395
each
0
H
!
, there exists a positive constant
G
such that whenever
1
1
{([ , ], )}
n
i
i
i
i
D x x
[
is a
G
-fine division of
[ , ]
a b
with respect to
g
, we have
| ( , , )
|
S f g D A
H
d
,
where
1
1
( , , )
( , , )
( )( ( ) ( ))
i
n
i
i
i
S f g D J f g D f
g x g x
[
¦
. We denoted a constant
A
by
( )
b
a
RS fdg
³
.
Theorem 1.5. (Cauchy’s Criterion for Riemann-Stieltjes integral)
Let
f
be a real valued function defined on
[ , ]
a b
. Then
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
if and
only if for each
0
H
!
, there exists a positive constant
0
G
!
such that whenever
1 2
,
D D
are two
G
-fine
divisions of
[ , ]
a b
with respect to
g
, we have
1
2
| ( , , ) ( , ,
) |
S f g D S f g D
H
d
.
Theorem 1.6.
Let
D
be a real number. If two real valued functions
f
and
h
are
RS
-integrable on
[ , ]
a b
with
respect to
g
, then
(i)
f h
is
RS
-integrable on
[ , ]
a b
with respect to
g
and
( ) (
)
( )
( )
b
b
b
a
a
a
RS f h dg RS fdg RS hdg
³
³
³
,
(ii)
f
D
is
RS
-integrable on
[ , ]
a b
with respect to
g
and
( )
( )
b
b
a
a
RS f dg RS fdg
D
D
³
³
.
Theorem 1.7.
Let
D
be a real number. If a real valued function
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
and
h
, then
(i)
f
is
RS
-integrable on
[ , ]
a b
with respect to
g h
and
( )
(
) ( )
( )
b
b
b
a
a
a
RS fd g h RS fdg RS fdh
³
³
³
,
(ii)
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
D
and
( )
( )
b
b
a
a
RS f d g RS fdg
D
D
³
³
.
Theorem 1.8.
Let
a c b
. If
f
is
RS
-integrable on
[ , ]
a c
and
[ , ]
c b
with respect to
g
, then it is
RS
-
integrable on
[ , ]
a b
with respect to
g
and
( )
( )
( )
b
c
b
a
a
c
RS f dg RS f dg RS f dg
³
³
³
.
Theorem 1.9.
Let
f
be a bounded function and
g
an increasing function on
[ , ]
a b
.
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
if and only if it is
DS
-integrable on
[ , ]
a b
with respect to
g
.
