Carnegie Mellon University
Research Showcase @ CMU
Department of Mathematical Sciences
Mellon College of Science
1971
Vector valued absolutely continuous functions on
idempotent semigroups
Richard A. Alo
Carnegie Mellon University
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VECTOR VALUED ABSOLUTELY CONTINUOUS
FUNCTIONS ON IDEMPOTENT SEMIGROUPS
by
Richard A. Al&, Andre de Korvin,
and Richard J. Easton
Research Report 71-2
January, 1971
sd
HUNT LIBRARY
CARffEGIE-MELLON UNIYEBSITY
VECTOR VALUED ABSOLUTELY CONTINUOUS FUNCTIONS
ON IDEMPOTENT SEMIGROUPS
by
Richard A. Alo, Andre de Korvin, and Richard J . Eastern
ABSTRACT
In this paper the concept of vector valued, absolutely continuous
functions on an
idempotent semigroup is studied.
of bounded variation on the semigroup
values of
F
in the Banach space
S
X, let
of
For
F
a function
semicharacters with
A ~ AC(S,X,F) be all those
functions of bounded variation which are absolutely continuous with
respect to
F.
A representation theorem is obtained for linear
transformations from the space
A
to a Banach space which are continuous
in the BV-nornu
A characterization is also obtained for the collection
of functions of
A
which are Lipschitz with respect to
F.
With
regards to the new integral being utilized it is shown that all
absolutely continuous functions are integrable.
AMS 1970 subject classifications, Primary 46G10, 46E40; Secondary
Secondary 28A25,28A45
Key Words and Phrasess bounded variation, absolutely continuous,
v-integral, Banach space, linear transformation, dual space, Lipschitz
functions, convex set functions, semi-character, semi-group, setfunctions, Mobius function, positive definite, polygonal function,
characteristic function, simple function.
VECTOR VALUED ABSOLUTELY CONTINUOUS FUNCTIONS
ON IDEMPOTENT SEMIGROUPS
by
Richard A. Alo, Andre de Korvin, and Richard J. Easton
Introduction.
Absolutely continuous functions have been
extensively studied in the literature.
For example in [5] the
dual space of the space of absolutely continuous functions is
characterized.
In [7], T. pildebrandt gives a representation theorem
for the linear functionals on
weak topology.
BV[O,1] which are continuous in the
In [6] a representation theorem for linear functionals
continuous in the variation norm on
sentation is in terms of a so called
BV[O,1] is given.
v-integral.
This repre-
The techniques
of that paper, however, make strong use of the order on [0,1].
In [8]
absolutely continuous functions and functions of bounded variation
on idempotent semigroups are defined and these functions are identified
with a certain class of finitely additive set functions.
In [1], the identification in [8] is used to obtain a representation
theorem.
A characterization of the so called Lipschitz functions in
the setting of [8] is also obtained by the authors.
The techniques
of [1] depend on a result of Darst [2] which states that if
are two finitely additive real valued set functions with
then
v
functions
u
and
v < < u,
is the limit in the variation norm of finitely additve set
u
n
defined by
u (A) = I s du, where
n
J_ n
s
n
is a simple
v
function.
This result does not in general hold true when
u
and
v
are vector valued.
In this paper we study the concept of vector valued, absolutely
continuous functions on an idempotent semigroup.
We obtain a repre-
sentation theorem for linear transformations from the space
to
Y
which are continuous in the
BV-norm.
A characterization
is also obtained for the collection of functions of
are Lipschitz with respect to
F.
AC(S,X,F)
AC(S,X,F)
which
It is also shown that this new
integral being utilized has a "wide enough" class of integrable
functions.
In fact all polygonal functions are integrable (see
Lemma 6) and even more so all absolutely continuous functions
(Theorem 2) are integrable.
1.
Notations and Definitions
Let
A
be an abelian idempotent semigroup, and let
semigroup of semicharacters on
that a semi-character on
function on
A
A
A
containing the identity.
be a
Recall
is a non-zero bounded, complex valued
which is a semigroup homomorphism.
idempotent it is clear that every
characteristic function on
S
A.
f
in
S
Since
A
is
can be viewed as a
The notations used here will be
consistent with the ones used in [1] and [8]. We recall some of
these notations.
For
Let
T
f
in
S, A f = {a€A:f(a) = 1}
be the set of all
n-tuples
be a finite subset of
S, that is
If
i—
o('i)
denotes the
and
J f * {aeA:f(a) = 0 ) .
consisting of
0
and
1.
Let
Q n = { f.., f2,. . . , f }, and let
component of
a, for
Q = Q
let
Q
<j€T .
B(Q,O) = ( n
A. n
l
n
J- ).
i o(i)=O
r
i
Any set of this form will be called a set of
be any function from
S
to the reals.
B-type.
Let
F
Define
n
L(Q,o)F =
S m(o,t)F( w f.
T€T
i=l
n
where m
function
sup £
denotes the Mobius function for T R (see [9]). The
F is said to be of bounded variation if
<
|L(Q,O)F|
aeT
CD
, where
the supremum is taken over all par-
n
titions of
A
into sets
B(Q,a)
as
a
ranges over
T .
The
collection of all real valued functions of bounded variation on
will be denoted by
BV(S).
we mean all functions
exists a
subset
H
6 > 0
of
Consider
G€BV(S)
FeBV(S).
Then by
AC(S,F) ,
such that for each £ > 0, there
such that for every finite set Q = Q
T
S
of
S
and any
9
n
S |L(Q,a)G| <€
From now on
F
L(Q,a)F ;> 0
for all
Let
X
if
2 |L(Q,a)F| < 6.
will be assumed to be positive definite9 i.e.
such
Q
be a Banach space.
mean all functions from
S
to
and
a.
Then by the space
X
BV(S9X)
we
which are of bounded variation
in the above sense where absolute value is replaced by the norm
For
GeBV(S,X),
HG|LV
will denote
sup
S
||L(Q,Q) GilV.
in
Definition.
and
v
Let
£
be any field of subsets of some set and let
be finitely additive set functions defined on
is scalar valued and
v
is
X
valued.
absolutely continuous with respect tp
if
v
We say that
of the form
n
£ v.*x., where
i=l 1 x
x.€X, and each
*
finitely additive, set function on
continuous with respect to
Remark.
In the case that
X
where
v
1L and write
is the limit in the variation norm,of
£
u
u
is
v < < u
valued set functions
v.
is a scalar valued,
X
£, which is £ - 6
absolutely
u.
X
is the reals the above definition
reduces to the usual one.
Definition.
Consider
and
GGBV(S,X)
F
as above.
Tfhe function
is called absolutely continuous with respect to JF
ll#M_.y
limit in
where
BV
x.eX
of
X
if
valued functions of the form
G
and each
G.
i
is the
n
£ G.«x.
i=1
I
1
i
is a scalar valued function defined on
which is absolutely continuous with respect to
F
as in
[8] .
AC(S,F,X) .
2.
denote a scalar valued finitely additive set
Let
u
function defined on
£
set function defined on
Lemma 1.
m < < u
and let
be an
X-valued finitely additive
£.
if and only if
norm of finitely additive
whose
m
S
We
denote this space by
Results.
G
m
.is the limit in the variation
X-valued set functions defined qr^ £
range is finite dimensional. and which are £ - 6
absolutely
continuous with respect to
Proof:
Suppose that
m
u.
is the limit in the variation norm of
finitely additive set functions
dimensional) which are
to
u.
e-6
It follows that each
mi(where the ranges are finite
absolutely continuous with respect
mi
can be written as
n.
m. =
S m. .• x. .
where each
m. .
defined on
£
respect to
u, and where the
Hence
is a finitely additive, real valued, set function
each of which are
m < < u.
absolutely continuous with
are linearly independent.
The converse is clear.
Lemma 2.
m < < u
norm of
X-^valued
of
x..
€ - 6
if and
Qflly if. m
is. tTie limit in the variation
set functions which are represented by integrals
X-valued simple fMJiGfcijQiis with ye^pgqt to
Proof:
Prom lemma 1, m < < u
if and only if
u.
m
is the limit in the
variation norm of set functions of the form
Z ITU- Xj,
where each
m.
is real valued and
£ - 6
absolutely continuous with
respect to
u.
From a result due to Darst [I], each
m.
limit in the variation norm of set functions of the form
where
each
s.
v
is a real valued simple function.
is the
It is clear then that
m
will be approximated in the variation
norm by
Es. v» x.du.
1 , Jv.
From now on
as
f
ranges over
Let
To
m
m
w
£
1
ill denote the field generated by all .Jf
S..
be a finitely additve
we associate an
X-valued
set function defined on :
X-valued function defined on
S, denoted by ft,
which is defined by
m(f) = j fdm = m(A f ) .
Let
BV(Z,X)
denote the collection of all finitely additive
set functions of bounded variation.
Then
BV(2,X)
X-valued
is a Banach space
under the variation norm [5].
Theorem 1.
BV(s,x) .
x
in
The map
Moreover
m—^m
is a linear isometrv from
m < < u
if. andfltnlyAJL m < < u
BV(£,X)
£LX±£L£QX
onto
each
X, u»x = u^x.
Proof:
Clearly the map is linear, we now show that it is onto.
Consider
GeBV(S,X), then
G
can be extended to the linear span of
S
by the eguation
since
S
is a linearly independent set (see lemma 1.4 [7] ) . Since
is a semigroup, for each
EeG, it follows that
x«
is an element
hi
of the linear span of
eguation
S.
Thus we define a set function
u
by
S
u (E) = G (x_.) .
I t follows that
defined on
uG
S.
i s a f i n i t e l y additive
Furthermore for each
X-valued
set function
f€S,
ft (f) „ u (Af) - 6 ( f ) .
We now show that the map i s norm preserving.
We have
\\m\\ = sup s HmtB^H
where t h e
B . ? s are s e t s of
B-type
and form a p a r t i t i o n o f
A,
Now
11 mil ~ s u p S Urn(B.) 11
= sup Z
Note that we can now obtain the norm of G directly from the
equation
"SUP S ^G(XA) ^ *
MBV
Now suppose that
G < < F, Then
G
is the limit in the variation
norm of
G
~ ^ Sh^ .• x^ .
n
. n, i n« I
i
where each h^
. is real valued and Tin 5I. < < F. Also each
n^ i
h n ^ = .u ^ where
it follows that
u
n
u
n
i < < UF*
Hence if we let
converges to u^ in the variation norm once
G
8
we have shown that for each real valued finitely additive set
function
u
and each
xeX
since then we have that
This follows since
u»x(f) = u« x(A )
= u(A f ). x
= u(f)* x.
Lemma 3,
Proof;
The space
Since
AC(S,X,F)
BV(S,X) is A Banach space,from theorem 1
sufficient to show that
G€BV(S,X)
BV-norm.
jLs, a, Banach space.
and
AC(S,X,F)
G eAC(S,X,F)
Since each
G
is closed in
where
[G }
BV(S,X) o
converges to
is the limit in the
it is
G
Consider
in the
BV-norm of functions
n
of the form
EG^ .• x .
n,i
i n,i
where each
Definition,
in
S
G n, .
l< < F
Let
{ A ^ A ^ . . . ,A n )
and let
=
Define
it follows that
n
S XA
i
GeAC(S,F,X).
be a partition of
A
by sets
V s (E)
for each
E
P eAC(S,F,X)
s
in
s, then clearly
v
< <
s
which corresponds to
V
s
U
'nie function
F*
from theorem 1 will be
called a polygonal function.
Lemma 4.
The collection of polygonal functions is dense in
AC(S,F,X) .
Proof;
H =
Consider
G€AC(S,F,X)
n
S h.«x., where
x
and £ > 0.
h.€AC(S,F)
x
There exists an
and
x
Furthermore each
h- = u., where
x
u. < < u__.
1
Darst [1], there exist
1
simple functions
From the result of
r
s.
such that
IK - J s i d u A < £
Let
•f*
"•"•
V
Q
^T
and
V t (E) = J tdu p ,
then if
P.
is the polygonal function which corresponds to
we have
- pt||
BV
Uj,. x± - \ S S i
x^
10
which establishes the lemma.
Now to each
G
in AC(S,X,F)
function which, in the case that
we associate a special polygonal
S
is the set of characteristic
functions on half open intervals, coincides with the usual idea of
polygonal function, see [3]. Let
finite subset of
WY,G =
S.
Let
£
*
u (B(Y,a))
U (B(Y,O)) *
\J t X
G€AC(S,X,F) , and let
Y
be a
*B(Y,o)*
J.
n
Since ti (B(Y,a)) = 0 implies
to be zero in this case. Let
V
then since
Lemma 5.
the
of
V
Y,G * I W Y , G d V
< < u , we denote the corresponding polygonal function
The collection Q£ all
BV-norm.
S
u (B(Y,a)) = 0, we define the ratio
In fact for
pG
is dense in AC(S,X,F)
in
£ > 0> there exists ^ finite subset
YQ
such that if Y D y , then
llG - pGYH < £.
Proof:
Let £ > 0, then there exists an
such that
2 '
since
u Q < < up.
If
B(Z,o)
X-valued simple function
s
11
then for each
B(Z,o)
V s (B(Z,a)) = Jf
sduF
B(Z,a)
Thus
*a'u,,(B(z,o))-VB<z»''>>
F
Z 1 3 Z, we have
Similarily, if
S
~
up(B(Z',o)) '
which we shall write as
Vs(B(Z',o))
up(B(Z',o)) '
Now
3" ""F
V (B(Z' ,o) ) - u
r
- u
Hence the result follows from the triangular inequality and theorem 1«
We will denote the space of all bounded linear maps from a
Banach space
Definition,
X
Let
to a Banach space
K
Y
by
L(X>Y) •
be a function defined on all sets of
B-type with
12
values in
whenever
L(X,Y) • We say that
{B(Z!,T)}, TeTm
K
is convex relative to
is a partition of
K(B(Z,a)) =
£, if
B(Z,a) , then
SATK(B(Z' ,T))
T
where
\
The set function
L(X,Y)
K
=
up(B(Z',r))
up(B(Z,o)) •
will be called bounded if
norm over all sets of
B-type.
least upper bound of the bounds for
Definition.
For
with respect to
G:S—±X
and
K
By
K
is bounded in the
\\K\\, we will mean the
K,
convex, by the
v- integral of
G
K, we mean the limit, if it exists, of
EK(B(Z,o))L(Z,a)G,
a
where the limit is taken over the net of all finite subsets of
S.
We denote the integral when it exists by
v j GdK.
Lemma 6.
All polygonal functions are
every convex and bounded
K.
v-integral with respect to
In fact
v J p g dK = SK(B(Z,a))L(Z,a)ps
for all
Proof;
Z o Z , ZQ
some finite subset of
Suppose
s = SI
a
S.
13
then
V (B(Z,a)) = u
S
for all
(B(Z,a))*x
r
Z z> Z .
O
So
L(Z,O)pe = u (B(Z,0))tx .
Consider
Z
c Z c Z ! > then
L(ZST)PS
where
x
= x
T
if
= UF(B(Z',T))« X T
By convexity
B(Z»,T) C B(Z,O).
G
K(B(Z0,a)) = E^aB(Z!,0)
where
uT?(B(Z',a))
a
u p (B(Z o ,o)) *
Thus
rK(B(Z-,a))L(Z.,o)pe =
a
Theorem 2,
into
Y
Let
T
SK(B(Z',T))L(Z',T)P
be a linear operator from the space
which is continuous in the
BV-norm.
unicfue convex and bounded set function
such that every
G
.
T
in. AC(S,X,F)
is
AC(S,X,F)
iTien there exists JL
K, with values in
L(X,Y) ,
K-intearable. and moreover
T(G) = v J GdK.
Furthermore
\\T\\ = \\K\\ .
Conversely if
K
set function, then each
is any convex and bounded.
GeAC(S,X,F)
is
defines a continuous linear operator from
L(X,Y) valued,
K-integrable and
AC(S,X,F)
into
v 1 GdK
Y.
HUHT LIBRARY
GAMEG1E-MELL0H UH1VEKIH
14
Proof;
Let
Z be any f i n i t e subset of
V
then
V
Z,a
(E)
=
S and l e t
up[B(Z,0) 0 E]
u p B(Z,a))
i s f i n i t e l y additive and
V_ „ < < u_.
Z,0
£> O
be the corresponding function in AC(S,F).
r
K(B(Z,a))« x = T(,|)z^a.x) ,
then
\|K(B(Z,O)). x|lY = ||T(,|, z ^ a .x)|| y
we have that
114 1 |lTl|.
Now,
.
V
Z,G
(E)
U (B(Z,o))
" ] I U F (B(Z,c))'
Su (B(Z,o))V_ (E)
Z a
a G
'
Thus by theorem 1,
From Leirana 5 , we have
»b_
be
4,a
Define the function
by the equation
Since
Let
K
15
T(G) = lim T(pGz)
z
= lim T(EL(Z,a)G&_
z
)
^"a
a
= lim 2K(B(Z,o))L(Z,a)G
z a
= v J GdK.
Also
=
sup HK(B(Z,O))»X\L,
= HK(B(Z,0))H L(XfY) .
Hence
\M = ||K||.
Conversely suppose that
valued set function.
K
is a bounded, convex, L(X,Y)
Tfrien
flv j P G Z dK - V JP G Z dKH 1 l|pGz - P G Z H \\K\\.
Since Y is complete this shows that G is K-integrable and moreover that
v f GdK = lim v J pG dK.
We now define the concept of a Lipschitz function and characterize
the space of all such functions in terms of convex and bounded set
functions.
Definition,
Let
g
be a real valued function defined on
is called Lipschitz with respect to
F
S.
Then
if there exists a constant
g
P
such that
|L.(z,o)g| < PL(z,a)F
for all sets
B(x,a).
We denote this space of functions by
Lip(F) .
16
Definition.
By the space
GeBV(S,X, F)
which are approximable in the BV-norm by functions
n
z g.*x., where x.eX and g.e LIP(F)
for i = l,2,...,n,
i
1
i
i=1 i
of the form
Lip(X.F)
we mean all functions
We now want to give a characterization of the space
in terms of convex and bounded set functions.
Lip(X,F)
For this purpose we
introduce a special class of convex and bounded set functions which
we denote by
M (X,F) .
Definition.
Let
We say that
KeMri(X,F)
collections
K
be a convex and bounded
X-valued
set function.
if and only if for each £ > 0, there are finite
{K..,K2,...,K )
and
{x-,x , . . . , x n ) f where each
scalar,convex,and bounded and each
Ki
is
x.eX, and such that
n
for all partitions
M (X,F)
set
The spaces
Consider
[h^,h2,... ,h n )
*L
u^
into sets of
n
M (X,F)
where
l
correspond to
and
B-type.
Clearly
Lip(X,F)
are linearly isomorphic.
and € > 0, then there exists a finite
each
h^eLip(F)
and a finite set
such that
n
i=l
Let
A
He Lip(X,F)
(x, ,x 0 ,. . . ,x }, x.eX
l
of
is a linear space.
Theorem 3.
Proof;
{B.}
H
1
1
and define
rl
K^
rl
VU<B)
by the equation
17
and if m. corresponds to h. define
k. by the equation
u i (B)
k (B) =
i
for each
functions
that
i, for all sets
K ,Kn.....K
li I7
n
K^eM (X,F).
ViT '
B
of B-type.
It follows that the set
are all convex and bounded.
Let {B.} be a partition of A
We now show
into sets of the
B-type, then
n
2UWF(BJ
SU
(B..) UKjjtBj)
llK^B.) - I2 k
k,i ((B.)-xJ
B j ) . x±\ =
n
Z n±(B.). x±\\ < i .
i=l
Conversely, consider
K€MC(X,F) .
Then for
£ > 0
there e x i s t s
B-type.
If we define
, K 2 , . . . , K } and {x.^x^ . . . ,x n ) such t h a t
..) \\KCBj) for a l l p a r t i t i o n s
{B.}
of
E ^(B^.x^l < £
A into sets of
u ( B ) = u (B)K(B)
and
m i (B) = u p (B)K ± (B)
then it is easy to check that
u__ and the m. ! s
Jx
are finitely additive
1
and absolutely continuous with respect to F. Let HT. correspond to
u__ where
j\
H.-.eAC(S,X,F)
J\
and h. correspond to m. where
l
hieAC(S,F)3 then
X
.Vi
i
1=1
l
18
Now
the maps
K—>H
and
H—J>K^
are inverses of one another.
Consequently the theorem is shown since linearity is immediate.
Remark.
It should be pointed out that the above
characterization is
rather different from the scalar case as in [1] . While the map
H
>K^
was straight forward in the scalar case, we have seen that in our
vector setting a weighted-type of variation is needed.to define the
map.
19
BIBLIOGRAPHY
[1]
Richard A. Alb and Andre de Korvin, "Functions of Bounded
Variation on Idempotent Semigroups", (submitted).
[2]
R. B. Darst, "A Decomposition of Finitely Additive Set Functions",
Journal Fur Reine Angewandte Mathematik, 210 (1962), pp. 31-37.
[3]
A. de Korvin and R. J. Easton, "A Representation Theorem for
AC(R n )", submitted Scripta Mathematica.
[4]
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Carnegie-Mellon University
Pittsburgh, Pennsylvania
Indiana State University
Terre-ftaute, Indiana
sd