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1 Introduction
Definition 1.1.
For a bounded function
f
on
[ , ]
a b
and a partition
1
1
{[ , ]}
n
i
i
i
P x x
, we use the notation
1
1
( , ( , )) sup{ ( ):
( , )}
i
i
i
i
i
i
M f x x
f
x x
[
[
and
1
1
( , ( , )) inf{ ( ):
( , )}
i
i
i
i
i
i
m f x x
f
x x
[
[
.
Defined
( ) lim ( )
t x
g x
g t
o
and
( ) lim ( )
t x
g x
g t
o
,
and let
1
( , , )
( )( ( ) ( ))
n
i
i
i
i
J f g P f x g x g x
¦
.
The
upper Darboux-Stieltjes sum
is defined by
1
1
1
( , , )
( , , )
( , ( , ))( ( ) ( ))
n
i
i
i
i
i
U f g P J f g P M f x x g x g x
¦
and the
lower Darboux-Stieltjes sum
is
1
1
1
( , , )
( , , )
( , ( , ))( ( ) ( ))
n
i
i
i
i
i
L f g P J f g P m f x x g x g x
¦
.
The
upper Darboux-Stieltjes integral
is defined by
( , ) inf{ ( , , ): is a partition of [ , ]}
U f g
U f g P P
a b
and the
lower Darboux-Stieltjes integral
is defined by
( , ) sup{ ( , , ): is a partition of [ , ]}
L f g
L f g P P
a b
.
We say that
f
is
Darboux-Stieltjes integrable
or
DS
-integrable on
[ , ]
a b
with respect to
g
if
( , )
( , )
L f g U f g
. In this case, we write
( , )
( , )
b
a
fdg L f g U f g
³
.
Let
G
be a positive constant,
[ , ] [ , ]
u v a b
then an interval
[ , ]
u v
is said to be
G
-fine with respect to
g
if
| ( ) ( ) |
g v g u
G
.
Let
1
{[ , ]}
n
i
i
i
P u v
be a finite collection of intervals. Then
P
is said to be a
G
-fine partial partition
on
[ , ]
a b
with respect to
g
if
P
is a partial partition of
[ , ]
a b
and each
[ , ]
i
i
u v
is
G
-fine with respect to
g
. In
addition, if
P
is a partition of
[ , ]
a b
, then
P
is said to be a
G
-fine partition on
[ , ]
a b
with respect to
g
.
Theorem 1.2.
(Cauchy’s Criterion for Darboux-Stieltjes integral)
A bounded function
f
is DS-integrable on
[ , ]
a b
with respect to
g
if and only if for each
0
H
!
, there exists a
positive function
G
such that for any
G
-fine partition
P
of
[ , ]
a b
we have
( , , ) ( , , )
U f g P L f g P
H
Definition 1.3.
A real valued function
g
defined on
[ , ]
a b
is said to satisfy
J
-condition if for every
0
G
!
,
there exists a
G
-fine partition of
[ , ]
a b
with respect to
g
.
Definition 1.4.
Let
g
be a real valued function defined on
[ , ]
a b
and satisfy
J
-condition. A real valued
function
f
is said to be
Riemann-Stieltjes integrable
or
RS
-integrable to
A
on
[ , ]
a b
with respect to
g
, if for
