1 (Middle point property). Let f ∈ D [a, b]. If c ∈ [a, b]. then

f(x) dx =

f(x) dx +

f(x) dx.

Eﬀectively, this means that you cannot get more area by adding parts of the

area together. We now take care of a few special cases with two deﬁnitions.

Deﬁnition 2.8 (Special cases). Let f ∈ D [a, b]. We deﬁne

f(x) dx = 0 and

f(x) dx = −

f(x) dx.

It is important to realize that these are deﬁnitions, this just ensures we don’t

have degenerate cases down the line. They’re also motivated geometrically, since

the area of a point c

times f(c

) is a rectangle of width 0, so the area of the

”rectangle” is 0. The next theorems will help us pin down exactly what functions

can be integrated with the Darboux-Riemann integral.

Theorem 2.3 (Continuous functions). Let C [a, b] be the vector space of continu-

ous functions on [a, b]. Then C [a, b] ⊂ D [a, b]. In other words, every continuous

function is integrable.

This is a major achievement. Since continuous functions can be manipulated

with a very nice kind of arithmetic, we now realize just how big D [a, b] is.

Consequently, arbitrary compositions of continuous functions are still integrable.

Thus a function like f(x) = e

−x

is integrable. This is important: the theorem

is a statement about existence, it does not provide us with a way to calculate

integrals. This is the content of the two following theorems. These are the

cornerstones of calculus.

Theorem 2.4 (Fundamental Theorem of Calculus, Part. I). Let f be a contin-

uous function deﬁned on [a, b]. Let F be the function deﬁned, for all x in [a, b],

f(t) dt. Then F is uniformly continuous on [a, b], diﬀerentiable on (a, b)

and F

(x) = f(x) for all x ∈ [a, b].

Theorem 2.5 (Fundamental Theorem of Calculus, Part. II). Let f be a func-

tion integrable on [a, b] and let F be an antiderivative of f on [a, b]. Then,

f(x) dx = F (b) − F (a).

It is this second part that is widely used in applications. We now understand

why this method fails for f (x) = e

−x

. One can show that this function has

no antiderivative in terms of elementary functions, hence why we use numerical

methods to approximate it.

We will end this chapter with a criterion for integrability that will tell us exactly

which functions are Darboux-integrable. To do that, we ﬁrst need a deﬁnition.

Deﬁnition 2.9 (Sets of measure zero). Let N be a subset of the real numbers.

We say that N has measure zero if it can be covered by a countable union

of intervals whose lengths are arbitrarily small. More formally, for every  > 0,

there exists a collection {U

}

∞

n=1

such that

N ⊂ ∪

∞

n=1

and

∞

n=1

| < 

This leads us to Lebesgue’s wonderful classiﬁcation theorem.

Theorem 2.6. Suppose f is a bounded function deﬁned on [a, b]. Then, f is

integrable if and only if it is continuous almost everywhere, that is, the set

of discontinuities of f is a set of measure zero. Thus, any function with at most

a countably inﬁnite number of discontinuities is integrable.

3 The Riemann-Stieltjes Integrals

3.1 Intermezzo: The Riemann-Stieltjes Integral

After having deﬁned the Darboux-Riemann integral, we have seen a few of its

properties. It is now time to generalize this integral to get a more general

integral. The idea here will be to integrate f with respect to another function

g. Since the Riemann-Stieltjes integral is a generalization, we can characterize

it using either modiﬁed Darboux sums or modiﬁed Riemann sums. We will go

through both deﬁnitions.

Deﬁnition 3.1 (The Riemann-Stieltjes integral: Riemann sums). Let P

be a

tagged partition of [a, b] and let α(x) be an increasing function from [a, b] → R.

Let S(P, f, α) denote the modiﬁed Riemann sum of f with respect to g:

S(P, f, α) =

i=1

f(c

) [α(x

) − α(x

i−1

)]

The Riemann-Stieltjes integral A =

f(x) dα(x) then exists if for every

 > 0, there exists δ > 0 such that for every partition P whose mesh is less than

δ and for every c

∈ [x

i−1

, x

] we have that

|S(P, f, α) − A| < .

While this deﬁnition is ﬁne on its own, it is standard to generalize this

integral by looking at reﬁnement of our initial partition. A reﬁnement breaks

down intervals and adds tags where needed.

Deﬁnition 3.2 (The Generalized Riemann-Stieltjes integral: Riemann sums).

The Generalized Riemann-Stieltjes integral is a number A such that for

every  > 0, there exists a partition P



such that for every partition P that

reﬁnes P



, we have that |S(P, f, α) − A| <  for every c

∈ [x

, x

i−1

] .

We now give a more palatable deﬁnition using a generalization of the Dar-

boux integral.

Deﬁnition 3.3 (The Generalized Riemann-Stieltjes integral: Darboux sums).

Let α(x) be an increasing function from [a, b] → R. Let f : [a, b] be a bounded

function. The modiﬁed upper Darboux sum (resp. the modiﬁed lower

Darboux sum) is the sum

i=1

[α(x

) − α(x

i−1

)] resp.

i=1

[α(x

) − α(x

i−1

)] ,

denoted U

[f,P,α]

and L

[f,P,α]

respectively. Then, the generalized Riemann-

Stieltjes integral A exists if and only if, for every  > 0, there exists a partition

P such that

[P,f,α]

− L

[P,f,α]

< .

3.2

Etude: Properties of the Riemann-Stieltjes integral

We have successfully generalized the Riemann integral to a wider class of in-

tegral. This must mean that the Riemann integral is a special case of the

Riemann-Stieltjes integral. The next lemma tells us precisely what the Rie-

mann integral is with respect to the more general integral.

Lemma 3.1 (Special case: The Riemann integral). Let f be Riemann-Stieltjes

integrable on [a, b]. If we integrate f with respect to the function α(x) :=



[a,b]

= x



[a,b]

we get that:

f(x) dα(x) =

f(x) dx

where

f(x) dx is just the Darboux-Riemann integral of f and x



[a,b]

This is rather easy to see directly from the deﬁnition, because [α(x

) − α(x

i−1

)] =

[(x

− x

i−1

)] and thus S(P, f, α) = S(P, f) in the notation of Riemann sums.

We now state a neat theorem regarding what happens to the integral when we

integrate with respect to a continuously diﬀerentiable α.

Theorem 3.1 (Continuously diﬀerentiable α). Let f be a Riemann-Stieltjes

integrable function, and let α(x) be a continuously diﬀerentiable function on R.

We have the following identity

f(x) dg(x) =

f(x)α

(x) dx,

where the right-hand side is simply the Riemann integral of the function.

We now take a look at the famous integration by parts formula from calculus.

A similar formula exists for Riemann-Stieltjes integrals.

Proposition 3.1. Let f be Riemann-Stieltjes integrable. Then we have the

following integration by parts formula:

f(x) dα(x) = f(b)α(b) − f (a)α(a) −

α(x) df(x).

We conclude this section with an existence theorem that will guarantee us

the existence of the integral under certain conditions.

Theorem 3.2 (Existence of the integral). Let f and α be functions of bounded

variation, that is, they are the diﬀerence of two monotone functions. Then the

integral

f(x) dα(x) exists.

4 The Henstock-Kurzweil Integral

4.1 Cavatina: Gauge functions

In this section, we turn our attention to one of the most powerful integral deﬁned

on R, the Henstock-Kurzweil integral, also called the gauge integral. when we

discussed the Darboux-Riemann integral, we used ﬁxed intervals [x

i−1

, x

] which

do not take into account how the function behaves at a particular point. A useful

example is the function f(x) = 1/x sin(1/x

) (see ﬁgure 4. We can see that we

would need a ﬁner partition around x = 0 and a coarser one around x = 2.

We have already seen one way to circumvent the problem, namely that of the

Riemann-Stieltjes integral. The gauge integral will try to give a satisfactory

way to deal with the problem of varying the ”length” of the intervals. First, we

rewrite the deﬁnition of the Riemann integral in a way that lets us generalize it

more easily.

Deﬁnition 4.1 (The Riemann integral, revisited). A function f : [a, b] → R on

[a, b] is Riemann-integrable if there exists an A ∈ R such that for every  > 0,

there exists a δ > 0 such that if

P = {(c

, [x

i−1

, x

]) : 1 ≤ i ≤ n}

is any tagged partition of [a, b] satisfying

i−1

, x

] ⊂ (c

− δ, c

+ δ) , ∀i : 1 ≤ i ≤ n,

then

|S(P, f) − A| < .

This is equivalent to the deﬁnition given in section 2.3.

We will now change our notation a bit, we deﬁne γ(t) = (t − δ, t + δ). This

is just notation meant to simplify the theory. Our condition in the deﬁnition of

the Riemann integral is then:

i−1

, x

] ⊂ γ(c

We remark that in the case of the Riemann integral, the intervals are of length

2δ. The generalization is then to allow δ(t) : t ∈ [a, b] to be a positive function

on its own. That is to say, γ(t) = (t − δ(t), t + δ(t)). More formally, we are led

to the following deﬁnition.

Deﬁnition 4.2 (Gauges). Let E be a subset of R. A gauge on E is a function

γ that associates with each point t ∈ E an open interval γ(t) that contains t.

Any constant function δ = C deﬁnes a gauge on E. We now introduce the

ﬁnal deﬁnition we need in order to deﬁne the Henstock-Kurzweil integral.

Deﬁnition 4.3 (γ-ﬁne). If γ is a gauge on [a, b] and P = {(c

, [x

i−1

, x

]) : 1 ≤ i ≤ n}

is a partition, then we say that P is γ-ﬁne if [x

, x

] ⊂ γ(c

) for i : 1 ≤ i ≤ n.

4.2 Sonata: The Gauge integral

Before deﬁning the integral itself, we will state an existence theorem for γ-ﬁne

partitions.

Theorem 4.1 (Existence of γ-ﬁne partitions). Let γ be a gauge on [a, b]. Then

there exists a γ-ﬁne tagged partition of [a, b].

We can ﬁnally move on to the most important deﬁnition in this lecture.

Deﬁnition 4.4 (The Gauge Integral). The function f : [a, b] → R is gauge

integrable on [a, b] if there exists A ∈ R with the property that for every  > 0,

there exists a gauge γ on [a, b] such that for every γ-ﬁne tagged partition of

[a, b], we have that

|S(P, f) − A| < .

This deﬁnition precisely captures what we mean by ”varying” the length of

the intervals in the partition P depending on the function itself. We now drop

the adjective ”gauge” and simply refer to the integral as ”the” integral. We will

henceforth denote the set of gauge integrable functions as G [a, b]. One might

wonder, even though the integral exists, is it unique?

Theorem 4.2 (Unicity of the integral). A function f : [a, b] → R can have at

most one integral.

This integral can actually handle more functions than the standard Riemann

integral. For those of you that have taken a course in measure theory, the gauge

integral is more general than the Lebesgue integral too. For instance, the gauge

integral can handle the function f(x) = 1/x sin(1/x

) from earlier. Notice that

neither Riemann, nor Lebesgue can integrate it.

-2 -1.5 -1 -0.5 0.5 1 1.5 2

-1.5

-1

-0.5

0.5

1.5

Figure 4: f(x) = 1/x sin(1/x

)

4.3 Concerto: Properties of the Gauge Integral

This section will discuss the various properties of the gauge integral. We start

with an important theorem.

Theorem 4.3 (Hake’s theorem). We have that

f(x) dx = lim

c→b

−

f(x) dx

whenever either side exists, and likewise for the lower bound of integration.

In practice, this means that ”improper” integrals whose bounds are not

inﬁnite are proper integrals too. One can still consider improper integrals of the

form

∞

f(x) dx however.

Theorem 4.4 (Integration by Substitution). Let f : [a, b] → R and φ : [α, β] →

[a, b] be diﬀerentiable functions. Then

φ(β)

φ(α)

(x) =

◦ φ)φ

(φ(x)) φ

(x).

This is just integration by substitution as seen in any calculus course.

Proposition 4.1 (Positivity). If f ∈ G [a, b] and f(t) ≥ 0 for all t ∈ [a, b], then

f(t) dt ≥ 0.

Using this result, we can show two interesting corollaries.

Corollary 4.1 (Order). If f, g ∈ G [a, b] and f (t) ≤ g(t) for all t ∈ [a, b], then

f(t) dt ≤

g(t) dt.

Corollary 4.2 (Absolute integrability). If f : [a, b] → R is absolutely integrable

over [a, b], then we have



f(x) dx



≤

|f(x)| dx.

Using the above results, we get the following well-known theorem.

Theorem 4.5 (Integration by parts). Let f, g : [a, b] be diﬀerentiable functions

on [a, b]. Then f

g is integrable over [a, b] if and only if fg

is. Furthermore,

we have that:

(x)g(x) dx = f (b)g(b) − f (a)g(a) −

f(x)g

(x) dx.

Theorem 4.6 (Subdivision of the interval of integration). Let f : [a, b] → R and

let P be a partition of [a, b]. If f is integrable on I = [x

i−1

, x

] for i = 1, 2, . . . , n,

then f is integrable on [a, b] and

f(x) dx =

i=1

i−1

f(x) dx =

i=1

f(x) dx.

The next theorem generalizes the Cauchy criterion common to sequences

and other limits to the gauge integral.

Theorem 4.7 (Cauchy Criterion for integrability). Let f : [a, b] → R. Then

f ∈ G [a, b] if and only if, for every  > 0, there is a gauge γ on [a, b] such that

if P

and P

are γ-ﬁne tagged partitions, we have:

|S(P

, f) − S(P

, f)| < .

We have seen that we can ”subdivize” the interval of integration. We now

see what happens with subsets of the interval of integration.

Theorem 4.8 (Subintervals of integration). Let f ∈ G [a, b]. Let I be a closed

subinterval of [a, b]. Then f ∈ G [I] .

Notice how we require the subinterval to be closed. This comes from the

fact that a closed subinterval of a compact interval is still compact.

Theorem 4.9 (Continuous means integrable). Let f be continuous. Then f is

integrable. In other words, C [a, b] ⊂ G [a, b].

This tells us that all continuous functions are integrable, which is nothing

new since the Darboux integral too had this property.

Theorem 4.10 (Mean value theorem: Integral form). Let f ∈ C [a, b]. Then,

there is a t ∈ [a, b] such that

(b − a) f(t) =

f(t) dt

This is simply the mean value theorem from calculus, but taken in its integral

form. We end this section with a theorem that will set the stage for the next

section.

Theorem 4.11. If f ∈ G [a, b] and F is the indeﬁnite integral of f , then F is

continuous on [a, b].

4.4 Finale: The Fundamental Theorem of Calculus

We conclude this lecture with three statements. We recommend that the reader

compare these with the theorems in the chapter on Darboux integration. These

are beautiful theorems on their own, so we will quiet ourselves and just state

the theorems in all their grandeur.

Theorem 4.12 (First Fundamental Theorem of Calculus). Let f : [a, b] → R.

If f and |f| are integrable over [a, b], continuous at x ∈ [a, b] and if F is the

indeﬁnite integral of f, then F is diﬀerentiable at x and its derivative is given

by F

(x) = f(x).

Theorem 4.13 (Second Fundamental Theorem of Calculus). If F : [a, b] → R

is diﬀerentiable on [a, b], then F

is integrable on [a, b] and

(x) dx = F (b) − F (a).

Theorem 4.14 (Integrals and Derivatives). Let f be a diﬀerentiable function.

Then, f is, up to a constant, the integral of its derivative.

A Further Reading

• Stephen Abbott: Understanding Analysis [The Darboux integral]

• Terence Tao: Analysis I [The Riemann-Stieltjes integral]

• Robert G. Bartle: A Modern Theory of Integration [The Gauge integral]

• John DePree, Charles Swartz: Introduction to Real Analysis [The Gauge

Integral]