Pub
.
Ma
.
UAB
Vol
.
29
Ns
1
Ab il
1985
ACHARACTERIZATION
OF
THE
RADON-NIKODYM
PROPERTY
Osca Blasco
de la
C uz
§l
.
INTRODUCTION
.
The
aim
o
his
pape
is
o
gi e
a
new
cha ac e iza ion
o
he
Radon-Nikodym
p ope y
in
e ms
o ma ingales
in
X- alued
O liczspaces
.
Le
X
be
a
Banachspace
andpu
Z
:
,
he
Lebesgue
measu able
se s
in
00,11
.
I
is
well
known
(see
[l])
:
(1
.1)
.
X
has
he
Radon-Nikodym
p ope ywi h espec
o
[0,1]
i
and
only
i
e e y
boundeduni o mly
in eg able
ma ingale
in
LX[0,1]
,
(
n
,B
n
)
whe e
o(UB
n
)
_

,
is
con e gen
in

LX[0,
1]
.
We
a e
in e es ed
in a
gene aliza iono
his ac
.
In
his
pape
we
shall
p o e
he
ollowing
Theo em
(1
.2)
.
Le
~
be
a
Young
unc ion
wi h
he
¿S
2
-condi ion
.

X
has he
Radon-Nikodym
p ope y
i
and
only
i
e e y
bounded
ma ingale
in

LX

,

( n
,B
n
)

whe e

o(U
Bn
)

_

,

is

con-
e gen
in
LX
.
The
de ini ions
and
he
main esul s
ela ing
o
X-
alued
ma ingales
and
O liczspaces
may
be
oi
:nd
in
[1]
and
[2]
espec i ely
.
We
a e
going
o
deno e
by
0 a
Young
unc ion,
a
.nd
LX
=
{
:
[0,11

-,>-X~
s ongly
úcasiu able
wi h espec
Lebesgue
measu e
s
.
.
p( ,O)
=

0(11 (x)l1)dx
<
-}
.
0
Le
~
be
he
complemen a y
Young unc ion
o
~
.
We
1
~~ (x)~~
~g(x)1dx
wi h
{
:[0,11
-~
X
s ongly
shallw i e

1
I l
l
o =

sup
{
g
e
Lyá
,
p(g,~)
1
1}
and
L
o=
measu ablewi h

U
ilo
<
W}
I is
well
known
ha
Lo
is
a
ec o
space
and
ll li~
is
a
no mon
i
.
Besides,
LX
=
LX i
and
only
i
0
e i ies
he
lá
2-

_
condi ion
.

I
is
easy
o
p o e
ha
he
con e gen e
and
he
boundedness
in L~
and
L.
a e
equi alen using
he
X X
ollowing
ac
:
(1
.3)

Suppose

~

e i ies
0
2
-condi ion,
i
.e
.
he e
exis s

K
>
0

and

T
'k
0

such

ha

QS(2 )

4
K«
)

o
all
~!
T
hen


Tm+2

See
(2]
7
page
158
.,

o
a
p oo
.
2
§2
.
PREVIOUS
LEMMAS
.
Lemnia
1 .
I (
n
,
n e N)
is
a bounded
sequence
in
LX
,
hcss
(
n
,
n
e N) is a
boundeduni o mly
in eg ablesequence
in
L
1
X
I
he eexis s
m,
helongs
o
(N
.
wi i

p( ,0)
L'
1/K
m
P oo
.
Fo
a
Young
unc ion
we
ha e
(2
.1)
~
-
:> °°

as
+

,
and
by
(2
.1)
we
ob-
ain
li
n
11
1
¢
p(
n
,~)
+
A
,
whe e
.
A
is
a
eons an
.
LX
We
ha e
only

o
show
ha

j
E

Il
n(x)
Ildx

0

as
m(E)
->
0
.
Gi en
E
> 0
,
by
(2
.1)
,
he eexis s
T
>
0
such
ha
(2
.2)

l

>
sc

o

>T

whe e

sup

p(
n
m

4
c
.
n
Le
d
=
E/2T
.
I
m(E)
<ó
and
deno ing
An
={x
: j I
n
(x)
11

:1
T}
(1
E

and
Bn
= {x
:
ji
n
(x)11
>
T}
(1
E
we
ob ain
Lemma
2
.
P oo
:
E
11
n
(x)11
=
11
n
(x)~~dx
+
A
n
+
J
li
(x)jIdx

<
B

n

(2
.2)
TM(E)
+
E
2c
0
~(11
n
(x)11)dx<E
I
0
e i ies
he
~
2
-condi ion
hen
he
simple unc=
:
ions
a e
dense
in
LX
.
Gi en
e
LX,
sünce
is
s Ongly
measu able, he e
exis s
a
sequence

(
n

,
n
e
N)

o
co
un ably
alued
unc-
ions
such ha
(2 .3)

~ n(x)- (x)

<

ñ

o
almos
al l

x
e

[0,1]
and o
all
n e
N
.
Suppose
n =
-
57
-
x
n
m
XE
whe e
m=o n,m
x
n
m e X
and

XE

a e he
cha ac e is ic
unc ions
o
'

n,m
disjoin
measu ables
se s
.
Since
2JI
n
(x)jj
<
211 (x)II,
ñ
a
.e
.
and
0
is
a
con ex unc ion
.2
we
ha e
2
n
e
LX
.
The e o e, he e
is
a
numbe pn
e N
such
ha
Lemma
3
(2
.4)

Jl,
.

E

_

0(211
n
(x)
11)dx
<
m=pn
n,m
We conside
he
simple
unc ion
By
(2
.3)
and
(2
.4)
1
n
pn
gn
=

x

.
n,m
X
m=o
E
n,m
J
o
~1
I (x)
-
gn
(x)j
-
j)dx
¿
2

Jo
0(21
j
(x)
-
n
(x)
1
jdx
1
21
(2~~
n
(x)
-
g
n
(x)dx

2
0(ñ)

+
ñ
0
Since
« )
->
0
as 0+
he
p oo
is
inished
.
Le
(BT
,
T e
1)
be
a
amily
o
sub-a- ields
o

.
Suppose
~
.wi h
á2
-condi ion
.
I
n
con e gs
o
in LX
hen
E(
n
/B
T
)
con-
e gs
o
E( /B
T
)
uni o mly
in
BT
,
whe e
E(
./B
Z
)
deno es
he
condi ional
expec a ion
ela i e
o
Bz
.
P oo
I
may
be
p o ed
,
wi h-a
sligh ,
modi ica ion
in
he
a gumen
in [lj,pa
:g°
122
ha
i
B
is a
sub-a- ield
o
hen
p(E(G/B)M

p(g,0)
o all
g e
Lo
.
Now,
gi en
e > 0
,
le
m
o
be
a
numbe
such ha
max

(~
m
+2

,

m
)

< c

whe e

K,T

a e
he

cons an s

in
2
o

Ko
he
0
2
-condi ion
.
Sinc
.e

1
1
n
-
1 1
~
(n -)
hen
p(
n
-
,O)
->
0 (n
->
-)
so
he e
is a
numbe
n
o
such
ha
i
n
>
n
o
we
ha e
p( n- ,O)
<
m
.
This
implies,by
Ko
(1
.3)
and
he
i s
esul
in
he
p oo , ha
1
JE(
n
-
/B
T
)110
<
e

o

n ~ n
o
and
i
is
ue
o
all
Te
I
.
93
.
PROOF
OF
THE
THEOREM
(1
.2)
.
Suppose
X
has he
Radon-Nikodym
p ope y
and
le
(
n
,B
n
)
be
a
bounded
ma ingale
in
LX
wi h
a((JB
n
)
_
By
lemma
1
and
(1
.1),
he e
is a
unc ion
in
LX
such
ha
n
-~
in
LX
and
n
=
E( /B
n
)
as i
may
be
seen
in
[1]
.
Since
he
con e gen eo ma ingales
in
LX
implies
he
con e gen ealmos
e e ywhe e,
we
ob ain,using
he
con inui y
o
0
ha
«
ll
n
(x)il)
-~
«
li (x)l1)
a
.e
.
and
by
Fa ou's
Lemma
.
1«
11 (x)11)dx

l~

lim

in

j
¢(11
n
(x)11
dx

~=

M
The e o e
belongs
o
LX
J
which
coincides
wi h
LX
.
We
shallp o e
ha
n
--~
in
LX
.
F om
Lemma
2,
we
see
ha
gi en
e >
0
,
he e exís s
a
numbe
m
o
and
a

sequence
o
simple
unc ions
such ha
(3 .1)

lis
m
-
il a
<

e/2

o

m

>

mo
and
usingLemma
3
wi h
Bn
,
he e
is
m
1
in
!N
such ha
(3 .2)

11E( -sm/BnM
O
.

<e/2

o

m ~ m
l
and
o
all
n
Since

o(U
Bn
)

=
T-

,
we
can
ake
he
unc ions

s
m
on
measu able
se s
om
lJ
B
n
.
I
s
m
=
Em
i
G
Bn
 o
i
=
1,
.
.
.,p
and
in
his case
E(s
m
/B
n
)
=
s
m
o
n = n
o
.
The e o e
i
n
!>
n
o
,
by
(3 .1)
and
(3 .2)
To
p o e
he
con e se
we
a e
going
o
use he
cha ac e-
iza ion
o
he
Radon-Nikodym
p ope y
in
e mso
ope a o s
:
Fo
e e y
T
:L
1
[0,1]

~p
X
he e exis s
a
unc ion
in
Bn
he
o- ixed
gene a ed
by
he
dyadicin e also
leng h
2
n

,

i
.e
.

Bn
=

o(l
n

i

'

i

=

0,
.
.
.,2n-1)

whe e
Le
m
be
a
ixed
numbe
such ha
m
>
max
(mo ,m
l
)
.
le

n
o
be
'a
numbe
such ha
ji
-
n110
`

lI
-
s
m
11
0

+

lis
m- nil~

=
=
jj -s
m
11
0
+
11E(sm
- /B
n
)jj
<
e/2
+
c/2
= e
1
such ha
T(q)
_

9P(x) (x)dx
o
~o
e L*1 [0,
1]

(see
[j,
page
(i3)
.
Le

T
:L~
[0,1]
-,oX
be
a boundedo
,
pe a o
.
We
conside
ii±l
In,i
=
`2n
.
2n)
2
n
_1
Le

n
=


2n
T
(XI

_)XI

.

I
is
easy
o
p o e

ha
i=o

n,i
n,i
E(
n+1
/B
n
)

=
n
and
ob iously

o(U
Bn
)
=
E-

.
Since
n
2ni
1I
n
11

=

Z1

2

1
T(XI

x,

,

i is
clea
ha
i=o

n,i
n,i
11
n
(x)jj
11
JITII
o all
x
e
[0,1]
.
Then
p(
n
,¢)
11
0(11TI1)

o all
n,
and
we
can
ind
a
unc ion
in
LX
such ha
n
-~
in
LX
.
This
is
equi alen
o
«
li
n ID)
--

«
ji il)

in

L
1
and
he e o e
he e
is a
subsequence

~
(l
i
n
(x)
1
J)

->

(

1
(x)
11)

ái
:e
.

Hencé
.
k
(x)~~)

¡TI
1)

a
.e
.

and
belongs
o
To
conclude
he
p oo ,
we
mus
only
p o e ha
.
1
(3
.3)

T
(s)

_~

s(x) (x)dx

o
all
simple
unc ion
on
0
U
B
n
-measu able
se s
.

-Fi s ,
.we,Shall
.p o e
ha
(3 .4)

n
=

E( /B
n
)
.
I E is
a
B
n
-measu able
se ,
1
n
(x)dx
=í
E
n+k
(x)dx
o
k~,l
lE
and
hen
i
is
su icen
o
p o e
ha
(x)dx
-~~
(x)dx
as
n-*oo
.I
is
clea
om
he
Holde
S
E
n

'

E
inequali y
i
11
n

-
(x)

-

(x)
11

dx~

11
n
-

l1XEU
'F om

(3
.4)
~
J
I

(x)dx
=~I

n
(x)dx
=
T(XIn,i)
n,i

n,i
and
by
linea i y
we
ob ain
(3
.3)
and
inish
he
p oo
.
SPAIN
REFERENCES
[1]
J
.
DIESTEL
and
J
.J
.
UHL
(1977)
.
Vec o
Measu es
.
Ame
.
Ma h
.
Soc
.
Ma hema ica
l
Su eys,
15
.
[2]

A
.

KUFNER
e
al
.

(1977)
.

Func ion
spaces
.

Noo
dho
In e na ional
Publishing
.
Rebu
ee
.
9
de
'no embne
dei
1984
Depa amen o
de Teo ía
de
Funciones
Facul ad
de
Ciencias
Za agoza