11th ISC 2024 (Universitas Advent Indonesia, Indonesia)
“Research and Education Sustainability: Unlocking Opportunities in Shaping Today's
Generation Decision Making and Building Connections” October 22-23, 2024

Primitives of Essentially Bounded Henstock-Kurzweil
Integrable Functions
Abraham P. Racca1*, Precious R. Tayaben2, Peter V. Datoy3
Adventist University of the Philippines
apracca@aup.edu.ph

ABSTRACT
A full descriptive characterization of essentially bounded Henstock-Kurzweil integrable
function is given. More precisely, an essentially bounded function on , is HenstockKurzweil integrable if and only if there exists a function F satisfying the Lipschitz condition
on [a,b] with ′
almost everywhere. Some implications were given, including
integration by parts, substitution formula and a convergence theorem. These known results
were presented and proved using the existing results in the Henstock-Kurzweil integration.
Keywords: Henstock-Kurzweil integral, essentially bounded functions.

INTRODUCTION AND LITERATURE REVIEW
A sequential definition for the Henstock-Kurzweil (HK) integral is possible as pointed
out in (Lee, P. Y., 2007). A real valued function f on a compact interval [a,b] in R is HKintegrable to a real number A, written
, if there is a sequence of positive
functions ,
1,2,3, …, such that for every -fine division
, , , we have
!∑

#

→

& → ∞.

An HK-integrable function on [a,b] is also (HK)-integrable to any subinterval of
)
[a,b]. Hence we can define a function F such that F(a) = 0 and
for a ∈ (a,b].
This function is called the primitive of F. In Theorem 5.7 of (Lee, P. Y., 1989) and in Theorem
5.9 of (Bartle, R. G., 2001), it was shown that if F is primitive of in [a,b] then ′
almost everywhere on , .
There are several full descriptive characterization for the HK-integral. Among them is
that a function f on [a,b] is HK-integrable if there exists a function F satisfying the strong Lusin
condition with ′
almost everywhere. For the discussion please see (Lee, P. Y. &
Vyborny, R., 2000). It was then shown in (Bongiorno, B & Piazza, L., 1996) that a function
satisfying the strong Lusin condition is differentiable almost everywhere. Hence a function F
is the primitive of an HK-integrable function if and only if F satisfies the strong Lusin
condition.
For easy reference, we state the following results.
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Theorem 1.1. Let a function
almost everywhere.

on [a,b] be HK-integrable with primitive F. Then ′

Theorem 1.2. A function f on [a,b] is HK-integrable if and only if there exists a function
F satisfying the the strong Lusin condition with ′
#
.

almost everywhere. In this case

In (Calunod, I. D., and Garces, I. J. L., 2021) and (Thompson, B. S., 2014),
independently, a characterization of the primitives of Riemann integrable functions was given.
It was shown that a function F on [a,b] is the primitive of a Riemann integrable function on
[a,b] if and only if F satisfies the Lipschitz condition and is strongly differentiable almost
everywhere on [a,b]. We say that a function F on [a,b] satisfies the Lipschitz condition if there
exists a nonnegative number M such that for any subinterval [u,v] of [a,b] we have |
#
| , - # . A function satisfying the Lipschitz condition also satisfies the strong Lusin
condition. Then the following statement holds.
Theorem 1.3. If F satisfies the Lipschitz condition on [a,b] then F is differentiable almost
everywhere.
Recently in (Indrati, Ch. R. and Aryati, L., 2016), a new integral, the Countably
Lipschitz or CL-integral, was defined. A function f is CL-integrable on [a,b] if there exists a
function F satisfying the countably Lipschitz condition on [a,b] (F ∈ CLC([a,b])) such that
′
almost everywhere. A function F ∈ CLC([a,b]) if there exists a countable
collection {Xi} of subsets of [a,b] and a countable collection {Mi} of nonnegative numbers
#
| , - # . It was shown that
such that for any i, for any u,v ∈ Xi, we have|
every CL-integrable function is also HK-integrable.

RESULTS AND DISCUSSION
We start this section by stating the definition of the main subject, that is, the set of
essentially bounded functions. Considering the proof that Riemann primitives satisfy the
Lipschitz condition suggests intuitively that the primitives of other bounded integrable
functions would also satisfy the Lipschitz condition. Recall that a real valued function f on [a,b]
is said to be essentially bounded if there exists a bounded function g such that f = g almost
everywhere on [a,b]. We will denote B([a,b]) as the set of all essentially bounded function on
[a,b]. The following result immediately follows from the definition of B([a,b]).
Proposition 2.1. A real valued function f ∈ B([a,b]) if and only if there exists a subset Φf of
[a,b] with full measure such that f is bounded on Φf.
For any subset S ⊂ [a,b] and for a function f which is bounded on S, we set /0 1
23
∶
∈ 1 and -0 1
& 5
∶
∈ 1 .
Theorem 2.2. Let f ∈ B([a,b]). If f is HK-integrable with primitive F then for any subinterval
[u,v] of [a,b], we have
#
/0 , ∩ Φ0 ! ,
, -0 , ∩ Φ0 !.
#

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Proof. Since F is the HK-primitive of and [u,v] ⊂ [a,b] then there is a sequence of positive
functions δj,j = 1,2,3,... such that for every δj-fine division Dj = {([u,v],ξ)}, we have
#

lim

!∑

→;

#

.

Therefore, since HK-integral is invariant under changing of values for measure zero
sets, we have
/0 , ∩ Φ0
#
,
#
, -0 , ∩ Φ0
# .
Corollary 2.3. Let f ∈ B([a,b]). If f is HK-integrable with primitive F, then F satisfies the
Lipschitz condition.
Proof. If ∈ < ,
and is HK-integrable with primitive F then there exists a
integrable bounded function g having F also as primitive. Then for any subinterval [u,v] of
[a,b], we have
/=

,

, /=

,

,

#
#

, -=

,

, -=

,

.

If M is the higher number between |mg([a,b])| and |Mg([a,b])|, then for any subinterval [u,v] of
[a,b], we have |F(v) − F(u)| ≤ M(v − u).
□
Corollary 2.4. If F is a differentiable function on [a,b] with bounded derivative, then F is
Lipschitz.
Corollary 2.5. Let f ∈ B([a,b]). If f is HK-integrable with primitive F, then ′
equal to
whenever f is continuous at x.

exists and is

Proof. Let f ∈ B([a,b]) be HK-integrable with primitive F and x0 ∈ [a,b] such that f is
continuous at x0. Then lim@
A and that for x close enough to x0, we have
)→)?

/0
Taking the limit as

A,

,

→

A , we get

A

, lim

)→)?@

#
#

A

#
#

A

A

, -0

A,

A

A .

,

.

Using similar argument, one can show that

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A

Therefore

C

A

, lim

)→)?B

#
#

A

A

,

A .

A .

Theorem 2.6. If F satisfies the Lipschitz condition then there exists a bounded Henstock
integrable function f with F as primitive.
Proof. If F satisfies the Lipschitz condition then ′
exists almost everywhere on [a,b] and
for some M ≥ 0, ′
, - whenever ′
exists. Define a function f on [a,b] such that
′
when ′
exists and
0, otherwise. Then f is a bounded HK-integrable
with F as primitive.
Theorem 2.7. Let f be HK-integrable function on [a,b] with primitive F. Then f ∈ B([a,b]) if
and only if F satisfies the Lipschitz condition.
Proof. Let f be HK-integrable with primitive F. If f ∈ B([a,b]) then by Corollary 2.3, F
satisfies the Lipschitz Condition.
For the converse, let be HK-integrable with primitive F satisfying the Lipschitz
condition. Then F is differentiable almost everywhere. Let X ⊂ [a,b] such that ′
.
Also by F being Lipschitz, there exists a nonnegrative number M such that for any subinterval
[u,v] of [a,b], |
#
|,- #
. Hence for any ∈ E,
′
, -.
Therefore is essentially bounded.
□
Lemma 2.8. If a function F on [a,b] satisfies the Lipschitz condition then it satisfies the
strong Lusin condition.
Theorem 2.9. Let f ∈ B([a,b]).
Then f is HK-integrable if and only if there exists a
function F satisfying the Lipschitz condition such that ′
almost everywhere. In
this case,
#
.
Proof. Let f ∈ B([a,b]). If f is HK-integrable then by Theorem 2.3, F satisfies the Lipschitz
condition. Furthermore, by Theorem 1.1, C
almost everywhere.
For the converse, suppose there exists a function F satissfying the Lipschitz
condition such that C
almost everywhere. By Lemma 2.8, F satisfies the strong
Lusin condition.
Therefore, by Theorem 1.2, f is HK-integrable and
#
.
We also have the following as auxiliary results.
Theorem 3.1. (A verion of Cauchy extension). Let f be a function on [a,b] such that for all c
∈ (a,b), f ∈ B([a,c]) and is HK-integrable on [a,c]. If
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#∞ F lim
then f is HK-integrable with

F I∞

H

G→

.
)

Proof. Let
0,
when ∈ , and
. Set an increasing
sequence JK in [a,b] such that cn → b as n → ∞. By HK-integrability of on [a,cn] with
GL
JK
, it follows from Theorem 1.1 that for each n, ′
almost
everywhere in [a,cn]. Furthermore, by Corollary 2.3, for each n, F satisfies the Lipschitz
condition in [a,cn]. With F being Lipschitz on each [a,cn] and on {b}, then F satisfies the
countably Lipschitz condition in [a,b]. Therefore F satisfies the countably Lipschitz condition
and ′
almost everywhere in [a,b]. Therefore f is HK-integrable. Furthermore,
H

#

lim

G→

H

.

Theorem 3.2. (Integration by Parts). If F and G satisfy the Lipschitz condition in [a,b] and
′
, M′
N
almost everywhere then
H

M

NI M

#

M

.

Proof. Note that both F and G are continuous in [a,b], hence bounded. Let K be a common
bound for F and G. Then for any [u,v] ⊂ [a,b], we have
|

M

#

M

|,|
, |M

M
#M

#
M |I|
|I|
|
#

M

#

M

|

It follows that M satisfies the Lipschitz condition in [a,b]. Since M ′
N
I
M
almost everywhere and product of two bounded functions is a bounded function,
the result follows from Theorem 2.9.
Theorem 3.3. Let g ∈ B([a,b]) be HK-integrable with primitive G. If : M
B([a,b]) and is HK-integrable then (f ◦ G)g is integrable and
H

Q

Q

H

,

→ ℝ is in

∘ M N.

Proof. Note that G is Lipschitz and M′
N
almost everywhere on [a,b]. Let MG ≥ 0
such that for any [u,v] ⊂ [a,b], we have |M
# M | , -Q | # |. Also, since f is
integrable on G([a,b]), its primitive, say , is Lipschitz and differentiable almost everywhere
on G([a,b]) to f. Let -S ≥ 0 such that for any [u,v] ⊂ G([a,b]), we have |
#
|,
-S | # |. Hence
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|

∘M

#

∘M

| , -S -Q |

#

|

Therefore ∘ M satisfy the Lipschitz condition, and hence differentiable almost
everywhere, and in fact, differentiable almost everywhere to (f ◦ G)g. By Theorem 2.9, the
result follows.
A sequence (Fn) of functions on [a,b] is said to be equi-Lipschitz if there exists M ≥
0 such that for any [u,v] ⊂ [a,b], for any n, we have | 3
# 3 | , -| # .
Theorem 3.4. Let (Fn) be a sequence of functions on [a,b] which is equi-Lipschitz. Then there
exists a function f ∈ B([a,b]) that is HK-integrable and a subsequence of (Fi) of (Fn) such that
such that
lim

T→;

T

#

T

H

.

Proof. Since (Fn) is equi-Lipschitz, it is equi-continuous and therefore uniformly bounded. By
Arzela-Ascoli’s theorem it has a uniformly convergent subsequence. For convenience in
notation, let us just denote the uniformly convergent subsequence by (Fn). Let F be the limit
of (Fn). Then F is Lipschitz and therefore differentiable almost everywhere. Define a function
f on [a,b] such that
′
when ′
exits and
0 otherwise. Then is
boounded and
-integrable with primitive . In particular lim T
# T
.
T→;

CONCLUSION, IMPLICATION, SUGGESTION, AND LIMITATIONS
The paper suggests that a full discussion of integrability of essentially bounded
functions via Henstock-Kurzweil theory of integration is possible. One may explore
convergence theorems for this set of integrable functions in the future.

REFERENCES
Bartle, R. G. (2001). A modern theory of integration. Graduate Studies in Mathematics, 32.
American Mathematical Society.
Bongiorno, B., & Piazza, L. (1996). A new full descriptive characterization of the DenjoyPerron integral. Real Analysis Exchange, 21(2), 656-665.
Calunod, I. D., & Garces, I. J. L. (n.d.). Strong derivative and the essentially Riemann integral.
Real Analysis Exchange. (To appear).
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Indrati, Ch. R., & Aryati, L. (2016). The countably Lipschitz integral. Global Journal of Pure
and Applied Mathematics, 12(5), 3991-3999.
Lee, P. Y. (1989). Lanzhou lectures on Henstock integration. World Scientific Publishing.
Lee, P. Y. (2007). The integral à la Henstock. Scientiae Mathematicae Japonicae Online, e2007, 763-771.
Lee, P. Y., & Vyborny, R. (2000). The integral: An easy approach after Kurzweil and Henstock.
Cambridge University Press.
Thompson, B. S. (2014). Strong derivatives and integrals. Real Analysis Exchange, 39(2), 469488.

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