Applied Mathematical Sciences, Vol. 9, 2015, no. 28, 1369 - 1372
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.5113
The Limit Theorems for Gauge Integral
Hwajoon Kim*
Kyungdong University
School of IT Engineering
Yangju 482-010, Gyeonggi, Korea
Corresponding author
c 2015 Hwajoon Kim. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
The gauge integral is a kind of generalized Riemann integral and this
is substituted for the place of Lebesgue integral to some degree. In this
article, we have checked limit theorems such as Beppo Levi’s theorem,
Fatou’s lemma and dominated convergence theorem for gauge integral.
Mathematics Subject Classification: 78M12, 28A25
Keywords: gauge integral, limit theorems
1
Introduction
There are many functions which are not Riemann integrable, and the integral
has no strong points for convergence theorems. While, Lebesgue integral is
more official integral, but the integral is demanded complicated measure theory
and exists which is not Lebesgue integral[1]. Hence Kurzweil and Henstock
proposed a generalized Riemann integral around 1960, but it has not well
known. This is the gauge integral and it comes into spotlight lately. Of
course, the gauge integral is easier than Lebesgue’s, and changed a factor in
the definition of Riemann integral. That has replaced a constant δ with a
positive function δ, which is called a gauge.
To begin with, let us see some needed definitions. A tagged partition
P = {([xi−1 , xi ], ti )} is a finite set of ordered pairs, where Ii = [xi−1 , xi ] form
a partition of I and the numbers ti ∈ Ii are called the corresponding tags. If
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Hwajoon Kim
P = {([xi−1 , xi ], ti )} is a tagged partition of I and f : I → R is a function,
then the Riemann sum S(f ; P ) of f corresponding to P is the number
S(f ; P ) =
n
X
f (ti )(xi − xi−1 ).
i=1
The number A ∈ R is the Riemann integral of f : I → R if for every > 0
there exists a constant δ > 0 such that if P = {([xi−1 , xi ], ti )} is any tagged
partition of I satisfying 0 < xi − xi−1 < δ , then
|S(f ; P ) − A| ≤ .
Next, let us see the definition of a gauge and δ-fine. A strictly positive function
δ on I is called a gauge on I. If δ is a gauge on I and P = {([xi−1 , xi ], ti )} is
a tagged partition of I, we say that P is δ-fine in case
0 < xi − xi−1 < δ(ti )
for i = 1, 2, · · · , n. Finally, a number A ∈ R is the generalized Riemann
integral(or gauge integral) of a function f : I → R if for every > 0 there
exists a gauge δ on I such that if P = {([xi−1 , xi ], ti )} is any partition of I
that is δ-fine, then
S(f ; P ) − A| ≤ .
In this case we will write f ∈ R∗ (I) and denote A = I f = ab f . Consequently,
the main characteristic of gauge integral allows δ to be any strictly positive
function instead of a constant on I.
We would like to take a look into preceding researches of this topic. Thompson has dealt with Taylor’s theorem for gauge integral[4], Bartle has given legitimacy to the definition of gauge integral in [1], Fremlin has researched the
gauge integrals of vector-valued functions[3], and [5] checked the gauge integral
in HOL4 which is the latest version of the HOL interactive proof assistant for
higher order logic.
In this article we have explored the limits theorems for integration, and
these results are ascertained well account for the strong points of gauge integral.
R
2
R
The limit theorems for gauge integral
In this section, we would like to deal with Beppo Levi’s theorem, Fatou’s
lemma and dominated convergence theorem.
Lemma 2.1 Monotone convergence theorem. Let
(fn ) be a sequence
Rb
∗
in R ([a, b]) that is monotone increasing and f (x) = limn a fn for all x ∈ [a, b].
Then f ∈ R∗ ([a, b]) if and only if
sup
n
Z b
a
fn < ∞.
1371
The limit theorems for gauge integral
In this case
Z b
f = lim
Z b
n→∞ a
a
fn
holds[1].
Theorem 2.2 Beppo Levi’s theorem[2]. Let
ries whose terms are in R∗ ([a, b]). Then
Z bX
∞
a k=1
fk =
∞ Z b
X
k=1 a
P∞
k=1
fk be an infinite se-
fk .
Proof. By the monotone convergence theorem,
Z bX
∞
a k=1
holds. Since
Pn
k=1
fk = n→∞
lim
Z bX
n
a k=1
fk
fk is finite,
Z bX
∞
a k=1
fk = lim
n→∞
n Z b
X
k=1 a
fk =
∞ Z b
X
k=1 a
fk .
Theorem 2.3 Fatou’s lemma. Let {fn } be a sequence in R∗ ([a, b]).
Then
Z
Z
lim inf fn ≤ lim inf fn .
n
n
Proof. For each positive integer n let gn = inf k≥n fk . Since {fk } is measurable,
each gn is in R∗ ([a, b]) and it satisfies the relations
g1 (x) ≤ g2 (x) ≤ · · ·
and
lim inf fn (x) = lim gn (x).
n
n
Thus, by the monotone convergence theorem,
Z
lim inf fn = lim
n
Z b
n→∞ a
gn ≤ lim inf
n→∞
Z b
a
fn .
Theorem 2.4 Dominated convergence theorem. Let {fn } be a sequence in R∗ ([a, b]) and let g ∈ R∗ ([a, b]).
If −g(x) ≤R fn (x) ≤ g(x) and
R
f (x) = limn fn (x), then f ∈ R∗ ([a, b]) and ab f = limn→∞ ab fn .
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Hwajoon Kim
Proof. The integrability of f and f1 , f2 , · · · follows from those of g and h. Since
{g + fn } is a sequence in R∗ ([a, b]) such that (g + f )(x) = limn (g + fn )(x), by
Fatou’s lemma,
Z
Z
(g + f ) ≤ lim inf (g + fn )
n
and so that
Z
f ≤ lim inf
Z
fn .
n
A similar argument applies to the sequence {g − fn }. Then we easily obtain
lim sup
n
and so
Z b
a
Z
fn ≤
f = n→∞
lim
Z
Z b
a
f
fn
for g(x) < ∞. Next let us consider the case of g(x) = ∞. Since g(x) < ∞,
the integrability of g follows and by the similar way, we can obtain the result.
R
The condition of f (x) = limn fn (x) is needed in the above theorem. For
example,
consider functions fn (x) = n forR x ∈ (0, 1/n), and f (x) = 0 otherwise.
R1
Then 0 fn = 1 for each n, and limn→∞ 01 fn 6= 1.
References
[1] R. G. Bartle, Return to the Riemann integral, To J. T. Schwartz, on his
65th birthday, (1996), 625-632. http://dx.doi.org/10.2307/2974874
[2] D. L. Cohn,
Measure theory Birkhäuser,
http://dx.doi.org/10.1007/978-1-4899-0399-0
Boston,
1980.
[3] D. H. Fremlin, The Henstock and McShane integrals of vector-valued
functions, Illi. J. of Math. 38 (1994), 471-479.
[4] H. B. Thompson, Taylor’s theorem using
Riemann integral, Amer. Math. Mon., 96
http://dx.doi.org/10.2307/2324092
the gerneralized
(1989), 346-350.
[5] Z. Shi, W. Gu, X. Li, Y. Guan, S. Ye, J. Zhang and H. Wei, The
gauge integral theory in HOL4, J. of Appl. Math. 2013 (2013), 1-7.
http://dx.doi.org/10.1155/2013/160875
Received: January 17, 2015; Published: February 23, 2015