281
INTEGRATION OF CONUCLEAR
SPACE VALUED FUNCTIONS
(dedicated to Professor Igor Kluv§nek)
Susumu Okada
In the classical theory of the Lebesgue integral, the space of
integrable functions is introduced as the completion of the space of
simple functions (or of continuous functions), with respect to the
topology of convergence in mean, that is, the uniform convergence of
indefinite integrals.
However, if the space of simple functions
taking values in a Banach space
X is considered, then its completion
can only be realized as the space of integrable functions with values
in some locally convex space larger than
X
(see [6]).
In this note, it is shown that the space of integrable functions
with values in a conuclear space
Y
is sequentially complete with
respect to the topology of convergence in mean.
simple functions form a dense linear subspace.
Further, the
Y-valued
In this case, there is
no necessity to integrate functions taking values in a locally convex
space larger than
spaces
Y .
The class of conuclear spaces includes the
V, V', E, E', S and S'
which arise in distribution theory,
as well as the product space and the locally convex direct sum of
countably many copies of the real line.
282
1.
THE ARCHIMEDES INTEGRAL
The real or complex numbers will be referred to simply as
scalars.
Let
n be a non-empty set.
n will be identified with their characteristic functions.
subsets of
Let
A be a
let
EnA = {EnF : FEOA} •
a-algebra of subsets of
negative measure on the
Let
To save subscripts and circumlocution,
Let
n .
LEMMA 1.
E of n ,
A be an extended real valued non-
a-algebra
A and let
X be a Banach space with norm
well-known
For a subset
I· I .
AA
=
{EEOA: A(E) < oo}
The following lemma is
(cf. [6,Lemma 1]).
Suppose that
aj EO X are veato:r>s and Ej EO A sets,
j
1, 2, .•• '
suah that
L Ia ·I A(E.) <"'
(1)
j=l
J
J
and also suah that the equality
"'
L a.E.(w) = 0
j=l J J
holds fo:r> eve:r>y
satisfying the :r>elation
w EOn
"'
L la.!E.(w) <oo
(2)
j=l
J
J
Then, the equality
L a .A(E .)
j=l J
J
0
holds.
Recall that a strongly
be Bochner
A-integrable
A-measurable function. f
on
n is said to
A-integrable i f the non-negative valued function
(cf. [2,Chapter II]).
lfl
is
283
According to
if and only if there exist vectors
such that
(1)
f : Q +X is Bochner
[ 4] , a func: tion
c. EX
A-integrable
and sets
J
holds and such that the equality
I
f(w)
holds for every
c .E .(w)
j=l J J
satisfying
wE Q
fA. : A+ X
indefinite Bochner integral
"f.
(fA.)(E)
(3)
(2)
For such a function
f , the
is defined by
c.A(E.nE)
J
j=l J
fA.
The set function
is well-defined by Lemma 1, and it is
by the Vitali-Hahn-Saks theorem
its variation
JfA.J
(cf. [2,Corollary 1.5.10]).
a-additive
Furthermore,
is finite.
The proof of the following lemma is omitted because it is proved
similarly to Theorem 3 of
LEMMA 2.
Suppose that
[6] .
fn :
Q +X
,
n
=
1, 2, ... , are Bochner
functions for which
L lfnAJ (Q) <co
n=l
Let
f :
Q+
for every
X be a function such that
wE Q
for which
I
lfn(w)
I <co
n=l
Then, the function
f
is Bochner
A-integrable and
N
limJfA.-
N->-co
If
n=l n
AJ($1)
0.
A.-integrable
284
Let
Denote by
sequence
Y be a locally convex Hausdorff space and Y'
P(Y)
the collection of all continuous seminorms on
{Wn}~=l
wn"Wn' n
~-+
there exist vectors
c.
Y
is said to be Archimedes
E
J
{wn}~=l
the sequence
= 1,2, •.• ,
is
Y
unconditionally summable in
f:
A
Y •
Y is called unconditionally summable
of subsets of
if, for any choice of
A function
its dual space.
and sets
Y
E.
E
J
A, j
/..~integrable
if
satisfying the
= 1,2, ... ,
following two conditions:
(i)
the sequence
{c.A.(E.nA)}"': 1
J
J
J=
of subsets of
is uncondition-
Y
ally sumrnable; and
y'
(ii) i f
E
Y' , then the equality
L (y ',c .)E .(w)
(y' ,f(w))
wE n
holds for every
j=l
J
J
for which
L l(y',c.)IE.(w)<co
j=l
J
J
The indefinite integral
fA
the measure
integral
fA.
A.
of such a function
is defined as in (3).
is a well-defined
f
with respect to
By Lemma 1, the indefinite
Y-valued set function on
A .
It is
a-additive by the Vitali-Hahn-Saks theorem and the Orlicz-Pettis lemma.
Let
y'
E
f:
Y' , let
0.-+ Y
be an Archimedes
(y ,f)
1
A.-integrable function.
For every
be the function defined by
(y',f)Cw)
For a continuous seminorm
p
on
sup
Y , define
I(y ',f)A. I (~) '
where the supremum is taken over all vee tors
y'
E
Y
1
satisfying
285
l(y',y)l
~
for every
p(y)
The space of all
will be denoted by
Y-valued Archimedes
convergence in mean.
Y
If
pA , p
E
Q
, that is, the topology of
P(Y)
The set of all
AA-simple functions on
Q
is then
L(A,Y)
is an infinite dimensional Banach space, then the space
is not always complete.
L(!,,Y)
A-integrable functions on
L(A,Y) , and will be equipped with the topology
given by the seminorms
dense in the space
yEY.
\vith the space of all
functions on
Q
(see [8,p.l31]).
,
In this case, the space
L(A,Y)
Y-valued, strongly measurable, Pettis
which is not complete if
coincides
A-integrable
A is a non-atomic measure
Therefore, it is necessary to integrate functions with
values in a locally convex space which is larger than the Banach space
Y
needed to accommodate the values of indefinite integrals (see [6]).
2.
INTEGRATION OF CONUCLEAR SPACE VALUED FUNCTIONS
A balanced convex bounded subset of a locally convex space will be
called disked.
Let
Y
be a sequentially complete locally convex Hausdorff space.
The linear space
YB
generated by a closed disked subset
B of
Y
is a
Banach space, equipped with the gauge of B (cf. [l,Lemma III.3.1]).
Let
space
BE
Y
C
be a collection of closed disked subsets of
is said to be conuclear with respect to
C , there exists a set
canonical injection from
C E C which includes
YB
into
YC
C
B
Y .
The
if, for each set
and for which the
is a nuclear map.
For the
properties of conuclear spaces, see [7],
THEOREM 3.
a-algebra
Let
A be an extended real valued non-negative measure on a
A of subsets of a non-empty set
Q .
Suppose that the space
y
286
is
with respeat to a aoZZeation C of aZosed disked subsets
aonuaZear~
of Y, whiah has the property that every weakZy aompaat subset of Y is
Then~
inaZuded in a set from C •
the spaae
L(J.. ,Y)
is sequentially
aompZete with respeat to the topology of aonvergenae in mean.
Proof.
{fn}~=l
Take a Cauchy sequence
for every
E
10
~(E)
a vector
{ (fnJ..) (E) }~=l
A , the sequence
in
from the space
L(J..,Y) •
Then,
of vectors is convergent to
Y since Y is sequentially complete.
It follows,
from the Vitali-Hahn-Saks theorem and the Orlicz-Pettis lemma, that the
so-defined set function
l.I(A)
of
1.1
1.1: A-+Y
is a vector measure.
is a relatively weakly compact subset of
every nuclear map is compact, there exists a set
which contains
YB
is
l.I(A) , such that
1.1(A)
Hence, the range
Y
(cf.[S]).
B belonging to C
is a relatively compact subset of
Then, the set ·function 1.1: A-+YB
is a vector measure; that is, it
B .
a-additive with respect to the topology given by the gauge of
C from
Choose another set
the canonical injection
{~jlj=l
C which contains B and for which
is a nuclear map.
J: YB-+YC
{ajlj=l
absolutely summable sequence
of vectors in the dual of
of vectors in
Since
There exist·, an
of scalars, a bounded sequence
YB , and a bounded sequence
{yjlj=l
YC , such that the equality
00
l.
J(y)
holds in
YC
j=l
a
.(c.y)y.
J
J
J
(cf. [7, Chapter IV, Part II]).
For every
j
vanishes outside the union of countably many sets belonging to
1,2, ... ,
A).
the Radon-Nikodym theorem ensures the existence of a scalar valued
).-integrable function
g.
J
such that
Further, by the Nikod§m boundedness theorem, there exists a positive
number
M for which
Thus,
287
1, 2,
0
0
0
Now, the functions
every
h.(w) =
J
w E S1 , j = 1, 2,. . . , are Bochner
a.g.(w)y.
J J
for
J
f.-integrable, with the property
that
f : S1 +
Define the function
I
f(w)
for each
wE S1
h .(w)
j=l J
with
l,
jh.(w)j <oo,
j=I
and by
f(w) = 0
Bochner
YC , by
J
otherwise.
It
then follows from Lemma 2 that
f
is
f.-integrable and that
l,
(ft.)(E)
(hof.)(E)
J =I
I
a.(g.t.)(E)y.
j=l J
J
J
J
00
I
a.(t;.,]l(E))y.
j=I J J
J
for every
E
E
A •
It is now clear that the function
and the sequence
If
J(]l(E))
A is
{fn}~=l
f
belongs to the space
is convergent in mean to
L(f.,Y)
f .
a-finite, then the arguments in the proof of the above
theorem can be used to prove the following.
COROLLARY 4.
Suppose that the space
Y satisfies the assumption in
Theorem 3 and also that
Y is quasi-complete (respectively complete).
Then, the space
is quasi-complete (1oespectively complete) for
every
L(),,Y)
a-finite non-negative measure
f. .
288
A sequentially complete locally convex Hausdorff space is called a
conuclear space if it is conuclear with respect to the collection of all
closed disked subsets.
It follows from
[7, Theorem IV.l, Part II] that
V', f, E', S and S', used in distribution theory, are
the spaces
all complete conuclear spaces.
EXAMPLE 5.
Let
rl = JRn , n
A be the Lebesgue measure in the Euclidean space
Let
1, 2, ',.
S be the Schwartz space of rapidly decreasing,
infinitely differentiable functions on
rl , and let
space equipped with the strong dual topology.
transform
<P
for every
8
of a function
E
Q •
Let
¢
If
S
f : Q + S'
(f<w),¢)
for every
E
S'
be its dual
Define the Fourier
by
be the function defined by
¢<-w) ,¢
E
S
wE Q •
y
is a locally
\-integrable function on
Q
which grows at
infinity more slowly than a polynomial, then the function
Archimedes
subset
E
\-integrable.
of
Indeed, for every
¢
E
S
yf :
Q+
S1
is
and every Borel
Q ,
JE (f(w) , <P )y (w) dl- (w)
JE¢(-w)y(w)d:l.(w)
Since the Fourier transform
from
¢1+ ¢, ¢ E S
S into itself and since
measure
values in
~
y
, is a continuous linear map
belongs to
, which is defined on the Borel
S' , there exists a vector
a-algebra of
S' , such that, for every Borel set E ,
rl
and takes
289
(ll(E),.p)
In view of the proof of Theorem 3, there exists an Archimedes
A.-integrable function
g: S1+S'
such that
Since the space
Jl "'gA. •
S, which is the dual space of S' , has a countable total subset, it
follows that
Archimedes
yf
is
A.-almost everywhere equal to
Let
A.
be the Lebesgue measure, in the space
equipped with the Euclidean norm
as in Example 5.
1·1 ,
n"' 1,2, •..•
For every real number
yf
is
A.-integrable.
:.]Rn
Define the function
is Archimedes
t , the tempered distri-
Moreover, for each real number
(<Ytf)A.)(n)
S1
t , let
ytf : S1 + S'
then Example 5 implies that the function
bution
Hence,
A.-integrable.
EXAMPLE 6.
f : S1 + S'
g •
equals the difference of two oscillatory integrals
<I>(t,x) "'(27T)-nfn (2ilwl)-lei((x,w)+tlwl)d::\(w) - (27T)-nfn (2ilwl )-1 ei<(x,w)- tlwl) d::\(w)
with phase functions
(x,w) En
c0
X
(x,w)t+(x,w) + tlwl
n ' respectively.
Let
/';
<I>
0
(x,w)t+(x,w)- tlwl ,
denote the Laplacian in
the Dirac measure at the origin of
the distribution
and
n
As noted in
n
and
[3, Example 7.8.4],
is a solution of the Cauchy problem
in lR
x
n; <I> ( o, · )
c0
when
t
0 •
Finally, the author wishes to thank Brian Jefferies for valuable
discussions concerning conuclear space valued measures.
290
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[1]
N. Bourbaki, Espaces Vectoriels Topologiques, Elements de Mathematique,
Livre V, Hermann, Paris, 1966.
[2]
J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys, no. 15,
Amer. Math. Soc., Providence, 1977.
[3]
L. Hormander, The Analysis of Linear Partial Differential Operators I,
Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
[4]
J.L. Kelley and ToP. Srinivasan, On the Bochner integral, Vector and
Operator Valued Measures and Applications, 165-174, Academic Press,
New York, 1973.
[5]
I. Kluvanek, The extension and closure of vector measure, Vector and
Operator Valued Measures and Applications, 175-190, Academic, Press,
New York, 1973.
[6]
S. Okada, Integration of vector valued functions, Measure Theory and
its Applications, 247-257, Lecture Notes in Math., no. 1033,
Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
[7]
L. Schwartz, Radon Measures on Arbitrary Topological Spaces and
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[8]
G.E.F. Thomas, Totally summable functions with values in locally
convex spaces, Measure Theory, Lecture Notes in Math., no. 541,
117-131, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
School of Mathematical and Physical Sciences,
Murdoch University,
Murdoch, W.A. 6150,
AUSTRALIA.