Advances in Pure Mathematics, 2013, 3, 653-659
Published Online November 2013 (http://www.scirp.org/journal/apm)
http://dx.doi.org/10.4236/apm.2013.38087
Open Access APM
On the Differentiability of Vector Valued Additive
Set Functions
Mangatiana A. Robdera, Dintle Kagiso
Department of Mathematics, University of Botswana, Gaborone, Botswana
Email: robdera@yahoo.com, dintlek.kagiso@gmail.com
Received October 7, 2013; revised November 8, 2013; accepted November 15, 2013
Copyright © 2013 Mangatiana A. Robdera, Dintle Kagiso. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
The Lebesgue-Nikodým Theorem states that for a Lebesgue measure
:2 0,
,
 an additive set function
which is
:F
-absolutely continuous is the integral of a Lebegsue integrable a measurable function
; that is, for all measurable sets
:f,
A
d.
A
F
Af
Such a property is not shared by vector valued set
functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case
in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set
functions with values in an infinite dimensional normed space.
Keywords: Vector Valued Additive Set Function; Lebesgue-Radon-Nikodým Theorem; Fundamental Theorem of
Calculus
1. Introduction
This note can be considered as a continuation of the work
started by the first author in [1]. Throughout the paper
is a
2

-ring of subsets of a nonempty set .
A vector measure is a countably additive set function
defined on taking values in a Banach space
X
and
such that . We denote by

0
,
M
X the set
of all
X
-valued vector measures defined on .
Let be a nonnegative scalar measure.
If a function
:0,
 
:
f
X is
-Lebesgue-Bochner in-
tegrable, then it is easily seen that the function
:
F
X defined by
d
A
F
Af

FA
is a vector
measure. It readily follows from the property of the
Lebesgue-Bochner integral that whenever
Such a property is expressed by saying that
0

A
0.
F
is absolutely continuous with respect to .
Definition 1. A set function :
F
X
f
is said to be
λ-Lebesgue-Bochner differentiable if there exists a λ-
Lebesgue-Bochner integrable function such
:,
that for every
,A

d.
A
F
Af
Such a function
f
is called the
-Radon-Nikodým
derivative of .
F
It easily follows from the property of
the Lebesgue-Bochner integral that if
f
is a
-Ra-
don-Nikodým derivative of
F
then so is any function
g
such that
 
:0xfxgx
.
  A repre-
sentative of the class of such functions is denoted by
d
d
F
.
The classical Lebesgue-Nikodým theorem states that
for an additive real valued set function (not necessarily
countably additive) :F
,
-absolutely continuity
implies
-Lebesgue differentiability (see for example [2]).
The analogue of the Lebesgue-Nikodým theorem does not
extend to the vector valued case in general. For instance,
it is a well known fact that the

10,1L-valued vector
measure defined by
A
1A
for every Borel subset
A
of
0,1, is absolutely continuous with respect to
the Lebesgue measure
on
0,1 but fails to be
-
Lebesgue-Bochner differentiable (see for example [3]).
In this note, we reactualize the approach to the defini-
tion of the integral first introduced in [1] in order to ob-
tain a Lebesgue-Nikodým type theorem on the differen-
tiability of Banach space valued additive set function.
The exposition will be organized as follows. In Section
2, we introduce a new approach to integration theory that
does not require the elaborate machinery of Lebesgue
measure theory, and at the same time significantly sim-
M. A. ROBDERA, D. KAGISO
654
plifies the approach to gauge integral. In Section 3, we
shall see that the space of classes of integrable functions
(in the sense of the definition of integrability in Section 2)
can be naturally given a Banach space structure. In Sec-
tion 4, we state and prove our main result which can be
seen as an extended vector valued version of the Lebes-
gue-Nikodým theorem. The fifth section is devoted to
some extension of the Fundamental Theorem of Calcu-
lus.
2. Extended Notion of Integral
We begin by recalling the definition of limit in its most
general form, that is, the Moore-Smith limit, also known
as the net limit.
A nonempty set is said to be directed by a binary
relation , if has the following properties:
D
 
1) for ,,
x
yz D if
x
y and , then yz
x
z
(transitivity);
2) for ,,
x
yD
y
there exists such that
and (
upper bound property).
zDzx
z
Given a set ,
X
a net of elements of
X
is an
X
-valued function defined on a directed set
,D.
The notion of convergence can then be defined whenever
the set
X
is a metric space, with distance function .
d
A net

:, ,
f
DXd is said to be convergent if
there exists an element
x
X denoted by

,
:lim
D
x
f
such that for every >0,
there exists 0D
such
that for every 0,

d,<fx.
a
Ordinary sequences n, in which nD
di-
rected by constitute a special case of nets. The net
limit takes over all the essential parts of the theory of
limits of sequences; to name a few: the uniqueness of
limit, the algebraic properties of limits, the monotone
convergence property of limits, the Cauchy criterion, and
so on. For more details on net limits, we refer the reader
to [4].
>,
Next we introduce the notion of size-function that will
generalize the usual notion of measure in integral theory.
In the following, is a nonempty set. The power set
of , that is, the set of all subsets of will be de-
noted by
 
2.
Definition 2. Let By a size-function, we
mean a set-function
2.

:
 0, that satisfies the
following conditions:

0;

 
A
B whenever

A
B in (monotoni-
city);
n
n

n
n
A
A

for every sequence n
nA
in such that (countable subadditiv-
ity).
n
nA

Obviously, any measure defined on a
-ring is a
size-function. The length function defined on the
-
ring generated by the bounded intervals in is another
example of size function.
It is clear from the definition that an outer-measure is a
size-function defined on 2
. In fact, an arbitrary
size-function
:
0,
 defined on a
-ring
can be extended to an outer-measure defined on the
whole of the power set 2
as follows.
Proposition 3. Let
:0,
 be a size-function.
For every define ,A


*
1
inf:, .
n n
nn
AIAI

n
I


Then η* is an outer-measure such that

EE

*
for every .E
Definition 4. Let
be an arbitrary subset of 2,
and let .A
A
-subpartition of
P
A
is any fi-
nite collection
,;,,1,2,
ii i
I
IAI in
with the following properties:
1)
<
i
I
for all
1, ,in ,
i
I
A
2) ,
i
I
and ij
II
whenever . ij
We denote by the subset of
P
A
obtained by
taking the union of all elements of A
.P
-sub-
partition
,
i
PI n
ii
I
:1,i is said to be tagge d if a
point t
is chosen for each . We write
,
n1,i
::, 1,,
ii
P
It in if we wish to specify the tag-
ging points. We denote by
,A
the collection of all
tagged
-subpartitions of the set .
A
The mesh or the
norm of
,PA
 is defined to be

max :.
ii
P IIP

Definition 5. If

,,PQA ,

Q
we say that Q is a
refinement of P and we write if PQP
and
.PQ

It is readily seen that such a relation does not depend
on the tagging points. It is also easy to see that the rela-
tion is transitive on

,.A
If
,,PQ A ,
we denote by
:\, ,\:,PQIJIJJII PJ Q .
Clearly,
,PQ A
, and PQ.
Thus the relation has the upper bound property on
PQ PQ
,.A
We then infer that the set is dire-
cted by the binary relation .
,A
In what follows, X will denote either a real or a com-
plex normed vector space. Given a function :
f
X
,
and a tagged
-subpartition

,: 1,,,
ii
PItin A
,
we define the
,
-Riemann sum of
f
on to be
the vector
P
Open Access APM
M. A. ROBDERA, D. KAGISO 655


,
1
.
n
fi
i
PIf

i
t
P
Thus the function is an

,f
P
X
-valued net
defined on the directed set

,,A
d:
li
.
For conven-
ience, we are going to denote


,, ,
mA
f
Af


whether or not the net limit exists.
We are now ready to give the formal definition of the
resulting integral.
Definition 6. We say that a function :
f
X is
,
-integrable over a set
A
, if d
Af
repre-
sents a vector in .
X
The vector d
Af
is then called
the
,
-integral of
f
over the set .
A
In other words, :
f
X is
,
-integrable over
the set
A
with

,
-integral df
A
if for every
>0,
there exists such that for every

0,PA ,
,,PA we have
0
PP

,
d<
f
AfP.


(1)
We shall denote by

,,,AX
the set of all
,
-integrable functions over the set .
A
Many classical properties of the integral follow imme-
diately from the properties of net limits and therefore
their proofs are obtained at no extra cost. To name a few,
we have the uniqueness of the integral, the linearity of
the integral operator, the Cauchy criterion for integra-
bility, and so on.
It is also important to realize that no notion of meas-
urability nor notion of gauge is postulated in the above
definition.
We finish this section by noticing that if 12
in
then

2,
1
,AA
2
. Hence, we have the
following proposition stating the relationship between
,
1,
-integrability and
2,
-integrability.
Proposition 7. Assume that in
1

22.
Then
for every
2,A




12
,, ,,,,AX AX

 
and


12 1
dd for ,,,
AA
ff fAX.
 



3. Spaces of Integrable Functions
In this section, we show that the space

,, ,AX
.
can be given a structure of complete seminormed space.
We fix a (real or complex) normed vector space
X
In
view of Proposition 3 and Proposition 7, we can always
regard integrability only with respect to a size-function η
defined on the whole of and we shall write
2d
A
f
in stead of d,
Af

A
in stead of and

,2 ,A
,,AX
in stead of

,,2,AX
.
Clearly, a Riemann integrable (resp. Lebesgue inte-
grable) function can be regarded as an integrable func-
tion in the sense of Definition 6 with respect to the
outer-measure obtained by the Carthéodory extension of
the length function defined on the ring of bounded
intervals (resp. the Lebesgue measure
defined on a
-algebra of subsets of
).
It should also be noticed that if the set
A
is such that
0,A
then for all subpartitions

,PA
0,
fP
and thus A It follows that the
integral does not distinguish between functions which
differ only on set of size zero. More precisely,
d0f
.
.
 

dd whenever
:0
AA
fg
xAfx gx



We say that a function
f
is
-essentially equal on
A
to another function ,
and we write ,
f
g
if
:xAfxgx
0.

It is readily seen that the
relation ,
f
g
is an equivalence relation on
,,AX
.
We shall denote by
,,
I
AX
the quotient space

,, .AX
Definition 8. For every :,
f
AX we define the
variation of
f
on
A
to be
 
:sup :.
f
f
PP A

We say that the function
f
is of bounded variation
if <.f
The collection of all functions of bounded variation on
A
will be denoted by
,,BVA X
. It is clear that
,, ,,AXBVAX
and that
f
f defines
a seminorm on
,, .BVA X
For the particular case
where
A
is finite, a slight modification of the proof
of Theorem 3 of [5] leads to the following result.
Theorem 9. Let
:2 0,
 be a size-function
and
X
a Banach space. Let
A
 be such that
<.A
Then the function space
,,AX
is com-
plete with respect to the seminorm
.
Proof. Let n be a Cauchy sequence in nf
,,AX
with respect to the seminorm
. Fix
>0,
and let such that for >0N
,>mnN
in ,

sup :.
nm
nm ff
ffPP A
 (2)
In particular, if we consider the subpartition
,
A
A
 , then for ,>mnN
in ,

<.
nm
ff A
Since
<,A
we infer that the sequence
n
nf
is Cauchy in .
X
Since
X
is a Banach
Open Access APM
M. A. ROBDERA, D. KAGISO
656
space, we can define a function
.
limnn
ff


On the other hand, since
,,
nm
ff AX
,
, there
exist such that

,
nm
PP A

d< whenever,
n
f
n
A
Pf PP

n

d< whenever.
m
f
mm
A
Pf PP

Combining these inequality with (2), it follows that for
,>mnN
in and for every we have ,
nm
PPP
 

dd d
d<3.
n
m
nm fnff
AA A
fm
A
nm
f
fPf
Pf




 
P
This proves that the sequence d
n
A
nf
is Cauchy
in ,
X
and thus converges to say .aX
Now since for each ,
A

,
limnn
ff

there exists >NN
such that for ,>mnN
in ,
 
.
nm
ff
A


It follows that for



,: 1,,
ii
P
It ikA,
and for every we
have

,max:1,,:
i
t
mnNikN,
P

 
1
.
nm
k
ffi nimi
i
PIftft


If we let we obtain
,m

.
n
ff
P
Since
d,f
limmm
A
a

there exists >
P
NN such that
d<
m
Afa
whenever m > N. Thus for ,>,nm N


d<3.
nnm
fffff
m
A
Pa P
fa



 

P
Since ε > 0 is arbitrary, this shows that
,,fAX
and that
d.fa
A
It goes without saying that when the
-equivalent
functions taking values in a Banach space
X
are iden-
tified, the restriction of the seminorm
defines a
norm on
,, ,IAX
and gives the space
,, ,IAX
the structure of a Banach space.
4. Extended Lebesgue-NikoDým Theorem
Again, we fix a (real or complex) Banach space .
X
Let
:2 0,
 
:
a size-function. We say that a set
function
F
X is
-absolutely continuous on
if for every >0,
there exists >0
such that

<FA
whenever
A
 with

A<.
Loosely speaking, our main result states that
-ab-
solute continuity implies
-differentiability for additive
set functions. It naturally extends the Lebesgue-Nikodým
Theorem in its “full generality”. Namely.
Theorem 10. Let X be a Banach space,
:0,

a size-function. Assume that :
F
X an additive set
function that is
-absolutels on y continuou
. Then
there exists an

,,,fX
  such that
d
A
F
Af
,A
for all with
<.A
unlike thRema te that,erk. No Radon-Nikodým deri-
vative of a vector measure, the above density function
f need not be measurable nor Bochner integrable. It is
t even required that the scalar function no
f
be integrable in the sense of Definition 6. Su
f will simply be called the
ch a function
-derivative of the set
ction .
fun
F
Proof. In view of Proposition 3 and Proposition 7, we
may assume without loss of generality that 2.
 Fix
2,A
with
<.A
For every subpartition
P
, conunction defined on sider the f
by

 

1I.
P
IP
F
FI A
IA
Here
denotes the indicator function of the set .
I
1
I
Then it easily seen that

,,
P
is
F
X
. The
-
absolute continuity of
F
ensures that
dFFIA


P
AIP
IP
F
IA FA

(3)
as

.
IP
I
AA

We claim that the net
P
PF
auchy with respect to tis Che seminorm
. Fix >0.
By
F
, we-absolutely continuity of can find P and
Q so refined that


\<
3.
JQ
JQ
I
FIAIA
IP
FIAF A


 (4)
For such and there exists P ,Q
0
RA such
that for 0,RR
 
d<, d<
33
Q
P FQ
AA
Fand .
P
F
RF R

(5)


It follows from (4), (5) that for 0,RR




d
d
.
PQ P
FF FP
RR
Q
A
QF
A
IP JQ
RF
FR
FI
AF IA
 





(6)

Open Access APM
M. A. ROBDERA, D. KAGISO 657
This proves our claim.
By Theorem 9, there exists

,,fX
 such that
the net
P
PF
converges
seminorm
to f with respect to the
. Thus given >0,
there exists a tagged
that for 1
RP subpartition su

PA
1ch
 

sup P
fF
R:< .
3
RRA


(7)
other ho
su
On theand, it follows frm (3) that there exists

2
PA ch that for R
 
2
P
.
3
P
FRFA
(8)
Finally, by definition of the integral, there ex

ists
3A such that for 3
RP
P

df.
3
f
AR
 (9)
Combining (7), (8), and (9), we have for3
12
RPPP
 
 
 
dd
f
fFA fR

.
A
P
P
AA
fF
F
RR
RF





The desired result follows since >0
is arbitrary
chosen. The proof is complete.
As a direct corollary of our main Theorem 10, we do
have the following extended version of the vectoued
Radon-Nikodým Theorem.
Corolla ry 11. Let be a σ-algebra of a set
r val
and
:
X
 is a countably additive vemeasure that
is absolutely continuous with respect nite measure
ctor
to a fi
:0,. Then there exists an
,,fX

(not necessarily Bochner integrable) such that

d
A
A
f
for all .
Proof. We notice that the vector measure
A
is an ad-
ion and that the outer-meas
the Carathéodory extension of the measure
ditive set functurtained bye ob
de
fines a
size-function on. The desired ult isained by
applying Theoo the set nction
2
rem 10 t
res
fu
obt
and the
ction size-fun.
Remark. Note again that in the above Collary 11,
the density function
or
f
is not a Radon-Nikodým de-
rivative. It is also important to realize that in contrast
with the classical vector valued Radon-Nikodým Theo-
rem, the boundedness of the variation of the set function
is not required here.
It is a well known fact that if the
-density function
f
of a vector measur
dAf
A
for every .A
Recall that a Banach spce is said to have the Radon-
Nikodým Property if ey vecto measure :
a
ver r
X
of bounded variation that is absolutely cont
respect to a finite measure
don-Nikodým derivative. We
llary.
inuous with
has a Ra-
on with the
:0,
 
end th sectiis
following immediate coro
Corollary 12. A Banach space
X
has the Radon-
Nikodým Property if and only if for every
-algebra
of a set
, for every vector measure :
X
of
bounded variation that is absolous with re-
spect to a finite measure
:0,
, the
utelontiy cnu
-deriva-
tive of
is a Radon-Nikodým derivative.
5. Extended Fundamental Theorem of
Calculus
In this section, we give conditions under which a given
vector valued function (noteasurale) is the
derivati of a given finitely additive set-f
necessarily mb
veunction. Our
integral
th
result can be compared to that of Lu and Pee in [6].
It clearly follows from our definition of the
at
,,Xif and only if

,,fAX
f
e
exists and is Bochner inte-
grable with respect to ,
then
must have bounded
variation and its variation is given by
 for
every subsets
A
of .
This gives rise to a set function
:
F
X
:
d
A
F
Af
for all 2.A
 It im-
mediately follows from the properties of the integral that
such a set function is finitely additive. Such a
set
function
F
is called the
-indefinite integral of the
funct .
f
Of coble function
ion urse, an integra
f
is
always the
-derivative its of
-indefinite integral.
The next result gives a necessary and sufficient condi-
tion under which an additivon e set functi
F
defined on
2
is the
-indefinite integral of an integrable function
:
f
X
. In what follows the size-function
is
consideredo be finite, that is, t
<.

Theore13. Let m
be a nonempty set and :2
0,
b a finite size-functon. Then the following
st
e i
atements are equivalent for an additive set function
:2
F
X
and a function :.
f
X
) 1
,,fX
 and
F
is the
-indefinite inte-
gral of .
f
2) For every >0,
there exists
P
  con-
sisting of elements of
 
:, :IFIIfI

 
every ,PPsuch that for


<and<.FII ft
tt
tt
IP IP


 
ssity, fix ε > 0 and let

Proof. For the nece
P

as in 2. Then for PP
Open Access APM
M. A. ROBDERA, D. KAGISO
658
 
 

 

 

 




\
\
2.
t
tt
t
ff
ii
ii
IP
tt
IP IP
t
IP
F
t
IP
t
ii
IP
PF P
I ft
FIIft
IFI
Ift





 






Since
FI
FI I ft


>0
is arbitrary, this shows that

,,fX
 and
Conversely, assume that

d.fF


,,fX
. Let
.k
Given ε > 0 there exists a η-subpartition

:1
k
Ekf


,k
P

such that for every ,,
k
PP
 

2

1.
2
t
tt k
IP
FIt k
I f
For each ,
n let
 
if
0
n
k
1
otherwise.
k
n
f
E
f

Then for ,
1,
n
k
k
PP
we have



 

.
2
tt
t
tn t
IP IP
tt
IP
I
ft Ik
FII ft


k
 




On taking the limit as , we infer that
n


.
2
t
t
Ift
IP

It then follows that





.
22
t
IP
f
t
tt
ttt
IP IP
t
FIFII ft
I


 



The proof is complete.
Let now assume that is a topological space. We
say that a size-function

:2
of
0,

. For a fi
is regular if it is
no sxed n-zero on open set,
let

the set of all neiω. Tghborhoods of hen
ng
e followiis directed by inclusionce th
definition.
Definition 14. Let
We introdu
.
be a topological space and
:2 0,
 a re size-function. Let gular2
contain the topology of .
A set function :
F
X
is said to be
-differentiable at a point ,
if


,
m
F

exists in
X
.
In other words, :
F
li
X
is
-differentiable at a
point ,
if there exists a vector
f
X
such
that for every >0,
there exists open neighborhood
,U

such that for every
,U
,U
U
 
<.UU fUF

We call the set


,
: ts
lim
F
F

exis
 

the domain of differentiability of .
F
By the uniqueness
ofspondence net limit, the corre


,
lim
F

nc denoted by
F
defines a fution on
F
and called
the
-derivative of .
F
is compact. Let 2

Now assume that con-
taining the topology of ,
and :2
F
X an
\,E
addi-
tive set function such that F
 with
E
0.
Given >0,
let U
be an open set such that EU
and
<U
. On the other hanfor every ,d,
F
there exists an open neighbo
rhood

U


,,UUU

 such that for every
 
<.FUU fU
Clearly,

:UU
F
.

 is an open cover
of
By compactness, a finite subcover exy ists, sa

: .
iiF
UiU


 
Let
,1,,n
P
 ch that for su
 
1,, n. min ,:
i
PUUi



Then the set P

,

and any arbitrar
and 1. of Th 13 is
satisfied by F y extension o
eorem
f
F
to .
It follows th-function at since the size
is und
va hsion
of boed
riation, t is an element of en any such exten
,,,
X
 and
F
is its
-indefinite integral.
Hene following result which can be seen as ce we have th
a generalization of the Fundamental Theorem Cal-
cu rem 15. Lact topological space,
of
lus.
Theo et be a comp
2
containing the topology of and
:2 0,

additive set be a finite re
fun
gular size-function. Then every
ction :
F
X
omain of differentiability with d
Open Access APM
M. A. ROBDERA, D. KAGISO
Open Access APM
659
\,
FE with

0,E
is the
-indeite in-
gral of any arbitrary extension of its
fin te
- erivative.
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d
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the
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[2] S. Bochner, “Additive Set Functions on Groups,” Annals
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