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3 The space of regulated function and its properties
Definition 3.1.
Let
f
be a real valued function. Then
f
is said to be regulated if
f
has one side limits at
every point of
[ , ]
a b
,i.e.,
lim ( )
x c
f x
o
and
lim ( )
x c
f x
o
exist, for each
[ , ]
c a b
. The set of all regulated function
defined on
[ , ]
a b
is denoted by
[ , ]
RF a b
.
Theorem 3.2.
If
[ , ]
f BV a b
, then
[ , ]
f RF a b
.
Lemma 3.3.
If
[ , ]
f RF a b
, then for every
0
H
!
there exists a partition
1
1
{[ , ]}
n
i
i
i
P x x
such that for each
1, 2, ,
i
n
whenever
1
,
( , )
i
i
x x
[ K
, we have
| ( ) ( ) |
f
f
[
K
H
.
4 Integrable functions
Theorem 4.1.
If
[ , ]
f RF a b
and
g
is an increasing function, then
f
is
DS
-integrable on
[ , ]
a b
with respect
to
g
.
Proof
. Let
[ , ]
f RF a b
, we may assume that
f
is bounded and let
g
an increasing function. By Lemma 3.3
we have that for every
0
H
!
, there exists a partition
1
1
{[ , ]}
n
i
i
i
P x x
such that for each
1, 2,...,
i
n
whenever
1
,
( , )
i
i
x x
[ K
we have
| ( ) ( ) |
.
( ) ( )
f
f
g b g a
H
[
K
Then, we have
1
1
( , ( , )) ( , ( , ))
( ) ( )
i
i
i
i
M f x x m f x x
g b g a
H
.
Hence
1
1
1
1
[ ( , ( , )) ( , ( , ))]( ( ) ( ))
n
i
i
i
i
i
i
i
M f x x m f x x g x g x
¦
1
1
( ( ) ( ))
( ) ( )
n
i
i
i
g x g x
g b g a
H
¦
1
1
( ( ) ( ))
( ) ( )
n
i
i
i
g x g x
g b g a
H
¦
.
Since
g
is an increasing function,
1
1
( ( ) ( )) ( ) ( )
n
i
i
i
g x g x
g b g a
d
¦
.
Thus
1
1
1
1
1
1
[ ( , , )
( , ( , ))( ( ) ( ))] [ ( , , )
( , ( , ))( ( ) ( ))]
n
n
i
i
i
i
i
i
i
i
i
i
J f g P M f x x g x g x
J f g P m f x x g x g x
¦
¦
( ) ( )
( ) ( )
g b g a
g b g a
H
H
Hence, we have
( , , ) ( , , )
U f g P L f g P
H
.
