Functions of bounded variation on idempotent

Carnegie Mellon University

Research Showcase Department of Mathematical Sciences

Mellon College of Science

1-1-1970

Functions of bounded variation on idempotent semigroups Richard A. Alo Carnegie Mellon University

Andre De Korvin

Follow this and additional works at: http://repository.cmu.edu/math Recommended Citation Alo, Richard A. and De Korvin, Andre, "Functions of bounded variation on idempotent semigroups" (1970). Department of Mathematical Sciences. Paper 195. http://repository.cmu.edu/math/195

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FUNCTIONS OF BOUNDED VARIATION ON IDEMPOTENT SEMIGROUPS by Richard A. Alo and Andre de Korvin Research Report 70-39

October,, 1970

mm imm tKUEBE-MQlSn SHIVEMY

FUNCTIONS OF BOUNDED VARIATION ON IDEMPOTENT SEMIGROUPS by Richard A. Alo and Andre de Korvin 1.

INTRODUCTION. In

[2]3 Dunford and Schwartz state a characterization

of the dual space of the space of all absolutely continuous functions on an interval by

AC(I).

I.

This latter space is represented

In particular each element

be represented as

x*

of

gf T ds

x*(f) = a f(a) + J

AC(I)* where

can g

is

I in

L

(I) -the space of essentially bounded Lebesgue

measurable functions on

I.

For the more general class

BV(I)

bounded variation on the interval in

of functions of

I = [0,1], T. Hildebrand

[4], has given a representation theorem for linear

functionals on In

BV(I), continuous in the weak topology.

[5]3 S. Newman defined a more general space of

functions of bounded variation.

Such functions and absolutely

continuous functions are defined over an abelian idempotent semi-group and more particularly over a semi-group of semicharacters which themselves are defined over an abelian idempotent semi-group. The main purpose of our paper is to study in more detail the space defined in

[5]. In this respect let

S

denote

a semigroup of semi-characters defined over an abelian idempotent semi-group

A

tion of bounded variation.

and let Then

F

denote a fixed func-

AC(S,F)

denotes all

[2]

functions of bounded variation which are absolutely continuous with respect to

F.

The first main result of

this paper deals with the representation of the dual space of

AC(S,F).

This representation is obtained via a

fundamentally bounded convex set function T(G) = vjG • dK

for all

G

in

is the "variational integral of

AC(S,F), G

K

by the formula

where

vjG • dK

with respect to

Kff.

Our second main result deals with the so called Lipschitz functions.

It turns out that there exists a

one-to-one and onto correspondence between Lipschitz functions and convex bounded set functions.

Moreover the

Lipschitz bound is equal to the bound of the corresponding '•.

convex function.

The first two results are a generaliza-

tion of some of the results contained in used in

[3]

[3]. The techniques

depend strongly on the fact that the functions

are defined on

[0,1].

Thus our first main result would,

for example, yield a representation for the dual of AC(I x I)

while the techniques developed in

[3]

would

not apply to this case. Our third result is a generalization of the more classical results contained in, for example, [6]. It will H

be shown that if

F

and

variation in the sense of

G

are functions of bounded [5]

and if

F

satisfies some

[3]

side conditions then

-=•=•

in some sense exists almost

CLr

everywhere. Throughout the entire paper it will be seen that much is to be gained when functions of bounded variation are considered as bounded measures over some algebra.

[4]

2.

BASIC DEFINITIONS AND RESULTS. In this section we will define the notations and

give some of the basic results of used throughout the manuscript.

[5]

which will be

The latter are used

considerably to obtain the first two main results. will make frequent use of notation in To this end let group and let

S

A

function on

A A

[6].

denote an abelian idempotent semi-

be a semi-group of semi-characters on

containing the identity. a semi-group

We

A

Recall that a semi-character on

is a non-zero, bounded, complex valued which is a semi-group homomorphism.

A

semi-character on an idempotent semi-group is an idempotent function and hence it can assume only the values zero and one. If

f

is in

{a e A : f(a) = 1}

S, then and

f(a) = 0 } . Now the sets Algebra

G

Jf

{f...... ,f }

will denote the set

will denote the set

Jf(feS)

of subsets of

algebra of all

Then if

A-

A.

Let

c(i)

generate a Boolean T

be the Boolean

n-tuples of zeros and ones, let

be a finite subset of denotes the

Bix.tr)

= ( n cr(i)=l

i

£

S, and let

component of

A

) n ( i

{a e A :

X

=

f B(X ,Q) If and

Q

(for some F

X

B- type if it is of the form and for some

is a real valued function

g). on

S

and if

are as above we can define an operator

(2)

L(X ,CT)F =

L

X n

by

n m( 0

T

X

of

in

BV(S) of

>

S

F e BV(S), by

= (f-,f2j...,f }

if

F e BV(S)

0

and for all corresponding

AC(S,F)

S

and for any subset

is given, there exists a

|L(X - 0

£ eR

H

Let

M(C)

| L ( X , 0

there exists a finite set

"

- i i i .

such that for all finite sets _______

_____»____.

i

_______

the difference

Proof.

In

i

____>_____.

| G - pG

|

[5]

Y

of xn

S

such that

•

•

•

j^

X

c Y n

= Y , n

< 6 .

it is shown that

in the variation norm by

X

_—__—_

V__ ^ .

fx

is approximated

This establishes the lemma.

Continuing in an analogous fashion we may define the concept of convex set functions.

Definition of

A

of

3.6.

Let

K

be a function from all subsets

B-type to the reals.

convex (relative to

F)

The function

if for every

is a disjoint union of sets

K

is called

BfX^c 1 ) (a € T )

B (Y , T - ) (X

c Y

and

which

r- € T ) .

[11]

we have

K(B(X n , a ))

=

E Aj K(B(Y m , Tj ))

where

A

Thus

K (B (X , a) )

The function |K(B)| < C of

C

K

j

-

is a convex combination of

is called bounded if for some constant

for all sets

B

of

is called the bound of

Definition

3.7.

Let

bounded set function £

L(Xn , e=)H

K

B-type.

K

H

and is denoted by

H

Ym

of

S ,

|v J H dK -

X

T T

Xn c Y

n

m

L(Y m

BV(S).

relative to the convex

This limit is denoted by «jv|H* dK .

Thus if the limit exists in

subsets

||K|| .

we mean the limit if it exists of

K(B(Xn , g)).

there is a finite subset

C

The least value

be a function from

By the variational integral of

m

K (B (Y , T •) ) •

3.7

of '

S we

r)H

then for every

e > 0

such that for all finite have

K(B(Y

[12]

We can now show that all polygonal functions are integrable relative to the convex bounded set function.

Lemma

3.8. JLf P q jus ja polygonal function, then

v J P dK exists.

In fact for some finite set

v J P g dK =

£

L(X n ,g)P g

Z c^f:

S^

K(B(X, 0

E

B

C e 5

such that

It will be said that E c A

A

and for every

3

in

A

D e B

and

then there is

and

Up(D!) < k M E c A

then 2?

x e E ., e > 0 x e C c B

and

and

V-property

V-covering ff of {B1^...,B }

E

of pair-

such that

/iF*(E -

n U

B±) < € .

We shall denote the outer measure generated by jju, as

Mp* • Lemma

in

A9

then

Proof. of a subset

4.2.

_If B

A

has the

is expansive at each point

E

of

A

x

V-property.

Choose disjoint sets from a as follows:

A

if

k

has the

there exists a sequence

wise disjoint sets in

x

where

and if

if for every

B-type in

V-covering 3»

[20]

Let

B.. be arbitrary and say

B..,B9,...B

J-

Let of If

JL

K n = sup /iF(B )

where the

i£

sup

have been chosen. XJ.

is taken over all sets

I? which are disjoint from B-.,B2,...B . Clearly 0 K i=1

~ 3* ) .

n

(E -

U

Now for

B. N

is a set in

C

large enough

and thus contains a set of

£ ^ n (B.) < e . Let

For

x e R , there exists

disjoint from

B1:,B2J,...B

from

B 1 ,B 2 ,...B n ,

Since

E ^ fj, (B.) < e N+l * x

On the other hand some

Bn .

Let

k

Since k D e 6

L k>N

In particular there exists

that

(Q

.

i=l

such that

Suppose that

x G C ^ C

C

B. . X

is

is disjoint

u p (C) N) . in A, there is a Therefore

then

N R = E - U

X

N+l

by

B

B

is expansive at each x k such that /x (D ) < 2 pu(B.).

M F (B k ) < 2 * K

Q e G , the

g-field generated

is a countable disjoint union of sets of

M F * ( E ~ Q) = 0 .

G)

such

Corrections to Research Report 70-39 FUNCTIONS OP SOUNDED VARIATION 0® IDEMPOTENT SEMIGROUPS by Richard A. Ale and Andre de Korvln

Page 21, between line 3 and line 4, insert? -'For the remainder of this report it is assumed that the collection

B

of all sets of

each point

x

in

E-type in, A

A/ ?

Page 22, line 8$ Change

/pa

f;>

Les3aa 5'" to

4,2;V,

Is expansive at

[21]

It should be remarked that if the fundamental sets are taken to be half open intervals of has the

point

B-type. p e A

Let

A

[C } be a sequence of sets

The sequence

(cn3

converges to the

if

fj, (C )

(2)

For every set

converges to B

of

zero. B-type such that

for all but a finite number of

(3)

C

4.3.

(1)

Cn c B

If

then

V-property.

Definition of the

A = I,

p e C

for all

converges to

p

n.

n.

we shall write

C

-• p .

[22]

Lemma 4.4. Assume that for every p e H c A , L(C n )G lim inf T , , _ < r for some sequence [C } converging to

p.

Let

H c B

joint sets of such that

where

B- type.

of

Consider

sets

C

Then

r? forms a

exists a set

B

contains

and if

H c B

B- type. then

G" such that

r

B

u^(Q) > riin(Q)

[23]

Proof,

The argument follows that in Lemma 4.4 once we

let

3? be all sets

and

C a B.

Definition in

BV(S)

let

4.6.

of

Let

B-type such that

F

and

satisfying conditions

p e A

fc }

we mean

lim C n ^p

dF

of the sequence

Theorem

G

L(C)G > r L(C)F

be two functions

(i), (ii), and

converging to L(Cn)G , N L (cn)F

[C }

4.7.

p.

By the derivative

where the limit is independent

converging to

Assume that every

p.

p e A

JLS the limit

of at least one sequence (C }. Then for some set dG //p*(E) = 0 , dF^P) exists for all p e A - E. Proof. that

G

(iii) 3

and assume that there exists at least

one sequence dG -r=r(p)

C

E

with

Due to the Jordan decomposition we may assume

is positive definite.

Let

L(C n )G E = {p € A: lim sup , , C n -*p L ( C n ) F

>

lim inf Cn« - p

L(C n ')G • ,y L(C n )F

[24]

where to

{C }

p.

and

{C '}

denote two sequences converging

Let L(C )G

If

jip*(E) 7^ O

1

Let k = inf

and there exists

By Lemma 1 ^

4.4 P

F

L(C

m m

then for some positive integers

u *(E. •) ^ 0.

k > o

m.1 m 1

^(B)

B2 e C

where

such that

there exists also a set x

^l y

i,i

i

and

B e G .

j , Then

jUF (B ) < k + e

Q,

such that

^

< f /iF(Qi) < J Now consider

E i . fl Q 1 .

union of disjoint sets of set

Q9 e G ,

Q

z

U (Q9) > — r -

c Q ^

Since

. D Q1 - Q2 .

provides a

n* (E. . p Q n - Q o ) = O ,

•

1., J

Now

the

the disjoint union of the sets Ei

is itself a countable

B-type, Lemma 4.5

X

JU^CQO)

Q^.

Consequently

set

±

E. .

Q2> E±

and

Z

is contained in

. - Q-. ,

and

[25]

i u F ^(E i ^ j )

every

£

»F*{Q2)

and

A

»F*(Q2) > * •

l s o for

e > 0 ,

^

k
0

p € A to

aLnd

g

there exists a. set B

£f

B~ type

M F (B) < e . the function

Then

X

G

jLs Lipschitz relative to

F.

^r(p) exists,, except on a null set, if for every there is at least one sequence

[C }

converging

p.

Proof.

With the above hypothesis

jur extends as

a countably additive set function to G , the generated by

G .

Q-field

[26] Bibliography 1.

R. B. Darst, tfA decomposition of finitely additive set functions", Journal Fur Reine and Angewandte Mathematik, 210 (1962), p. 31-37.

2.

N. Dunford and J. Schwartz, Linear Operators, Vol. I , New York - Interscience Publishers, (1957).

3.

J. Edwards and S. Wayment, "Representations for functionals continuous in the BV norm", unpublished.

4.

T. Hildebrandt, "Linear continuous functionals on the space (BV) with weak topologies", Proceedings of American Mathematics Society, JL7, (1966), p. 658-664.

5.

S. Newman, "Measure algebras and functions of bounded variations on idempotent semi-groups", Bulletin of the American Mathematics Society T5m3 (1969), p. 1396-1400.

6.

G. C. Rota, "On the foundations of combinatorial theory: I. Theory of Mobius functions", Z. Wahrscheinlichkeitstheorie und Veriv. Gebiete 2. (1964), p. 340-368.

7.

H. Royden, Real Analysis, New York - Macmillan Company, (1967).

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