In. J. Math. & h. Si.
721
Vol. 2 No. 4 (1979) 721-724
WEAKLY COMPACTLY GENERATED FRECHET SPACES
SURJIT SINGH KHURANA
Department of Mathematics
The University of Iowa
Iowa City, Iowa 52242
U.S.A.
(Received on March 12, 1979 and in revised form July 13, 1979)
It is proved that a weakly compact generated Frechet space is
ABSTRACT.
LindelBf in the weak topology.
As a corollary it is proved that for a finite
measure space every weakly measurable function into a weakly compactly generated Frechet space is weakly equ’ivalent to a strongly measurable function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
Secondary 28A20, 46A99, 54E50.
Primary 28A40, 60B05,
KEYWORDS AND PRHASES. LindelBf spaces, weakly measurable functions, strongly
measurable functions, universally measurable functions, Frechet spaces, weakly
compact s
i.
INTRODUCTI ON.
If
E
is a weakly compactly generated Banach space then it is proved in
that E, with weak topology, is Lindelf.
(A
[7]
topological space is said to be
Lindelf if its every open covering has
a countable
we extend this result to the case when
E
subcovering.)
In this note
is a weakly compactly generated
722
S.S. KHURANA
Also, some consequences are obtained.
Frechet space.
All locally convex spaces
By a Frechet space we mean a Haus-
are taken over the field of real numbers.
dorff, metrizable, complete locally convex space; we use the notations of [4]
E’
for locally convex spaces.
A locally convex space is said to be weakly compactly
E.
locally convex space
will always denote the topological dual of a
G(E,E’ )-compact
generated if there exists an increasing sequence of
E
(E,G(E,E’))
Let
E
PROOF.
(i)
be a sequence of
[An]
E’
R
a subspace of
dorff space
JA
(g+Vn)
=h + z
n
take an
polar of
is a
n
c H
n>n 1.
Vno
follows that
O-nbd.,
n.
Since
([4]).
[hn]
Fix
n_
6N
x 6
E
D
E
e >0.
1
for each k
6 (H+Vn)
n=l
for every
[ (H+V).
n
n=l
with
and
f(x-hn)--+O,
is Cauchy in
n
If(z n)l
Vn0D nVn,
Thus
H+V
[z n]
and a sequence
for each
’
We claim that
in
n
E’,
and a compact set
is a compact subset of
n
V
AcRE’
y6n’, x+y6n’
and
E.
n
as
compact Haus-
is a subset of the
E’
x6R
For an
Conversely, take an
[hn]
n
V
and
E.
E’
R
(E,G(E,E’))
We identify
E.
is dense in
R= [-,].
+
&
K
Thus
3
=H
For a compact set
being the closure of
n=l
sequence
n=l n
where
E
n.
with product topology.
the natural meaning.
dense in
for every
for an increasing sequence of weakly compact, absolutely convex
subsets of E such that
is compact.
O-nbd. base having the properties"
is absolutely convex and closed,
n
(n+l)Vn+ 1 cVn,
(ii)
We take
V
each
Em
(En,G(Em,E’)),
is a Borel subset of
E.
[Vn]
Let
Then
be a weakly compactly generated Frechet space.
Lindelf space and E
is a
being the bidual of
X
E.
whose union is dense in
THEOREM 1.
subsets of
has
BC
A+B
and n,
V
H
Since
n
is
and so
This means there exists a
Zn6Vn
for each
Choose an
< 1 < e,
II
uniformly for
which is complete.
n_
I
>
n,
such that
max(n^,)
and
o
for every
o
f6
If
Vno.
hn--*y
f6Vn0
the
From this it
in
E
it
Y COMPACTLY GENERATED FRECET SPACES
is easy to verify that, as elements of
Thus, in weak topology,
([6]).
that
(E ,O(E,E))
Also
(Ak+Vn)
(H+Vn)
E
is Borel in
(Ak+Vn)
, ,
(E (E E
).
Es.
(E,o(E,ES))
is closed in
E’’
I H+vn
E
Since
n=-
and so
it follows
(E u,O(E ,E’)).
is Borel in
REMARK.
This proves the claim.
x=y.
is analytic and so is Lindelf
n=l k=l
Since
can be considered as a subspace of R
(+Vn)NE
R
is compact in
E
E=
E
723
[2, Cor. 3.2].
Similar results for Banach spaces are proved in
In the following result, some results and notations of ([3]) are used.
(X,/,)
be a
function
f
h6E
every
.
finite
X--E
measure space,
It is proved in ([2]) that if
(E,((E,E’)), S
-R, )(B)
ho
equivalent if
fl =h f2
o
f
(f-l(B)),
a.e.
[la],
f.
1
X--E,
for every
f
n
COROLLARY 2.
Let
X--*E,
(X,/,)
such that
f
n
-
(E’,G(E,E))
E
If
hE
f :X-- E
is Prechet then
[fn]
pointwise
E
a.e.
of
[].
a weakly corn-
a weakly measurable function.
is weakly equivalent to a strongly measurable function.
f
By
PROOF.
(E,O(E,E ))
sures are
([3], Cot. 5)
is tight
it is enough to show that image Baire measure on
(cf. [2]).
Since
(E,o(E,E ))
T-additive (normal in the terminology of
is
Lindelf,
Baire mea-
[5],[8]). By ([5],
3.3, 3.4) every Frechet space is universally measurable and so every
measure is tight.
REMARK.
88(4),
([2],
are said to be weakly
1,2
be a finite measure space,
pactly generated Frechet space, and
for
is a Baire measure on
i
f,
A
is weakly measurable
X--*E
f
-measurable
is
is called strongly measurable if there exists a sequence
l-simple functions,
Then
h
being the class of all Baire subsets of
Two weakly measurable functions
f" X-*E
a Hausdorff locally convex space.
is called weakly measurable if
that the image measure
[8]).
E
Let
Theorem
Theorems
T-smooth
This proves the result.
In case E is a Banach space, this result is implicit in ([2], p.
5.4);
if in addition f is bounded this is proved in
([i], p. 88).
S.S. KHURANA
724
REFERENCES
i.
Diestel, J. and J. J. Uhl Jr.
veys, Number 15 (1977).
2.
Edgar, E.
Vector Measures, Amer. Math. Soc. Math. Sur-
Measurability in Banach Spaces, Indiana Univ. Math. J.
26(1977),
663-677.
3.
Khurana, S. S. Strong Measurability in Frechet Spaces, Indian J. Pure
Appl. Math. (to appear).
4.
Schaefer, H. H.
5.
Schwartz, L. Certaines Propriete*s des Measures sur les Espaces de Banach,
Sere. Maurey-Schwartz, 1975, Ecole Polytechnique, No. 23.
6.
Sion, S.
Topologica.l Vector Spaces, Macmillan: New York, 1971.
On Analytic Sets in Topological Spaces, Trans. Amer. Math. Soc.
96(1960), 341-354.
7.
Talagrand.
(1975),
8.
Sur Une Conjecture de H. H. Corson, Bull. Sci. Math.
(2), 99
211-212.
VaradaraJan, V. S.
Measures on Topological Spaces, Amer. Math. Soc. Transl.
48(1965), 161-228.