330
BEBR
ST
B385
No. 1601 COP
FACULTY WORKING
PAPER NO. 89-1601
2
Set- Valued Functions of
In
Two
Variables
Economic Theory
Nicholas C. Yannelis
J
College of Commerce and Business Administration
Bureau of Economic and Business Research
University of
Illinois
Urbana-Champaign
6 1969
BEBR
FACULTY WORKING PAPER NO. 89-1601
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
September 1989
Set-Valued Functions of Two Variables in Economic Theory
Nicholas
C.
Yannelis, Professor*
Department of Economics, University of Illinois, Champaign, IL
61820
Digitized by the Internet Archive
in
2011 with funding from
University of
Illinois
Urbana-Champaign
http://www.archive.org/details/setvaluedfunctio1601yann
ABSTRACT
Several properties of set-valued functions of two variables are studied.
(i) Caratheodory- type selections,
Specifically, we study the existence of
An application to
(ii) random fixed points and (iii) random maximal elements.
the problem of the existence of a random price equilibrium is also given.
:
Table of Contents
1.
Introduction
2.
Preliminaries
3.
Elementary Measure Theoretic Facts
4.
Carathe'odory-type Selection Theorems
5.
Random
6.
Random Maximal Elements and Random
Fixed Points
Equilibria.
INTRODUCTION
1.
Research
in
economic theory
[see for
example the
The
has necessitated the use of set-valued functions.
have been useful
this paper,
may be
it
open
/ (x
4>(x)
X.
in
)
G
<p(x
)
-1
(y
{<£
finite
:
We
for
y
argument
show
will
all
x € X,
£Y
}
that there
i.e.,
•
l
<f>~
(y)
exists
is
open and
an open cover of X.
is
refinement F' = (V a
:
6 A} of F.
a
we
X—
8a'-
*
can
[0, 1],
find
a
g a (x) =
eY
the set
_1
<£
nonempty
<f>(x) is
X
Since
We
is
£ Va and YiSa( x
.x
X—
2 Y (where
(y) = {x
e X: y e
f:X—*Y
all
4>(x))
is
such that
for
all
x G X,
paracompact there
is
the collection
an open locally
can use a standard result [see for instance
continuous
of
set
for
<f>:
a set-valued function such that for
Dugundji (1966)] on the existence of a partition of unity subordinated
i.e.,
above properties.
to describe the
continuous function
a
main purpose of
admits a continuous selection.
)
<f>(
discuss the
be a linear topological space and
convex, nonempty and for each y
is
we
Before
selections.
nonempty subsets of X) be
all
Since for each y e Y,
F =
Y
be a paracompact space,
A'
basic properties of set-valued which
existence of fixed points, the existence of continuous
instructive to outline a basic
2 X denotes the set of
x e X,
The
are:
and the existence of measurable
selections
Let
economics
in
Debreu (1959)]
classical treatise of
)
functions
=
1
for
all
{g a
x 6
to the
a e A
:
above covering,
such
}
Since F
X.
'
is
that
a refine-
a€A
ment of F,
/
X— Y
*
:
for each a
e A we may choose y a e Y such
by f (x) = J] £ a U'
)y a
Given the
-
fact that
that
F
'
Va c
is
-1
(y a
<£
).
locally finite
Define the function
and
<f>(
)
is
convex
a€A
valued, one can easily verify that /
4>(
•
)
:
X— Y
*
in
continuous and / (x
)
6
<p(x
)
for
all .v
eX,
i.e.,
admits a continuous selection.
Note now
that if
topological space and
X
<f>:
is
a conpact,
X—
2
X
is
a set-valued function
convex, nonempty and for each y e X,
there exists an
x*
eX
convex, nonempty subset of a locally convex linear
such that x* E
-1
4>
(v)
4>(x').
is
open
in
The proof
such that for
X, then
4>(
)
all
x e X.
<f>(x)
has a fixed point
is
i.e.,
follows directly by combining the
above
result together with the
(1966,
Theorem
Finally,
-* 2 Y
«^:
x
{x
e X:
(X,a)
measurable
To
and
see this let {v 1? y 2
{<?„:
n =
of
n
it
4>
is
b, (»
to a.
4>
n
=
(n
1,
eX
x
each
+^i(x) =
—
X
to a,
Y such
such that for each
and
4>(
•
)
f(x) e tfx)
that
closed
V
subset
Y
of
the
nonempty closed valued, then there
is
and
a complete separable metric space
is
for
all
x e X,
i.e.,
exists a
admits a
)
<f>(
set
selection.
for each «,
For
function
n V f 0} belongs
measurable function /:
Y
measurable space,
a
is
set-valued
a
4>(x)
Dugundji
[see for instance
2.2, p. 414)].
if
is
Tychonoff fixed point theorem
,
.
.
2
}
.
let
)
set
be a countable dense subset
(a-)
<^
B n (i) =
= ^(x)
(x)n B n ^(Mn (x)\
1. 2,
.
.
}
.
goes to zero. Define /
verified
easily
that
= 1,2,...) and
for
/
is
<
Y: dist(.x, v,)
—
},
(where
and
inductively
define
A/n (x) = min{/:
n (x)n
<j>
= 1,2,...)
(?
=
dist
<f>
n+l
B n+1 (i)
sequence of nonempty closed subsets
X—
:
6
i,
distance).
n
where
a decreasing
is
{.x
For each
in Y.
in
X
-* 2 Y
X
:
by
Then
f 0}.
and the diameter
00
Y bv /
(.x )
4>
n=i
measurable.
each closed subset
= n
In
V
n (x
fact,
of
Y
Then /
).
is
it
a selection
is
e X:
<f>,
and
check that for each
easy to
the set {x
from
4>
n (x)
n V f 0} belongs
Since
/"1 (F)
=
{a-
eX: f{x)eV}= n{x e
X:
n=0
we can conclude
that
Recent work
/~*(F)
is
an element of a,
i.e.,
/
economics and game theory
in
is
4>
n
{x)nV
^q>)
measurable.
[see for instance
Yannelis (1987), Kim-
Prikry-Yannelis (1989). Balder-Yannelis (1988) and Yannelis-Rustichini (1988)] has necessitated the use of set-valued defined either on the product space
TxX,
space and
U
p:
A'
TxX —
each fixed
is
2
>
/
y
e
.
a topological space) or in an arbitrary subset
(where
7",
4>(t
,
>'
•
)
is
is
a linear topological space)
lower semicontinuous, ^(
, •
is
of
(where
TxX.
T
a
measure
In particular,
a set-valued function
) is
is
if
such that for
nonempty, convex, closed valued
and lower measurable, one would
tion
from
and
for
able.
4>,
i.e.,
a function /:
each fixed
t
like to
TxX
e T, /(/,•)
is
-»
know whether
Y such
that
f
there exists a
(t
,
x) e
4>(t
,
Carathe odory -type
x)
for
(t,x)eT
all
continuous and for each fixed x e X, /
(
selec-
•
,
x
)
is
x
X
measur-
Thus, the concept of a Carathe'odory-type selection combines the notion of a continuous
selection
It
and measurable
selection, via the setting of a
product space.
turns out that under appropriate conditions one can obtain several Carathdodory-type
selection results adopting a similar
argument with the one outlined
continuous selection.
one can carry out a "parametrized" version of the above
argument [which
Specifically,
in turn
over the measure space
Our main concern
variables.
of
random
In particular,
is
to
prove the existence of a
based on an idea of Michael (1956)] where the parameter
/
ranges
T.
in
this
we
will
paper
is
to study several properties of set-valued
maps of two
prove the existence of Carathdodory selections, the existence
fixed points and the existence of
the above results to prove the existence of a
random maximal
random
elements.
price equilibrium.
Finally,
we
will use
PRELIMINARIES
2.
Notation
2.1
2A
denotes the
conA
denotes the convex hull of the
I
denotes the set of theoretic subtraction.
If
all
R
X—
<f>:
image
2Y
set
of
all
non empty), then
,
v
U
:
denotes the open ball centered
e)
int4
denotes the interior of A.
o\A
denotes the closure of A.
dist
denotes distance
diam
denotes diameter.
proj
denotes projection.
If
\
->
2
is
which
a set-valued function for
Y denotes the restriction of
U.
to
4>
denotes the /-fold Cartesian product of the set of real numbers R.
denotes the empty
X
If q
g X* and y
eX
at
x of radius
e.
set.
a linear topological space,
is
tional on X.
its
dual
is
the space
the value of q at y
is
X*
of
continuous linear func-
all
denoted by q
y.
•
Definitions
2.2
Let
Gq =
X
and Y be
G
i(x, v)
X
V
of 7;
open
in
X—
<t>:
X
sets.
x Y: y e
lower-semicontinuous
is
<f>
set A.
set A.
a correspondence (a correspondence
is
sets are
l
B(x
set
nonempty subsets of the
4>(x)}.
(l.s.c.) if
2Y
for every
is
The graph G^ of
If
X
the set {x
V
of
Y.
correspondence
and Y are topological spaces,
G X:
<j>{x)
n V f 0}
is
open
X—
<j>:
X—
<f>:
in
X
open subset
V
Obviously
of F;
if
if
Y
is
the set [x
u.d.c. is a
2Y
-
2Y
4>(x)
c
I'} is
weaker requirement than
the
set
said to be
for every
open sub-
G X:
a linear topological space.
G X:
is
is
said to be upper-semicontinuous (u.s.c.) if the set {x
said to be upper-demicontinuous (u.d.c.)
open half space
a
open
u.s.c.
4>:
in
4>(x)
c V)
X —
2y
»
X
is
for every
(X, a) and (Y, P) are measurable spaces and
If
have a measurable graph
to
able
i.e.,
if
say that
endowed with
is
above,
we
if
measurable
if
4>(x
)
We now
Y
is
We
(g) p.
4>
said
is
are often
and
a topological space
from a measurable space into a
<f>
understood that the topological
it is
Borel a-algebra (unless specified otherwise). In the same setting as
n V f 0} 6 a
{x e X:
correspondence,
a
is
a measurable space,
For a correspondence
Y
(X, a) a measurable space and
€ X:
{x
its
is
has a measurable graph,
<f>
2Y
>
belongs to the product a-algebra a
the Borel a-algebra of Y.
is
topological space,
space
G^
where (X, q)
interested in the situation
P = P(Y)
if
X—
<t>:
<f>(x )
a topological space,
for every
n B ± 0} e a
V
open
Y
in
B
for every closed
4>
said to be lower measur-
is
Furthermore,
.
said to be
is
<f>
in Y.
define the concept of a Carathdodory selection which combines the notion of
continuous selection and measurable selection.
Let
X
4>:
U
X—
x
2Y
*
= {(*, z
)
G
£/
— y
Us
=
{r
U,
={x eX:
f
:
eZ:
X
(x, 2)
(x t
/ (x z ) €
f/ia/
e
,
£/},
,
z
/or
);
if
z
eacr/
G
Z
instances
/
U
is
is
a
correspondence.
A',
/(
Let
spaces.
from
,
/ (x
•
,
z)
is
)
a
is
<j>
let
function
continuous
measurable
is
on
on
z)eU).
Z
is
is
for every
B e
)
is
l
f~ (B) = U n A
proper subset of
it
Y
is
U c X
for
x
Z
some
X
(where P = P(Y), F =/?(Z)).
x Z, and
this situation is
It
more
can be shown (see Proposition 3.1) that /
and f
/4
metric and /:
continuous and for each fixed :
jointly measurable
delicate situation
P,
a separable metric space,
each fixed x e X, /(x,
able, then
topological
selection
x e
eac/7
(X, a), (y, £) and (Z, F) are measurable spaces,
dard result that
more
<£(•*
be
empty-valued)
A Carathe odory -type
/or
c/zd
Y and Z
and
space
(possibly
a
x Z: ^(x, z)^0}.
/ jointly measurable
for
measurable
a
be
sue/?
*
If
be
(X, a)
X
€ a
x
:
U
we
—> Y,
F.
It is a
Z — y
eZ, /(•,:)
is
call
stan-
such that
is
measur-
turns out. that in several
delicate.
is still
However,
in this
jointly measurable.
Basic Theorems
2.3
We
close this section
by recalling some interesting
results,
[whose proofs can be found
in
Castaing-Valadier (1977)] which will be of fundamental importance in this paper:
PROJECTION THEOREM:
Let (T,
r,
p) be a complete finite measure space and
complete, separable metric space. If// belongs to r(g) £(T),
Y
be a
projection proj r (//) belongs to
its
T.
AUMANN MEASURABLE SELECTION THEOREM:
finite
Let (7\
measure space, Y be a complete, separable metric space and
valued correspondence with a measurable graph,
able function /
:
T
-+
Y
such that f(t)e<f>(t) p.-
G^erfg)
i.e.,
T—
<f>:
P(Y).
be a complete
r, /z)
2
Then
Y be a nonempty
there
is
a
measur-
a.e.
KURATOWKI AND RYLL-NARDZEWKI MEASURABLE SELECTION THEOREM
Let (T,
be a measurable space,
t)
Y be
a separable metric space
measurable, closed, nonempty valued correspondence.
tion
f:T-+Y
such that /
(t)
e
4>(t)
for
all
e
t
<f>:
T—
*
2
Y
T—
4>:
2 Y be a lower
there exists a measurable func-
T.
CASTAING REPRESENTATION THEOREM:
be a separable metric space and
Then
and
Let (7\
r)
be a measurable space,
Y
be a closed, nonempty valued correspondence.
Consider the following statements:
lower measurable, and
(i)
4>
(ii)
there
is
exist
cl{/ n (0:
Then
(i) is
measurable
n = 1,2,
equivalent to
functions
...}«M)forall/ e
(ii).
T.
fn
:
T—
*
F, (n =
1, 2,
.
.
.
)
such
that
:
ELEMENTARY MEASURE THEORETIC FACTS
3.
This Section contains several elementary results of measure theoretic character, which
are going to be useful in the sequel.
PROPOSITION
Y
3.1
U c T
be a metric space and
each
(i)
for
(ii)
for each
Moreover,
x eZ, f
/
let
(
,
x)
the a-algebra r
:
e T
t
the set
be such
Ux
i.e.,
£/x
V
for every
(/,
x)€U)
is
= {teT:
(t,
x)eU)
belongs to
open
e7\ /(/,•)
/
is
PROOF
e£/,
see that
if
:
«
Let x n (n =0, 1,2,
...
the smallest integer such that x
is
f f,(t,x) = f
(t
,
xn )
xn
xn e
(/,*)€
=0,
(n
U
l
.
We
}'
and
set
1,2,
U.
...)
By
(i),
(ii),
U*
are dense in
Ux
is
e
Z, there
,
now show
that
fp
t.
open.
Hence x e B(x n \/p) and
will
and
r.
continuous on
£/*
and
for
each
measurable with respect
to
U
n A
is
For p >
in Z.
1
set
e B(x n \/p) and
,
f p (t,x) =f(t,x n ),
(/,
xn ) €
U.
for
easy to
It is
belongs to the set
if (/, jc)
Observe that by assumption
let
Z
in Y,
be dense
)
[UXnn x(B (x„
this,
is
in
r® ^9(Z).
some A e
(/, A')
open
relatively jointly
{(t,x)eU: f(t,x)eV) =
for
be a separatable metric space,
{xeZ:
Then /
.
Z
that:
be such that for each
measurable on
/9(Z),
Z
x
be a measurable space,
r)
the set U* =
Z
x €
U— Y
is
(g)
Let (7\
:
(t
,
-)/u B(x m -)))nU.
,
p
m<n
Note
that
Thus,
is
p
let
fp
is
€>0
some n such
To
defined everywhere on U.
be such that
that
,v„
€ B(x, min(e,
x n ) e U, and therefore, / p (/, x)
relatively jointly measurable.
B(x,e)c U
To
this
is
1
l
.
//?)).
see
Since
Thus
defined.
end
let
V
be open
in
8
UT n
Since
checked
e
and /(
r
,
•
xn )
Ux
measurable on
is
= u [S n x (B(x n
-)
,
n=0
Thus, / p
is
{t
x
,
)
p goes
as
to infinity.
n A where A e
LEMMA
»
2
p
continuous on
Thus /
x)
(t,
is
is
U we
l
,
conclude that f p (t,x) converges
X be
3.1
is
measurable
relatively jointly
Let (T,
r)
be a measurable space,
U
which are of the form
X
be a separable metric space and
Consider the following statements:
lower measurable,
(a)
<t>(
(b)
for each
(c)
the set-valued function V:
is
clear, since
just the ordinary measurability with respect to an appropriate
a set-valued function.
)
is
fact that a
0(Z ).)
r (g)
:
The
relatively jointly measurable.
our case, with respect to the a-algebra of subsets of
cr-algebra; (in
T—
is
)
measurable functions
relative joint measurability
<t>:
u m<n B(x mt -))] nU.
I
p
€ T, f (t,-
t
limit of relatively jointly
U
can be easily
It
relatively jointly measurable.
Since for each
/
r.
that
f-\V)
to
Sn e
follows that
it
,
ft
x e X, the function
-» dist(x, 4>(x))
t
T—
2
»
is
X defined by
measurable
o\4>(t
)
i>(t
in /,
)
and
has a measurable
graph.
Then a <=>
b
PROOF
in
X
the set
<=>
(a
:
c.
<=>
4>~l (B(x, 6))
x e X, the function
belongs to
r for
t
each
<5
<=>
position 3.1,
c
):
=
—
>
can conclude that a <=>
(b
Note
b).
{t
eT:
dist(x,
0.
that
Since
4>(t)
is
lower measurable
,
6)
? 0} belongs
measurable
e T:
<fi(t)
in
t
if
nB(.xJ)^0)
if
to
for
{/
eT:
B(x
dist(.x, <f>(t))
e T: dist(x,
{i
ball
,
S)
Also note that for each
r.
the set
=
each open
<£(0) <
<^}.
<
6}
we
b.
Define the function /
/(•,•)
is
n B(x
is
<f>(t))
{/
)
<f>(
:
T
jointly measurable.
x
X
->
[0, oo]
by /
(/
,
.v
Hence, we can conclude
)
=
that:
dist(.v
.
^(/
)).
By Pro-
/ "HO) =
and
this
{(*
,
x)
3.2
Let (7\
:
there exist [f k
(ii)
for almost
all
,
fk
1, 2,
=
0}
{(*
x)
,
x e cl(M
:
)}
y)€ T x
For each n,
=
i
1,
1, 2,
each
«,
/'
=
of
...
1,
(£*:
2,
.
=
1, 2,
define
and
JT(
.
}
/
,
.
n
E,
k = 1,2,
(^ ):
fc
T? =
)
•
n
a dense subset of X(t).
be an open cover of
}
...
is
}
...
(/
has
a
It
can
be
define the set-valued function
X{t)nE?
if
/er,n
A-(r)
if
t<±Ttn
Y such
eT: X(t)n E? f
n
£,
theorem.
projection
the
,
{/
let
...,
€ X(t) n
Y: y
2,
that:
6 proj r (Gx ),
t
virtue
t
complete
a
2 Y be a set-valued function with a measurable graph.
*
such
}
measure space, and Y be
finite
a measurable function from proj r (Gx ) into Y, and
is
all
For
T n er by
T—
...
For each n -
:
= proj r {(/,
x Y,
k
for
diam(£,n) < 1/2*.
T
k =
:
(i)
PROOF
t
=
4>(t ))
be a complete
r, /z)
Let X:
separable metric space.
Tn
,
completes the proof of the Lemma.
LEMMA
Then
dist(*
:
X*:
0}.
measurable
T—
2y
Since
graph
checked
easily
that
in
that
by
A7XO =
Since the graph of
X?
the correspondence
AT has
X(t
)
is
{(r,
1
a
:
X
^ 0, hence the graphs of X* and
S
-*
Y
each m, {£,n
such that fft
:
i
=
1, 2,
...
y € X(t) n £*. Therefore,
e Xftt
)
}
t
{(/,
)
measurable graph. Also, for each
is
{/,*(/
)
/
y) €
r
=1,2,
i
for almost
all
/
€
...
T.
:
n,
i
=
1.
2,
after a suitable reindexing. gives the desired
.
.
.}
is
7? x
F: y
T.
if
By
G
*(/)},
and only
the
if
Aumann
there exists a measurable function
,
Fix
t
in T.
Let y e X{t
an open cover of Y. for each n. there
)
/
e F, AV(0 ^
have the same projection onto
measurable selection theorem, for each n,
/,"
eX(t)n E n u
y) e T? xY: y
dense
sequence f k
.
in
X(t
).
is
some
).
i
Since for
such that
Hence, the sequence
/,".
This completes the proof of the
10
lemma.
LEMMA
3.3
and
able function
Let (S
:
A € ax
a,) for
f ,
a2
<g)
=
i
n ^
proj s .(Ch
proj Sl (C7fc
If
proj
5l
(C
LEMMA
3.4
£
T\
®r ®r
for
rt
)
proj
H'
x
=
i
r 2 —>2
has a measurable graph,
PROOF
3
Sx —
Tx
x
7*
ax =
2,
able function and
rx
r2 ,
A € ax
<g)
Cw =
LEMMA
If
B,
n -
1, 2,
.
3.5
.
.
:
Let (7\
where
,
^,
(G n /T) € a lt
e
a,,
=
/
then
1, 2,
a2
then
2
1,
eT
r)
Gw €
T
T3
X):
rx
),
a2 =
,
(f
,
X,
2
y(t)e<Kt t
:
t2
(g)
(
For,
•
(GB n
proj Sl ((7B
(u~^ n ))
n A
)
e a lm
e a2
T x —* T 3
for
For,
all
be a measur-
i.e.,
r3 ,
2
),
and
x)}.
.
-^T 3
by
and A = G^.
Lemma
/l(/,
X )) € A } €
T
Z/Wn
into
(
•
ft
Then
(*,
x) =
.Si
— S2
/?:
Let
)>(/)•
is
a
measur-
3.3,
be a measurable space,
nB H B
5x
be measurable spaces, y:
xT
l
Hence, by
{(/,
proj
Therefore,
3
1, 2,
h:
.
S 2 /A
be defined by
i.e.,
S^ =
/I^^x
where
fc
be correspondences from
correspondences u n Wn (
.
be a set-valued function with a measurable graph,
2
2
Define
:
5l
=
w
all
W(t) = {x
Then
ax
^).
for
r —
Let W:
3-
2
(7*,-,
7\ x
</>:
n^)€
((7B
= u^iprojs^Cfc n A n ).
Let
:
able function and
G<t>
then
n ^n) € ox
(u~^ n ))
n
fc
a measur-
(^ 2 ) e a i-
n ^ c ) = Si/proj^CGfc n
proj Sl «7„
(c)
_1
/j
n^leaj,
proj Sl (Gh
If
(b)
= Ax n
)
5l
A - Ax x A2
If
(a)
:
S 2 be
Then
.
proj
PROOF
be measurable spaces, h: S x -*
1, 2,
)
<*!
= rx
®r
2
.
Z
be an arbitrary topological space and
Z
with measurable graphs.
have measurable graphs.
Then
the
11
PROOF:
Obvious.
LEMMA
3.6
Let (T,
:
able metric space
for every
x e Z,
PROOF
{s
e S:
•
Now
dist(x, W(s)) < A}
=
S =
G T: W(t
[t
be
A
let
dist(x,
± 0} belongs
)
positive
a
e S: W(s)n B(x,
{s
where
a measurable function,
)) is
First observe that
:
theorem.
tion
W{
dist(x,
Z
measure space,
finite
be a complete separ-
—* 2 Z be a correspondence having a measurable graph.
T
and W:
,
be a complete
t, /x)
0) =
oo.
by virtue of the projec-
to r
number
real
and
note
0} = proj r [C7vyn (T x 5(x,
A) ^
Then
that
A))].
Another application of the projection theorem concludes the proof.
LEMMA
3.7
Let (7\
:
able metric space,
and W:
correspondence V:
T—
r, /x)
T—
2
be a complete
Z
2 Z defined by
(where A any real number) has a measurable graph,
Lemma
3.6,
Define
:
g(
By Proposition
•
x)
,
the
r<g)
function
T
g:
x
i.e.,
G v € r®
— [0, oo]
Z
,
/?(Z).
by g (f , x ) = dist(x , W(t)).
measurable for each x, and obviously g(t,
we have
3.1
product a-algebra
is
be a complete separ-
be a correspondence having a measurable graph. Then the
V(t) = {x e Z: dist(x, W{t)) > A)
PROOF
Z
measure space,
finite
P(Z).
that
g
is
Hence,
jointly measurable,
Gv
= £ -1 ([A,
oo])
i.e.,
)
is
continuous for each
measurable with respect
er0 i9(Z),
i.e.,
By
V{
•
)
/.
to the
has a measurable
graph.
LEMMA
S —
If':
3.8
:
Let (5, a) be a measurable space,
2 X be a lower measurable correspondence.
J
Then
be a separable metric space and
the set-valued function V:
S —
2X
defined by
V(s) = {x e
(where A
is
any
PROOF
H'(
•
)
is
:
real
dist(.x,
A":
number,) has a measurable graph,
Define the function g: S x
lower measurable,
it
follows that g(
,
X
x)
^(s)) <
i.e.,
Gv
-» [0, oo]
is
A},
belongs to a
(x)
/3(X).
by g(s, x) = dist(x, W(s)). Since
measurable for every fixed x, for
12
{s
and the
fixed
latter set
G S,g(s,
5
G S:
•
)
is
•
)
))
< A} =
{s
e S: W(s )nB(x,\)^e>)
Hence by Proposition
continuous.
Gv
V(
W(s
,
belongs to a by the assumption of lower measurability. Obviously, for each
the product a-algebra a
i.e.,
dist(x
® P(X).
eV{s))=g-\(-oc,\))ea®0(X),
3.1
Let (T,
:
r)
S c T, S G
be a measurable space,
T—
*
<f>:
complete,
2 Y be a lower measurable correspondence and
f:S—*Y
^)M(/)n(/(O
lower measurable. (Here
PROOF:
g 7\
/
let fCl)
{i
0(/)
by
(7
= (v G
Lemma
eT:
is
y":
we understand
We must show
G
{/
V(0 n
r
2 r defined by
+ B(0,O)
=
e)
^ 0} g
£/
Observe
e).
—
7*
0:
that /(/) + #(0,
7":
= (4>(t)n U) + 5(0,
if
t for
/^
eT: (#/) n
=
{/
G 5: / (0 g 0(0} = proj T (Gf n
4>(t)nU
proj r (G,
nU)
<
n G e) e
e},
r.
0(
•
)
£/)
n
(f(t) + B(0,
lower
is
{t
eT:
V>(0
e))
of
Y.
+ 0}
<7„).
measurable,
has a measurable graph by
Therefore
£/
that
{t
open,
5").
every open subset
rl>(t)nUr<2) =
dist(v, 4>{t)
3.3,
that
and Y be
a
be a measurable function. Then the set-valued function
Since
measurable with respect to
Therefore,
= {(s,x): x
separable metric space. Let
For each
is
has a measurable graph.
FACT
is
g
3.1,
n
U
Lemma
? 0} G
r,
and
and
3.8.
this
since
Therefore
completes
the proof of the Fact.
LEMMA
0:
S—
»
3.9
:
Let (S a) be a measurable space Y, be a separable metric space and
,
2 y be a lower measurable correspondence.
Then
the
defined by
0(5) = (v
g K:
dist(v,
rl>(s))
=
0},
correspondence
0:
5
—
2
Y
13
has a measurable graph,
PROOF
Since
\p(
•
)
Ge G a
i.e.,
P(Y).
(g)
Y
Consider the function g: S x
:
lower measurable
is
it
—
[0, oo]
•
defined by g(s, y) = dist(y, ^(s)).
follows that for each fixed y
G Y, g(
•
,
y)
is
measurable,
for
€ S: dist(y,VCO)<e} =
{5
and the
e S, g(s,
5
able,
i.e.,
seen
that:
g
•
)
G9 = {(s,y)eS
Consequently,
8(
LEMMA
4>\
S —*
is
continuous.
•
x Y: y e 6{s)) =
PROOF
:
Y.
5 x
Define the correspondence
P(X\
Since
i.e.,
4>{
0(
•
)
•
)
is
= {q
7:
•
)
is
P(Y).
jointly
It
measur-
can be easily
0)
Y
be a separable metric space and
then have that for
all 5
e.
EX:
6:
S —
Ibea
2 X by
q -4>(s)>0).
lower measurable and closed valued, there exist measurable
cl{Ui(s):
5"
:
i
e
— Y
,
i
e
I
(where
I) = 4>(s)
S
oo
6(s)=
ug n (s),
n=l
where
g(s,y) =
,
has a measurable graph.
functions (Castaing representation) u {
We
y) e
(g)
•
nonempty compact valued and lower measurable correspondence. Let
9(s)
(g)
{(s,
g(
3.1,
product a-algebra a
to the
Let (S, a) be a measurable space,
:
nonempty subset of
Then G e e a
^0}
has a measurable graph as was to be shown.
)
3.10
2 Y be a
fl(y, e)
Therefore, by Proposition
measurable with respect
is
€ S: VCO n
belongs to a by the assumption of lower measurability. Obviously for each
latter set
fixed
{s
for
/ is a
all 5
e
S.
countable
set)
such that
14
g n (s) =
now
We
hi
:
i
show
5x1^
Ga e a
that
[0, oo]
by
e X:
{q
n hrl ((—
n
i&
,
oo)).
,
It
q
/,
—
w.CO >
end,
this
}.
each
for
easily seen that for
is
h~ {{—
l
,
oo)) belongs to
can be easily checked that G„ n = n
It
•
each
e
i
define
I
e S, h { (s
s
,
•
measurable, and therefore by Proposition
is
Consequently,
jointly measurable.
is
q)
•
€
i
To
h,(s).
•
continuous and for each q e X, h { (
/!,(-,•)
all
® P(X),
q) = q
fc,(s,
for
-1
/i,
((—
,
«»))•
a
® fi(X)
is
3.1
and so does
gj
Therefore,
"
•'€/
)
)
has a
(x)
/3(X).
•
oo
measurable graph,
Since
Ge
= u
the
we
G Un
a
B|i
Ga®
yS(A').
conclude that
,
n=l
C
i.e.,
0(
It
•
)
follows from
Lemma
3.5 that
U GQn ea
has a measurable graph. This completes the proof of
Lemma.
LEMMA
3.11
Let
:
(5",
a) be a measurable space and
pact convex valued and lower measurable correspondence.
nonempty subset of R
l
.
Define
S —
6:
6(
)
is
:
By
virtue of
S —
the correspondence h:
Since
(1975,
4>(
)
Theorem
*
Theorem
eB:
seS
is
measurable.
•
<f>(s)
>
be a compact, convex,
0).
by h(s) =
B /9(s)
lower measurable and compact valued,
is
3.1, p. 55)].
clings):
h{s) = {q £ B: for
and so
q
Himmelberg (1975,
4.4 in
2 B defined
Hence,
there exist measurable functions
all
be a nonempty com-
lower measurable.
PROOF
able.
B
Let
2R
B by
2
6(s) = {q
Then
S —
4>:
all
B/h(-).
/
e
/,
Since
•
=
B /h(s)
Since measurability of
l
,
<
i
it is
€/, (where /
We
<j>(s).
u,(s)
= {q € B: q
also
0).
It
is
a
)
<
0}
is
that
measur-
measurable [Himmelberg
)
countable
set)
can be easily checked that h(
•
)
is
we conclude
implies lower measurability of
such that for
have
then
= d(s) = {q 6 B: q-<t>(s)>0)
6(
4>(s
show
follows from the Castaing representation theorem that
S — R
u,:
iel)
<?
it
p. 59), it suffices to
0(
),
that
measurable,
that
0(
[Proposition
)
is
2.1
15
in
Himmelberg (1975,
Fn
If
,
=1,
(n
2,
Lemma
set of its limit superior
x = limx^, x^ e/7 ^,
{x e X:
now
is
LiF n = {x € X: x = limx R ,x n e F n
,
complete.
nonempty subsets of
a sequence of
is
...)
denote by LsF n and LiF n the
LsF n =
proof of
55)] the
p.
and
X, we
limit inferior points respectively,
will
i.e.,
and
=1,2,...},
A:
a metric space
n = l,2,...},
n—*oo
LEMMA
3.12
Let (7\
:
Let {F n
metric space.
n =
:
1, 2,
correspondences. Then LiF n (
PROOF
[see
•
)
if
An
a
is
Kuratowski (1966,
has a measurable graph,
LiF n (
sequence of
p. 335)],
X
be a separable
be a sequence of nonempty valued and lower measurable
...}
First notice that
:
336-337), that
n) be a complete finite measure space and
r,
)
sets,
LiF n {t) =
is
GLiF € r®
i.e.,
r
/9(A
).
closed valued [recall from Kuratowski (1966, pp.
LiA n and LsA n are both closed
{/
e X: lim
F n (t))
dist(/,
=
sets].
0}.
By
definition
Since by assump-
n—*oo
tion the
sequence of set-valued functions
Fn
complete measure space,
dist(/,
F n (t))
is
(
)
Fn
(
)
have
respect to the cr-algebra
t®0(X).
r (g)
F n (t))
is
Hence,
/, i.e., dist(
lim dist(/,
{(/,/)
e
T xX
:
lim dist(/
n—oo
jointly measurable, the set
n—*oo
a
It
follows from
•
,
•
F n (t))
)
is
is
Lemma
r,
n)
3.1
is
a
that
jointly measurable with
jointly
measurable with
p(X). Notice that
Gurn =
Since lim dist(/,
measurable graph and (T,
lower measurable.
are
continuous in / and measurable in
respect to the a-algebra
a
,
C LiFn
F n (t ))
=
0}.
belongs to r(g) P(X),
i.e.,
LiF n has
measurable graph. This completes the proof of the Lemma.
REMARK
graph
as
LsF n (() =
{f
3.1
well.
gX:
:
Under
the
Simply
Li dist(/
,
assumptions of
notice
Fn {t )) =
0).
that
Lemma
[see
3.12,
LsF n (
Kuratowski
•
)
has a measurable
(1966.
p.
337)]
16
Bibliographical Notes:
Lemma
3.1 is
an earlier result of Kuratowski (1966).
also Castaing-Valadier (1977)
due
to
Debreu
The argument
and Himmelberg (1975)
is
in
Proposition
3.1
taken from Kim-Prikry-Yannelis (1987, 1988) and
3.12
taken from Yannelis (1989).
generalizes
essence that of Kuratowski, [see
for similar arguments].
3.8 are
is
(1967).
Lemmata
3.8-3.11 are
Lemmata
new.
3.2-
Lemma
17
CARATHEODORY-TYPE SELECTION THEOREMS
4.
Below we
theorem implies the
that neither
The reader can
state three Carathe'odory-type selection theorems.
THEOREM 4.1
Z
Banach space and
Let (T
:
easily see
other.
be a complete
r, /i)
',
finite
measure space, Y be a separable
T—
be a complete separable metric space. Let X:
valued correspondence having a measurable graph,
i.e.,
Gx £
>
2 Y be a
and
r<S> P(Y),
<f>
nonempty-
T
:
Z —
x
be a convex valued correspondence (possibly empty-valued) with a measurable graph,
G<*>
£
r
® £(Z)
i.e.,
^(y), satisfying the following conditions:
(g)
each
(i)
for
(ii)
for each
y e Y,
eT,4>(t,x)c X(t
t
t
4>(t
,
•
,
)
x) €
(/,
for
)
x e
all
Z.
has open lower sections in Z,
i.e.,
y6 ^(/, x)}
Z.
y) = {x e Z:
4>~\t,
each
for
(iii)
2Y
Tx Z,
if 4>(t
,
is
open
x) f 0, then
in
<f>(t,x)
for each
6
t
7*,
and each
has a nonempty interior
in
X{t).
Let
and
tion
((
,
(7
for each
from
x) e
e 7\
/
and
PROOF
:
l
.
It
function /
a
exists
x e Z, /
in
4> x
T
=
{t)
x
4>(t
,
(
•
x)
,
x
is
for
Observe
Y.
x) e U). Then there
(/,
Moreover, /(•,•)
Let
measurable graph
x Z: <£(',*) ^ 0} and for each x 6 Z,
= {x e Z:
for each
U
continuous on
£/'
there
i.e.,
<f>,
U
T
= {(t,x)e
)
Theorem
For each
/
e T,
let
Wn
(t
)
e
7":
x)e
(/,
tf}
C/x
(/
,
x) e
and for each
<f>(t
,
x
for
)
e T, f (t
t
•
,
all
)
is
Notice that for each x e Z,
Z.
<£ x (
•
)
has a
that
follows from the Projection
}
x e
all
={t e T:
...
{/
exists a Carathe'odory-type selec-
such that /
measurable on
is
=
jointly measurable.
Ux
functions (v n ( -):« = 1,2,
U— Y
:
(7X
<f>
x
{t)^2>) = proj r (G
that
Ux
such that for each
= {x e Z: y n (t
)
e
4>(t
e
/,
,
x
r.
By Lemma
(v n (/)}
)}.
).
is
a
3.2 there exist
measurable
countable dense subset of X(t).
By assumption
(ii) H" B (/
)
is
open
in
Z.
18
Since by
dense in X(t),
is
Wn
{
•
)
It
1,
m
):
n =1,2,
Wn
{
=
{vv
e W:
dist(>v,
in
T
let
Fn (/)
=
/
)
define the operator
Z/HO >
l/2
} is
U
(
{y n {t
:
n = 1,2,
By Lemma
l
)
)
.
...}
3.5,
m by
m
}.
JFn (/)/u£r}(ff*(0)»-
...
and
)
Vn (t)
Obviously,
is
open
in
a locally finite open cover of the set U*
has a measurable graph, so does
Vn
•
(
)
by Lemmata
and
3.5
Let
3.4.
be a partition of unity subordinated to the open cover [Vn (t): « = 1,2,
}
...
•
...
interior in X(t
a cover of the set
is
}
1, 2,
can be easily checked that {Vn (t): « = 1,2,
for instance, for
};
...
:
(If) m
and
2, ...
Since for each n,
{g n (t,
{Wn (t) «=1,2,
follows that
it
x ) has nonempty
4>{t,
has a measurable graph. For each
For each « =
Z.
(t,x)e U,
for each
(iii)
each n =
1, 2,
...
let
...,
dist(x,
g n (t,x) =
Z/Vn (t))
£dist(x,Z/n(0)
Then
{,§„(/,
•
)
n =
:
1, 2,
...
}
a family of continuous functions
is
# n (/,
U —
l
•):
[0, 1]
such that
oo
g n (t, x) =
for
jc
^
Vn (t) and £#„(/,
=
jc)
for
1
all
(/,
x)eU.
Define
n=l
U^Y by f(t,x)=
/:
Since [Vn (t)
Y,8n(t, x)y n (t).
n =
:
1, 2,
}
...
x
locally finite, each
is
n=i
has
/
a
neighborhood
e T, /(/,•)
Nx
is
£ B (/,x)>0, x 6
a
f
a
convex
(t
a finite
Consequently,
.
,
x) e
Nx
,
x)
sum
for
all
(/
,
= {x e Z: y n (t) e
elements
x
)
G
£/.
Since
measurable function by Lemmata
measurable for each x,
follows from Proposition
i.e.,
3.1
/(/, a)
that
vn (
is
Ln
and
3.5
a
/(•,•)
•
<f>{t
,
from the
y n (t)
measurable function. Since for each n,
is
finitely
)
(
•
)
and
a
2)},
i.e.,
convex
it
is
)
any
6
set
M, x
such
n
,
Z/Vn
,
,
it
is
Consequently,
Hence, for each n and x g n {
measurable function,
that
So/(/,x)
).
<f>(t,x).
follows that
Caratheodory-type selection from
is
each
for
therefore continuous on
for
v n (/
Hence,
has a measurable graph, dist(x
3.6.
is
Nx
many Vn (t).
Furthermore,
continuous.
is
Vn (t)c Wn {t)
of
only
intersects
of continuous functions on
/(/,•)
combination
4>(t
which
4>\
v
.
)) is
(
x)
/(
is
•
,
a
x)
Finally,
it
jointly measurable. This completes the proof of
19
the theorem.
THEOREM 4.2
Z
Banach space and
Let (T,
:
be a complete
r, /i)
measure space, Y be a separable
finite
be a complete, separable metric space. Let
T
<t>:
Z —
x
2 Y be a convex,
closed (possibly empty-) valued correspondence such that:
(ii)
for each
We
:
Ux
let
show
given
>
e
=
Y
0,
i
that
for
(/. .v)
e U,
=
</>(r,
(
fi(Z)
(g)
U
-
{(*,
(t,
x) € U) and
for
<f>(t
y
/
x)
>
e
0, let
gT
for 5
for each
£
set
=
(*)
« =
and
x) + 0, the
e Y:
WR
{.v n
dist(v,
1, 2,
,
<f>(t
...,
x
<
))
<f>.
By Lemma
3.4,
W*
(
•
)
{
>v
e W:
each
for
1, 2,
[<^(f
x) e
(/,
each
jc)€
S
=T
x
Z,X
dist(vv,
m on
and
follows from
It
Since
that
= Y, a
•
subsets of
Z/W) >
—
,
<f>(
has a measurable graph.
)
Z?(0, e),
i.e.,
<f>,
continuous on
is
)
Z.
in
we conclude
(
x) +
for
f/
e
.
Z
•
As
)
(ii)
each
for
Note
that
=r® fi(Z)
and
an open cover of U*.
is
Setting
•
,
5(0,e)]}.
open
}
4>(t
We
f/}.
be a countable dense subset of Y.
x) +
,
is
...
(/,
e T, f%t,
/
to
}
...
e
vn
define the operator
(W) m =
U* = {x € Z:
let
such that / £
3.8
For
0}.
Caratheodory-type Selection from
H7ne (0
e).
Lemma
in
1,2,...,
e Z:
{jc
x) 4
4>(t,
e T,
t
e-
{W£(t): n = 1,2,
=(r,x)e5"
m=
n =
:
x Z:
each
and
£/z
T
x) €
—* Y
C/
:
measurable on
we may choose
e T and
each
and
is l.s.c.
let
jc) is
,
•
measurable graph.
theorem
)
there exists a function / £
4>{t,x) + B(0,e) = {y
If'(5)
•
,
there exists an approximate or
separable
is
For each
<j>{t
r
begin by proving the existence of an approximate Carathe'odory selection.
e T:
{/
that
each x e Z, / £
Since
e T,
/
end,
this
x e X,
)
there exists a jointly measurable Carathefodory-type selection from
PROOF
To
•
,
lower measurable with respect to the a-algebra
is
4>(
Then
will
•
(i)
+ B(0,
has a
e)
the previous
in
by
}.
L.
For
[V£(t
;?
= 1,2,
...
,
):n = 1,2,
...
measurable
graph,
K^f) = H^e(0 = W$t )/U^ (W^r )) B
let
}
is
a locally finite
by
Lemmata
open cover of the
3.5
and
3.7,
rn
It
set £/'.
£
(
.
)
has
can be easily checked that
Since for each n
a
measurable
.
W£(
)
graph.
has a
Let
20
n = 1,2, •••}
e
{g n (/,x):
{y*{t ):n = 1,2,
},
...
be
a
=
for instance, for each n
subordinated
unity
of
partition
1, 2,
...
open
the
to
cover
let
,
distU,Z/Fn<(0)
£n(',*) =
£dist(x,Z/KflO)
jt=i
Then
{g'(t
,
•
«
):
1, 2,
...
}
a family of continuous functions
is
g*
(t
•
,
)
-»
£/'
:
such that
[0, 1]
oo
g£(t,x) =
£*#.*)-
x £V*(t) and
for
for
1
(r,x)e*7.
all
Define
n=l
/
£
—
£/
:
y by f
e
x) =
(t,
%g&t,x)yu
[V&): n =
Since
.
1, 2,
...
locally finite, each
is
}
x
n=l
Nx
has a neighborhood
sum
g^t,x)>
fore,
f %t
x € V*(t)c
0,
So f\t, x)
Lemmata
is
x
,
continuous
a
is
)
G
each n, x g„(
,
T
A Z and
Wm =
4>(t
,
Therefore f (
x
,
)
+
B (0,
3.6, dist(.x,
x)
is
e)
for
U
all
Z /V*(
a measurable function.
is
an approximate or
e-
+ 5(0,
e)]},
U— Y
continuous on
£/*
(c)
/,(/,*)€
/,_!(/,
Y
4>{t,
which
x) n
=
1, 2,
/,(
=
/
(b) for
satisfying
satisfies
(/*(/,
and
(a),
x) + 5(0,
Fn
£
Consequently,
•
(
•
)
4>(t
f%
/
(a),
,
that
G<Kt, x) + 5(0,
e).
e).
There-
has a measurable graph, by
•
,
x)
is
x e Z. Hence,
for
measurable for each
<f>\
v
.
Now we
can
such that
...,
x)
such
n
x) + 5(0,
,
a finite
is
measurable on
Ux
,
1,2,...,
x) + 25(0,l/2'-1 ),
satisfying (a)
we have f lv ..,f k
I
t
and
/,(*,*)€*(/,*) +5(0,1/20,
is
Since
U.
vn
i.e.,
is
Consequently,
.
any
for
Carathebdory-type selection from
(b)
^t+iC* x) =
Moreover,
.
a measurable function for every
)) is
fiit,-)
-*
l
[<f>(t,z)
x) £
(/,
(a)
The existence of//
fk+v.
G Z: y n e
{_"
construct inductively, functions /,:
pose that
U
on
function
Nx
therefore continuous on
is
it
Hence, f%t,-)
Vtfjt).
a convex combination of elements from the convex set
and
3.5
many
intersects only finitely
of continuous functions on
e
f (t,-)
x.
which
=
=2,3,....
/
1, is
(b),
guaranteed by the above argument. Sup-
and
(b),
and
1/2*)).
Then
(c)
(c)
<t>
k+l (t,
for
/
for
x)
= 1,2
I
is
=k
k.
+
1.
We must
Now
find
define
nonempty, by the induction
21
hypothesis, and
Fact
can be easily checked that for each
k+l
that
3.1
it
•
(
<j>
,
Carathe'odory-type
approximate
By
lower measurable.
is
)
•
e T,
t
exists
+1
f k+x (t,x)e<f> k+1 (t,x)+B(0, l/2* )).
f k+1 (t,x)e
and
/(/, x)
e
converges
therefore
x)
for
all (r,
and for each x e Z, /(
is
THEOREM 4.3
T
<(>:
2Y
Z —
x
(i)
Y
(ii)
4>{t
PROOF
CLAIM:
)
We
:
uniformly
4>{i
,
U.
x)
measurable on
is
f7
£/
n
=
{(/,
n
(
•
Aftt) =
Note
by
{.v
that
Let
:
Z
has
)
U—
:
Ux
1, 2,
Since
Y.
»
Furthermore, for each
eT,
t
...
uniformly Cau-
is
}
/(/,•)
is
valued
closed
is
<j>
and
is (c)
continuous in U*
and therefore, by Proposition
3.1,
/(
•
,
•
)
remains true without closed valueness
3.1
dimensional or
has a nonempty interior for
all (/
,
x
€
)
U.
begin by proving the following claim:
Under
x)e T
= (x €
(/)
that
the conditions of
Theorem
4.3 there exists a countable collection
such that for every
<j>
(t,x)€U,
{/
(t
,
x)
:
F of
f e F}
is
x).
PROOF
Un
such
»
then
{/,:/ =
(c),
/
to
x) €
Carathebdory-type selections from
dense in
U— Y
:
if either
x
,
,
•
By
is (b).
The statement of Theorem
:
finite
is
f k+1
This completes the proof of the theorem.
jointly measurable.
of
follows from
It
is l.s.c.
)
+ 5(0, 1/2*)) +5(0, l/2* +1 ) c (f k (t,x) +25(0, 1/2*)) which
(f k (t,x)
<£(/,
•
But
+1
f k+1 (t,x)e 4>(t,x) + 5(0, l/2* ) which
chy,
k+i(t,
the above argument (the existence of an
there
selection)
4>
e Z:
Z
x
:
{£" n
:
n =
:
1, 2,
<f>(t,x)n
(/,jc)e
Un
).
dist(.x,
u^A^t
)
=
Z/Un (t))
Un
(t ),
}
En
±
Note
measurable
a
...
and
be a convex open basis of
<d
}€
r®
For each
B{Z).
that for each
t
e T,
graph.
For
>
1/2*}.
By Lemma
for
each
/
each
e T, A?
is
Y.
k
3.7,
t
U n {t)
is
For each n =
eT
and each w,
open
in
=1,2,...,
/*#
•
closed in Z.
)
1, 2,
and
...
,
set
Moreover,
Z.
/
eT,
let
has a measurable graph.
Define
4> k
T
x
Z
—
2
y
22
^
c\(<t>(t,x)nE n )
x eA£(t)
<fi?(t,x) =
4>U,x)
Since for each
t
V
subset
eT,
A£(t)
of
<f>£ (
•
,
•
)
/£(•,) from
^
•
•
,
V ^
<t>(t,x)n
<f>(t,x):
•
V
± ®) = {(t,x):
(/
x
,
eF }
/
):
We
dense
is
now need
will
in
4>{t
x
,
)
for all
an interior point of a segment
K
which are not
below are due
facts
FACT
If
:
FACT
(y,
=
1,
4.2
K
2,
...
}
We
-
are
all
\p(t
,
in AT is in
Michael (1956,
K
K) c
fj&
,
'"
•
),
<f>,
k
set
« =1,2,
,
and
is
a closed,
it
...
Then F
.
is
convex subset S of K, S ^ K, such
a
is
The
in S.
that if
set of all ele-
of
K
is
either closed or has an interior point or
will
be denoted by I(K). The following
p. 372).
K
Y
of
is
K.
1"/1
a
Banach space Y,
If
b
max(l
that
can be easily seen that
be a nonempty, closed, convex separable subset of
+
follows
convex subset of a normed
S, then the whole segment
be a dense subset of K.
"
K
a closed
any supporting
it
This completes the proof of the claim.
U.
If
V?Q>,
)
M
f° r
,,!,)
'
a" d
* =
£< l'-
e I(K).
now ready
to
complete the proof of Theorem
T
x
Z
Define
is
is
any convex subset
Let
:
-"•
then
in
dimensional, then /(cl
finite
and
4.1
to
*) €
,
the following notions.
linear space, then a supporting set of
ments of
(t
En n
x) n
,
there exist Carathe'odory-type selection
4.2
countable collection of Carathe'odory-type selections from
{/
c\(4>(t
B
Let F be the collection of
).
Moreover, since for every open
is l.s.c.
)
(/ )
x^4 (/)}€r(g)j3(Z),
0,
By Theorem
lower measurable.
is
<££(/,
{(/,*): 4>kU,x)c\
r,
x e A?(t)}u {(t,x):
Z,
closed in
is
x g A?
if
).
\p:
Moreover,
rp
—
is
2 Y by
rp(t
,
x
)
=
cl
lower measurable.
type selections {g k (t,x>. k =1,2,
...
}
dense
<f>(t
,
By
x
4.3:
Since for each
).
the
in ^(/,
t
eT,
above claim there
x)
for
all (/,
x) e
U.
4>(t
.
)
is l.s.c.
so
exist Carathe'odory-
For each k =1,2,
23
,
let
fkU>x) =g (t,x) +
1
:=1
lb:
By Fact
formly,
4.2,
it
e
I(ip(t,
By Fact
4.1,
f
(t,
/
^
all (t
,
x) €
e T, /(/,•)
x) e
max(l, \\g k (t,x)-g 1 (t,x)\\)
I(rp(t
,
x))
is
c
£/.
Since the series defining / converge uni-
continuous and for each x G X, /(
<£(/,
x)
if
either
(i)
or
(ii)
of
Theorem
•
,
x)
is
4.3 are
This completes the proof of the theorem.
Bibliographical Notes:
Less general versions of
Rybiiiski (1985).
in
x)) for
follows that for each
measurable.
satisfied.
/(/, x)
g k (t,x)-g 1 (t,x)
Theorem54.1
Theorem
-
4.3 are
4.2 are given
due
to
Kim-Prikry-Yannelis (1987, 1988).
by Castaing (1975), Fryszkowski (1977) and
Applications of these theorems in economics and game theory can be found
Yannelis (1987), Kim-Prikry-Yannelis (1989), Yannelis-Rustichini (1988), Balder-Yannelis
(1988) and Yannelis (1989b).
24
RANDOM FIXED POINT THEOREMS
5.
Let (7\
correspondence.
for
all
We
that
4>
be a metric space and
f:T—*X
measurable function
If there exists a
e T, then we say
t
I
be a measurable space,
t)
T
<f>:
2X
X—
x
*
such that /
(/)
e
<f>(t
be a
,
f
(/ ))
has a random fixed point.
begin by proving a random version of the Kakutani-Fan-Glicksberg fixed point
theorem.
THEOREM
5.1
Let
:
n) be a complete finite measure space and A' be a compact,
(7*, r,
convex, nonempty subset of a locally convex, separable, metrizable linear topological space.
Let
T
<p:
x
X—
>
(i)
Then
<f>
•
,
<f>(
{(/
2 X be a nonempty, convex, closed valued correspondence such that:
,x
)
lower measurable,
is
)
e T x X:
(ii)
for each fixed
has a
random
PROOF
t
4>{t
,
x ) n V ± 0} belongs
e T,
<f>(t
S
=T
X
x X,
conclude that F(
for
•
)
each fixed
/
)
to r
(g)
V
subset
of
6
Aumann
the
x
:
all
/
G
T
/
—X
e
T.
F
{x
r®
= Y, a =
€ X:
e T,
^(X) and
the correspondence
T — 2* by
:
dist(x
has a measurable graph,
ip(s
,
)
<£(/
=
GF e
i.e.,
4>{t
,
•
):
X
,
x
<£(/
,
))
x
=
)
0).
for 5
r(g) ^(A').
It
= U *
,
in
)
Lemma
we
can be easily checked that
—> 2 X satisfies
T —
3.9,
the conditions of the
all
2 X satisfies
all
Hence, for
all
the conditions of
measurable selection theorem and therefore there exists a measurable function
such that
Since
7", i.e., 4>(
set
is u.s.c.
F(t) f 0. Consequently, the correspondence F:
7*,
the
0(X),
Fan-Glicksberg fixed point theorem [see for instance Glicksberg (1952)].
/
X
fixed point.
F(0 Setting
,
Define the correspondence
:
every open
for
i.e.,
•
,
4>{
)
,x(/
•
,
has a
)
)
6 F(t
is
)
for almost
closed valued
random
all
/
e 7\
i.e.,
we conclude
dist(.v
(/ ), 4>{t
that x(t)
e
.v (/ )))
,
<t>(t
,
x
fixed point. This completes the proof of the
(/ ))
=
for almost
for almost
Theorem.
all
25
The
below
result
THEOREM 5.2
is
random vision of Fan's Coincidence Theorem, [Fan (1969
a
X
Let
:
compact convex nonempty subset of a
be
separable and metrizable linear topological space
Y and
X—
2 Y be
Let
ure space.
least
T
7:
x
X
—> 2 Y
/x:
T
x
one of them compact valued correspondences such
and i(
(i)
/x(
(ii)
for each Fixed
•
are
•
,
)
, •
•
/
)
every
€
t
«£7(i,jc), z e
all
PROOF
Since 7(
•
•
,
e
t
G T,
two nonempty, convex, closed and
•
):
X—
2 Y and 7(/,
T
and
/i(/
,
x € X,
every
x ) and
there
number A >
a real
r— X
exist
compact valued,
three
such that y
•
):
X—
2Y
y € A\
points
-x =
A(//
- 2
).
such that f(/, •x*!/)) n n{t x*{t)) f
,
T.
T
x
X—
2 Y by H7 (/,
and //(•,•) are closed valued and lower measurable and
)
at
that:
the correspondences n(t,
Define the correspondence W:
:
v) be a complete finite meas-
r,
are lower measurable,
there exists a measurable function x*:
for almost
(T,
convex
u.s.c.
for
(iii)
Then
and
let
locally
)].
follows from
it
Theorem
measurable. Define the correspondence
<t>:
= {x
4>{t)
4.1
in
T—
2
x
)
=
-y(r
,
that
n n
)
(t
,
x
one of them
at least
Himmelberg (1975)
x
W{
•
,
•
)
is
).
is
lower
X by
eX: W(t,x)t2>).
Observe that
G
4>
=
{(t
y
x)eT
x X: x
e<f>(t))
=
{(t,x)eT x X: W{t,x)±<2)
T
= {(t,x)€
and the
G<t>£ r
latter
® P(X).
set
It
belongs to r(g)/3(X) since
•
,
•
)
is
W(t,x)n Y
4>:
T
—> 2 X satisfies
Theorem and consequently,
all
7*0},
lower measurable.
follows from Fan's Coincidence Theorem, that for each
Thus, the correspondence
Selection
W(
x X:
the conditions of the
/
Therefore,
e 7\
4>(t)^<z>.
Aumann Measurable
there exists a measurable function x
':
T
—X
such that
26
x'(t)e<j>(t) for almost
all
in
t
T,
i.e.,
i(t
x
,
n
*(t ))
n(t
,
x
*(/
for almost
+
))
all
/
This
in T.
completes the proof of the Theorem.
An
immediate corollary of the above
COROLLARY
5.1
Let
:
result
Let
T
7:
that for each fixed
random
Y and
each fixed
=x
eT,
t
n(t,
Let x e
(iii)
)
and
) is u.s.c.
•
and
is u.s.c.
and
of
Theorem
t
/x(
•
x *{t ) €
f(f ,
tion that (7\
x *{i
5.1
v)
t,
is
))
5.2
•
r,
v
)
be a complete
meas-
finite
lower measurable. Then 7(
) is
•
,
each fixed
Theorem
:
complete
a
t
*
)
,
it
is
X—
x
2 X by n(t
has a
(simply
= {x
)
Clearly for
}.
z =
notice
x 6
n(t
,
x) and
since
that
X
A
/z(/,
finite (or cr-finite)
exists
all
remain true
5.1
if
we
•
)
is
/
a
€ 7\
measure space, by the
replace the assump-
fact that (7",
In this case one only needs to observe that in the proof of
e T, W(t,
(0, 1)
convex
is
for almost
x*(t)) f
e
eT.
and Corollary
5.2
x
convex, lower measurable, nonempty, compact
such that 7(/, x'(t)) n
t
,
Hence, by the previous theorem there
u.s.c.
dences) and therefore, the correspondence
valued and
)
T
n:
satisfied
is
for almost all
ply a measurable space.
for
^(
By choosing u €~t(t,x),
eT.
T— X
•
,
+ X(u - z) = \u +(1 - X)x e X).
REMARK
5.2
7<f,
X
measurable function x*:
i.e.,
(T,
2 X be a nonempty, convex, closed valued correspondence such
Define the correspondence
:
e T,
t
assumption
v
>
let
fixed point.
PROOF
valued.
X—
x
5.1.
be a compact, convex, non-empty subset of a locally convex
A'
separable and metrizable linear topological space
ure space.
Theorem
is
has a measurable graph by
(as
<j>:
is
it
T
Lemma
the intersection of
—* 2 X
3.1,
is
^(
•
closed valued.
) is
two
lower measurable.
4>(
sim-
Theorem
correspon-
u.s.c.
Since
r) is
•
)
is
closed
One can now
appeal to the Kuratowski and Ryll-Nardzewski measurable selection theorem to complete the
proof of Theorem
5.2.
THEOREM
5.3
:
Let (7\
compact convex subset of
r, /i)
be a complete
a separable
finite
Banach space
convex, closed valued correspondence such
that:
Y.
measure space, and
Let
4>:
T
x
X—
»
X
be a nonempty
2 X be a
nonempty
27
Then
<f>
4>(
(ii)
for each
has a
random
PROOF
that
/
(/
for each
It
:
x) e
,
•
,
4>(t
,
lower measurable and
) is
(i)
e T,
t
is l.s.c.
fixed point.
follows from
Theorem
x
x)e
)
for
e T, /(/,•)
t
4>(t,-)
all (/
,
T
,
t
)
= /
(r
,
x) -
x.
It
X
selection
T,
jointly measurable,
is
theorem there
x*(t)eF(t),
F
7-+2x
:
X
-*
is
,
T
:
x)
x
X
-*
X
such
measurable and
is
jointly measurable.
F(/) =
by
{jc
e
A':
g(/,
F
exists a
Therefore, for each
has a fixed point.
has a measurable graph.
measurable function x
x*(t) = f(t,x*(t))e
i.e.,
/(•,•)
•
/
x) =
0}.
where
follows from the Tychonoff fixed point theorem that for each fixed
e T, the function /(/,•):
Since g
x A\ and for each x € X, f (
continuous. Moreover,
is
Define the set-valued function
g (/ x
4.2 that there exists a function
*:
Hence by
T— X
the
e T, F(t) f 0.
t
Aumann
measurable
such that for almost
all
/
in
This completes the proof of the
<f>(t,x*(t)).
theorem.
REMARK
ness of
5.2
:
The statement of Theorem
x
X
—> 2 X if either
(i)
Y
is
finite
(ii)
<f>(t
<j>\
T
,
x) has
dimensional or
a
nonempty
The argument
is
now Theorem
4.3 instead of
REMARK
(i)
for
(ii)
4>
The argument
interior for
must now appeal
all (/
,
x
similar to that adopted in the proof of
5.3
:
Theorem
The statement of Theorem
each fixed
t
e T,
<t>(t
the
to
same with
Theorem
4.1.
that
)
e T x
X.
Theorem
5.3
except that one must use
4.2.
,
has a measurable graph,
is
remains true without the closed valued-
5.3
•
)
5.3
remains true
has an open graph in
i.e.,
G^ e
r (g) fi(X)
X
if
we
replace
(i)
and
(ii)
by:
x X, and
®/3(X).
adopted for the proof of Theorem
5.3
except that one
28
We
this Section
conclude
by proving a random fixed point theorem
weakly
for
(w-
u.s.c.
set-valued functions, which has found useful applications in mathematical economics.
u.s.c.)
However, before we
Let
we
our result
state
will
We
denote by S? the
F:
T
2X
,
X
n) be a finite measure space,
(7~, t,
denote the space of equivalence classes of
-
need some notation.
i.e.,
S} =
set
all
Let
<j>
:
T
X -valued
L x (/i,
let
Bochner integrable function on (T,
A')
r, /x).
Bochner integrable selections from the set-valued function
eL^X):
{x
THEOREM 5.4:
of
be a separable Banach space and
x(t)eF(t)
X—
x
»
n-a.e.).
2 X be a nonempty, convex, weakly
compact valued
correspondence such that
lower measurable,
is
(i)
4>(
(ii)
for each
t
(where
w-Ls
{x n
(iii)
•
,
4>{t
e T,
<£(/,
•
)
has a weakly closed graph,
weak
denotes
=1,2,...} converges
n
:
,
)
to
limit
x
w-Ls
i.e.,
4>{t
,
whenever
superior)
xn) c
<f>(t
,
x),
sequence
the
,
T—
x) c F(t) n-a.e., where F:
»
2X
a
is
lower measurable, integrably
bounded, weakly compact, convex and nonempty valued correspondence.
Then
4>{
PROOF
assumption
3.1] that
)
(iii) it
is
has a random fixed point.
Define the set-valued operator
:
nonempty.
for
•
a
V>
:
S?
— TF
by
t/>(;c)
= Sh. ,«(•)) In view of
follows from Diestel's theorem [see for instance Yannelis (1989b),
weakly compact subset of L x (n, X). Obviously S?
is
Theorem
convex and by virtue of
Kuratowski and Ryll-Nardzewski measurable selection theorem we can conclude that S?
the
is
Sf
,
We now show
any weakly open subset
valued,
{x n (
it
•
)
:
suffices
m =
we must show
1, 2,
that
.
to
.
.
}
that
V
show
V
is
of Sf.
that
w-u.s.c.
i.e.,
Since Sf
is
has
a
V(
be a sequence
"
)
in Sjr
the set
(.x
G S?
:
tp(x)
weakly compact and ^(
weakly
converging
closed
in the
graph.
c
•
T'} is
)
is
To
L^m. X) norm
open
in
Sf
weakly closed
this
to
x(
end
•
)
let
G Sp,
29
(5.1)
(By passing to a subsequence
w-Ls
Let z e
weakly
show
z
to r
rpix^) =
z eip(x).
follows
(5.2) z
w-Ls
Combining now
z
z(t)
e S#
.
il(
e
.))
=
(ii)
S^
x
,
.
})
(
1, 2,
,
.
.
.
4>(t
,
x^t))
w -Ls
<f>(t
each
for
t
x n (t ))
,
*
e
\p(x
*), i.e.,
(5.3)
€ T,
4>{t
and
this
,
)
•
has a weakly closed graph
*(t
)
G
4>(t
,
Bibliographical Notes:
Yannelis-Rustichini (1988).
Prabhakar (1983) and
fixed points
(1979).
is
(1989b)
that
proves
it is
x
*{t))
4>
(5.1).
n-a.e.
Theorems
Theorem
5.1
5.3
we have
that:
ii-a.e.
and taking into account that
Since
x(t)) n-a.e.
x
We must
n-a.e.
is
xp
is
convex valued we conclude
compact
weakly
Hence,
<f>
:
S?
—
2
F
valued
we
have
satisfies all the
of the Fan-Glicksberg fixed point theorem and consequently there exists
x
z^ converges
^-a.e.
Yannelis
in
4.1
to x(t) n-a.e.).
such that
in 5"/
}
z^t) G
i.e.,
Theorem
from
x n (t) converges
that
p-a.e. and therefore
and
i>(x),
k =
:
=^U).
Sfr. ,,(.))
may assume
if
<j>(t,x n (t))c<f>(t,x(t))
(5.2)
<f>(t,
It
)}
e con
(t )
Since by assumption
that
necessary
e S/ and z^ e
that
(5.3)
if
there exists {z^
,i.e.,
rf>(x n )
e con w -Ls {z^At
(t )
Mxjc
w-Ls S^. tXni)) = w-Ls
that
conditions
x' e S? such
that
This completes the proof of the theorem.
and
is
a
5.4
are new.
Theorem
random version of
a
5.2
result
is
taken from
in
Yannelis-
taken from Kim-Prikry-Yannelis (1987). The literature on random
growing rapidly, and perhaps one of the most complete references
is
Itoh
Applications of random fixed points in game theory can be found in Yannelis-
Rustichini (1988).
30
RANDOM MAXIMAL ELEMENTS AND RANDOM EQUILIBRIA
6.
6.1
Random Maximal Elements
Ibea
Let
nonempty subset of
We
ence correspondence.
read y eP(x) as "y
binary relation on
X
one may define P:
-» 2 X
is
said to have a
X
dence P:
Let P:
a linear topological space.
X
preferred to
is strictly
-» 2 X
by P(x) =
maximal element
if
{y
x".
For instance
€ X: y >
there exists
x €
-» 2 X be a prefer-
X
X
x).
if
>
is
a
The correspon-
such that P(x
)
= 0.
Several results on the existence of maximal elements with applications to equilibrium theory
have been given in the
Prabhakar (1983) among
by
be representable
depend on the
let
(T,
t, /i)
mapping from
of nature
states of nature,
/
T
we
will
allow for
now
x
X
into 2
X
large
We
.
Sonnenschein (1971) and Yannelis-
enough
read y
€
allow our preference correspondence to
random
measure space.
finite
is
We
preferences.
We
A random
P(t, x) as "y
is
T
function x:
x
X—
T— X
2X
as the states of nature of
we
consider to be
preference correspondence
strictly
is
said to have a
such that P(t x(t
,
The following two theorems on
eralize the ordinary (deterministric)
))
for almost
the existence of
maximal elements
and Yannelis-Prabhakar (1983). These theorems
all
/
is
a
is
the
maximal element. The correspon-
random maximal element
=
P
preferred to x at the state
can introduce the concept of a random maximal element which
natural analogue of the ordinary (deterministic) notion of a
dence P:
T
interpret
to include all the events that
denote the u-algebra of events.
We now
".
T
that
instance
Notice that the above preference correspondences need not
i.e.,
be a complete
r will
interesting,
others.]
for
[see
utility functions.
and suppose
the world,
literature
if
there exists a measurable
in T.
random maximal elements below genresults given in
will also play a
Sonnenschein (1971),
key role
in
proving random
price equilibrium theorem in Section 6.2.
THEOREM 6.1
:
Let (T,
convex, nonempty subset of
R
r, /x)
l
.
be a complete
Let P:
T
x
X—
finite
measure space and
A'
be a compact,
2 X be a correspondence (possibly empty-
31
valued) such
that:
V
open subset
for every
(i)
{(t,x)eT: con
of X,
P(t,
V
x) n
f 0} belongs
to
T®fi{X).
(ii)
for each
(iii)
for every
e
t
Then
€ 7\
t
P{t
•
,
is l.s.c.
)
T
measurable function x:
-*
X, x(t)$ con P(t, x(t))
for almost
all
T.
x
there exists a measurable function
T— X
*
:
such that P(t, x(t)) =
for almost
all
in
/
T.
PROOF
Define the correspondence
:
Proposition 2.6 in Michael (1956) for
lower measurable.
Carathdodory
f
x) €
(t,
=
£/*
{jc
rp(t,
e
A':
Ux ={teT:
t
e T,
£
(7
U
Let
for
all
(x)
By
fi(X).
and
r/f(
•
,
•
/
n (T x
e T,
ip(t
that
)
{x
}))
6\
T
x
X
-* 2 X
by
•
,
—> 2 X
X
) is l.s.c.
,
•
)
the
3.1,
U
V(',
=
{'
tfx
€T:
e
/(•,)
*) n
)
JT
7*
{(t
,
x) €
6
T
x X:
T
(i)
it
x X:
is
•
,
such
measurable
is
open
0(
continuous
is
•
)
is
that
on
on
each
for
0}
By
x).
4.3 there exists a
f:U—*X
/(•,*)
Furthermore,
Theorem we have
,
and by assumption
that
is l.s.c.
set
x) - con P(t,
,
(eT, /(/,)
each
xeA\
each
tff(t
function
a
exists
by
By Theorem
rl>(t,x)^<Z)}.
for
= proj T ({(/ x
=
Proposition
x
Notice
=
by
and
for
virtue of the Projection
proj T (C7
Hence,
U
x) e
ij>(t
there
^(/,x)^0} = {xeA':
= (x € X:
of
x X:
i.e.,
(t,x)eU).
measurability
€ T,
t
T
x) e
\p,
{t,
(l,x)E[/}
ogy of X, since for each
t
{(/,
from
selection
x)
=
all
T
\f>:
in the relative topol-
follows at once from the lower
4>(t
,
x) n
X
?
e>)
belongs to
that
rj>(t
,
x ) ? 0} n (T x
{x
}))
tl>(t,x)t<Z>)
r.
is
jointly
measurable.
Define
the
correspondence
32
{/(/,*)}
if
(t,x)eU
X
if
(t,x)(£U.
0(/,x)=
Lemma
By
9(t
•
,
):
that 6(
point,
X
,
•
i.e.,
—* 2 X
)
in T.
0,
x(t) = f
in
(t,
Hence,
T which
x(t))
e
x
5.1,
T— X
:
for almost
all
(t
x(t)) = con P(t, x(t)) for
rp(t,
in
/
7\
(/,
have
that
convex and nonempty valued and
is
Hence by Corollary
lower measurable.
is
we
(1983)
Suppose that for a non-null subset S of T,
/
/
Clearly, 6
is u.s.c..
there exists a measurable function
all
(iii).
Yannelis-Prabhakar
in
6.1
x(t))
implies that P(t, x(t)) =
U
$
T
6:
such that x (/
x(t
,
all
))
E
e S,
/
all
t
in
7*.
€
6(t
random
x(t
,
Then by
U.
e T,
))
fixed
for almost
the definition of
a contradiction to assumption
and consequently
for almost
)
t
can be easily seen
it
2 X has a
X—
x
each
for
rj>(t
,
x(t
))
=
for almost
all
This completes the proof of the
Theorem.
Theorem
tinuity
can be extended
6.1
assumption
More formally we can
(ii).
THEOREM 6.2
Let
:
(7", r,
fj.)
Banach spaces by strengthening the con-
to separable
state the following extension
be a complete
measure space and
finite
convex, nonempty subset of a separable Banach space.
of
Let P:
T
x
X—
>
Theorem
A'
6.1.
be a compact,
2 X be a correspon-
dence (possibly empty-valued) such that
(i)
{(/,
x, y)
(ii)
for
each
€Tx
/
e
T and
in the relative
(iii)
for each
(t
,
the relative
(iv)
for every
I
e
T.
X
x
x X\ y e con
each y e
X
P(t
,
the set
x
)}
e
r (g)
0(X)
P -1 (' y) =
,
{x
(g)
p(X),
e X:
ye
P(t, x)}
is
open
norm topology of X,
)
e
T
x X,
if
P(/
,
x
)
then P(t, x) has a nonempty interior in
^
norm topology of X,
measurable function x:
T
—>
X, x(t)$ con P(t, x(i))
for almost
all
33
Then
e
i
there exists a measurable function
:
T—
X, such that P(t, x(t)) =
for almost
all
r.
PROOF:
T
V>:
x
x
The proof
2 X by V(', *) = con P(/, x).
X—
*
(1983) for each
relative
e T and each y e
t
X
By
virtue of
4.1
in
5.1
e X: y
6.1.
Define
Yannelis-Prabhakar
x)}
erf>(t,
is
open
in the
there exists a Carathe'odory selection from
proof of Theorem
as in the
Lemma
the set rp-\t, y) = {x
norm topology of X. By Theorem
One can now proceed
Theorem
almost identical with the proof of
is
6.1 to
Below we indicate how versions of Theorems
6.1
\f>.
complete the proof.
and
6.2
can be easily proved by com-
bining the deterministic maximal elements results given in Yannelis-Prabhakar (1983) with the
Aumann
measurable selection theorem.
THEOREM 6.1
P(
(i')
PROOF
•
•)
,
:
is
Theorem
6.1
remains true
can be easily checked that for each fixed
all
t
the assumptions of
in
tion
Theorem
5.2 in
T, the correspondence P{t,
such that P(t
P(
,
•
)
x
,
is
t
)
=
one replaces assumption
for all
/
•):
in T.
lower measurable, the
/
M: T
-» 2 X
in T, the
Yannelis-Prabhakar (1983,
X—
2 X has a
X—
correspondence P(t,-):
p.
Therefore, for each
t
i.e.,
T
x X: x
2 X satisfies
there exists x t
e
X
e T, M(t) f 0. Since by assump-
set
P(t
,
x) n
which
X
is
? 0),
denoted by A
that
= {(t,x)e
It
239) and so for each fixed
maximal element,
belongs to r0j3(I), and so does the complement of the set A
GM
by
by M(t) = {x 6 X: P(t,x) = 0}.
A ={(/,*)€ T x X: P(t,x)?2>) = {(t,x)€ T x X:
Observe now
(i)
lower measurable.
Define the correspondence
:
if
eM(t)) = {(t,x)eT x X: P(t,x) = 0}
=
{(t,x)€T x X:P(t,x)?0) =A<,
c
c
.
34
and the
latter set
graph.
We
r® f3{X)
belongs to
as
,
x (t
=
))
for almost
THEOREM 6.2'
tion
(i) is
all
T -* X
Theorem
:
)
has a measurable
measurable selection theorem
such that x (t
e M{t
)
)
jP(
PROOF
for almost
all
t
in
T,
i.e.,
6.2
remains true
if
assumption
dropped and assump-
(iii) is
lower measurable.
) is
, •
•
The proof
:
M(t) = {x e X: P(t,x) =
can conclude that M(t
one can show that
GM
Using Theorem
0}.
for
f
)
similar with that of
is
all
6r® /3(X).
/
Theorem
Define
M: T —
Yannelis-Prabhakar (1983,
in
5.1
6.1'.
2 X by
239)
p.
we
Adopting the argument of the previous Theorem
in T.
Appeal now
to the
Aumann
measurable selection theorem
complete the proof.
REMARK
(T,
t, /x)
is
6.1
Theorems
:
a complete finite
6.1'
and
remain true
6.2'
measure space by the
if
we
fact that (T,
replace the assumption that
r) is
defined by M{t) =
{X e X:
(this is also true if for
closed valued and
By
virtue of the
it
P(t, x) =
each
e
/
0}
T and
is
closed valued since for each
each y e X,
has a measurable graph,
it
is
i.e.,
P(/,
Random
6.2
{
,
t ,
<?,)
v)
is
open
t
e T, P(t,
in X).
measurable
•
)
is l.s.c.
Since A/(
(recall
2X
M: T —
Lemma
•
)
is
3.1).
Kuratowski and Ryll-Nardzewski measurable selection theorem, one can
x *(/)) =
for
all
t
e
T— X
such that
x*(t)eM(t)
for
all
T.
Equilibria
An exchange economy
{X P
,
also lower
assure the existence of a measurable function x'\
e T,
P~l (t
The
a measurable space.
proofs remain the same provided that one observes that the correspondence
/
ensure the
to
replaced by
(i')
to
•
This completes the proof of the Theorem.
in T.
/
was noted above. Thus, A/(
Aumann
can not appeal to the
existence of a measurable function x:
P(t
it
where,
E = ((X
;
,
P i5
e
;
):
i
=
1, 2,
...
,
N)
is
a family of
ordered
triples
35
(i)
X cR
(ii)
/>,:
(iii)
e{
'
The
the consumption set of agent
is
{
X — 2^
»
{
is
is
the initial
pair (e,,
6.1,
we
i.e.,
read
the preference correspondence of agent
endowment of agent
The
preference correspondence.
Section
,
where
/,
the characteristic of agent
is
P,-)
i
€
i.e.,
X
for
{
and
all
his/her
i.
initial
endowment and
interpretation of the preference correspondence
G ^\(*,)
.v,
/,
e{
/',
as "agent
{
is
as in
consumption vector yt
strictly prefers the
/
P
to
x,".
A
Let
= [q €
£<?,'-
ft:
(where
denotes
ft
the
cone
positive
of
R
l
).
For
i,
and
i=i
p e A,
B
Di(p) =
{
(p) = {x
(*t
Define
€
(p)
p
:
•
aggregate
i=i
?
{
5,(p): P.Cx.)
the
As
X
e
x < p
n
£,(/?)
•
denotes
e,}
budget
the
= 0} denotes the demand
excess
demand
$
A—
:
of agent
set
2R
of
set
/'.
E
economy
the
for
agent
by
«=i
Debreu (1959)
in
n (~0)
A
0-
7^
a free disposal price equilibrium
pr/ce equilibrium
is
a vector
p e
A
is
a vector
G
such that
peA
such that
£ (/T ).
Conditions which guarantee the existence of either a free disposal price equilibrium or
now well-known
price equilibrium are by
We now amend
his references.
in the literature, see for instance
the deterministic
Debreu (1959) and
economy described above by introducing
randomness.
Let (7\
t, fi)
A random
triples (A't
(i)
,
Pit
finite
measure space.
exchange economy E = {(X
e { ),
Xi
be a complete
;
,
Pj, e
;
):
i
where
c R
l
is
the consumption set of agent
i,
=
1, 2,
•
•
•
,
N
}
is
a family of
ordered
36
(ii)
/*,-:
(iii)
£>,:
e
/
Notice that
T
x
A',
F
-»
/?'
now each
*
random preference correspondence of agent
random
the
is
the
is
initial
agent's characteristics,
Hence, randomness
framework,
nature
2
endowment of agent
/',
where
i,
e,(/)
€
X,-
for
all
T.
state of nature.
this
—
i.e.,
preferences and endowments depend on the
explicitly introduced into agents' characteristics.
is
£/>,(/,*,) means that "agent
>',
strictly prefers
i
at the state of
to x,
v,
In
t".
p e
A
S,(/,p) = (x e
X
For
Di(t, p) ={x t
demand
f
:
{
G
T
and
:
p
<p
x
A
—
define
•?,(/)}
the
and
the
Pf (t xt ) nBi(t p) *
£,(/, p):
x
€ T
I
,
,
0}.
random
budget
random
demand
Define
l
economy E by ?(f,p) = ££><(*•£) -
the
for
random
i
by
i
by
excess
N
N
2*
agent
of
set
aggregate
the
agent
of
set
E
i=i
*«•(*)•
We now
i=i
define the natural analogues of the ordinary concepts of price equilibrium.
A free
disposal
random
(-f2)
price equilibrium
is
a measurable function p:
Notice that
states of nature.
P (t
now
Hence,
environment
(deterministic)
for almost
^
for almost
))
the equilibrium price (or the
in this
notion
T— A
all /
»
in
—*
A
such that
in T.
/
such that
7".
market clearing price) depends on the
framework the market clearing price
price equilibria which
of
price
equilibrium
Bhattacharya-Majumdar (1973 Section IV,
recently to Weller (1982,
all
T
will
change from one
to another.
The concept of random
nary
a measurable function p:
n
,
€$(t, p(t
state of the
is
))
f (/
A random
price equilibrium
p. 75).
p.
45),
is
obviously a generalization of the ordiis
not
new.
It
Hildenbrand (1971.
can
p.
be
traced
to
427) and more
37
Below we provide conditions which guarantee
dom
price equilibria or
random
THEOREM 6.3:
Let
the existence of either a free disposal ran-
price equilibria.
£
T
:
2R
A—
x
be
random
a
aggregate
demand
excess
correspondence, satisfying the following assumptions:
(i)
For each
(ii)
$•
R
p
:
T
•
,
(iii)
for
all (t
(iv)
for
all
u.d.c,
is
)
measurable,
T
p) e
x A,
6.1
-+
A
e
there exists z
$(t, p(t)) such that p(t)
$ (t
:
,
pit
))
n
random
^0
(-ft)
equilibrium,
for almost
Observe that Theorem
i.e.,
all
tive
way
by
fixing
e T and considering
/
to
The proof we
prove the ordinary
PROOF OF THEOREM
F((,p) =
{q
e A:
We
will
6.1
and therefore
show
q- z >
that the
it
for
give
is
G-N-D
6.3
all
:
z
direct (starts
for
Theorem
f (/,
6.3
•
):
Debreu
A—
2R
.
does not use the
from "scratch") and provides an alterna-
result.
Define
the
correspondence
F:
7xA^2 A
by
e$(t,p)}.
correspondence F:
has a
[see for instance
the correspondence
important to note that the argument adopted to prove
theorem.
<
6.3 gives as a Corollary the ordinary (deter-
82)] simply
G-N-D
z
in T.
t
(1959,
it is
of
there exists a measurable function
Gale-Nikaido-Debreu (G-N-D) excess demand theorem
Also
V
subset
convex, compact and nonempty,
is
ministic)
p.
open
every
for
i.e.,
F.
such that
REMARK
p)
£ (/,
T
measurable p:
e
/
,
there exists a free disposal
—A
•
,
lower
is
)
$ (t
{(t,p)eTxA: c(/,p)nF^0}er(g)/?(A),
l
all
Then
,
•
(
GT,
t
T
x
A—
2 A satisfies
all
the properties of
random maximal element. By construction
the
Theorem
random maximal
ele-
ment, will turn out to be a random price equilibria.
(i)
Hie
p:
correspondence
T
—
F:
T
x
A—
2A
A, p(t) & F(t, p(t)),for almost
convex
is
all
t
in T.
valued
and
for
all
measurable
38
can be easily checked that for
It
from
directly
T
p:
By
all
t
(t,p)
F(t p)
,
T and
each q e
= {x: q
x >
set
W
convex. Moreover,
for
,
all
it
follows
measurable
in T.
eT,
t
A
F(t,-)
is
F~x (t,
the set
U.c.
q) = [p e A: q
= [p e A: ?(/,/>) C J7^}
open
is
in A.
A
6 T and each q e
t
R
an open half space in
0} be
Therefore, for each
is
p(t) £ con F(t ,p(t)) = F(t p(t))
that
(iv)
€ T x A,
virtue of Proposition 4.1 in Yannelis-Prabhakar (1983)
e
Vq
and
For each fixed
(ii)
t
A
-*
assumption
all
l
.
It
the set
suffices to
it
€ F (t,p))
is
Since for each
open
in
t
in
T,
show
A.
$ (/,
can be easily checked that
F _1 (/,
#)
open
is
that for
To
•
)
W
each
this end, let
is
u.d.c, the
=
F~x (t,
q).
in the relative topology of
A.
T
The correspondence F:
(iii)
S =
Setting
3.10,
T
x A, a =
we conclude
A—
2A
r(g) /9(A), <£(s
F(
that
x
•
)
,
is
)
=
for
and consequently, there
all
in T,
/
We now show
f (f
,
e
(f
,
p(t
))
))
exists a
A
T
n
n
e RL
r
= 0.
(-ft)
R
f0 and b e
(6.3)
x
e
$ (t
sup
ye-n
r
•
for almost
+
(-ft)
for
Since
closed convex cone, the sets
/•
0(5
)
F (t p
=
A
,
-+ 2 A satisfies
measurable function p
there exists z
then
otherwise,
p(/
p ) and
,
for 5 =(/,/?)
)
Lemma
in
:
all
the assumptions of
T— A
Theorem
such that F(t,p(t)) =
,
p(t
))
such that q
•
z
<0
for almost
all
/
in
7".
that (6.1) implies that
(6.2) f
F,
f (/
i.e.,
(6.1) for all q
Suppose
lower measurable.
lower measurable.
Therefore, the correspondence, F:
6.1
is
<;
f
(t
,
7"
A—
x
/T(f ))
and
such that
y <b <
inf
xe?(t.p(0)
t
teA,
all
:
all
r
•
x
2*
-ft
in F.
where
is
A
is
a
non-null
convex and compact valued and
can be
strictly separated,
i.e..
of
subset
-ft
is
a
there exist
39
and
Notice that b >
from
(6.3)
Hence,
that r
(6.2)
>
r
z >
•
Without
0.
for all z
loss of generality
es(t,p(t)) and
we may assume
for
all
that r
e A.
It
e A, a contradiction
t
follows
to (6.1).
holds and this completes the proof of the Theorem.
Notice that the dimensionality of the commodity spaces in Theorem 6.3
now provide an
extension of
Theorem
Banach space whose
ticular to a separable
Theorem below may be seen
dimensional commodity space and
6.3 to infinite
positive cone has a
We
is finite.
nonempty norm
in par-
interior.
The
as a generalization of the deterministic equilibrium results of
Florenzano (1983) and Yannelis (1985), but only
if
the underlying
commodity space
separ-
is
able.
THEOREM 6.4
Let
:
having an interior point u,
and
A
=
[p
e C*\ p
Y
be a separable Banach space,
C* =
u = -1).
•
e Y*\ p
[p
7"
Let ?:
•
A
x
x <
for
- 2y
all
C
the closed convex cone of Y,
x e C) ±
{0} the dual
cone of
C
be an aggregate random excess demand
correspondence satisfying the following conditions:
(i)
For
each
t
£(/,•): (A,
(ii)
$(',')
Y,
{(/
,
)
$ (t
lower
g T x A:
A
):
,
w*)^2 y is
is
p
e T,
—> 2 Y
in
the
weak*
topology,
open
subset
(i.e.,
u.d.c.),
measurable,
$ (t
u.d.c,
is
,
p) n
V
for
i.e.,
+ 0} €
r
<g)
every
u *(A), where P u *(A)
is
V
of
the Borel
a-algebra for the weak* topology on A,
Then
/
p)
(iii)
f (/,
(iv)
for
all
for
all
is
convex, compact and nonempty for
measurable p:
/
e
T—
>
We
T
x A,
0,
T.
T— A
such that j(/,p(/))nC
in T.
:
p) G
A, there exists x e$(t,p(t)) such that p(t)- x <
there exists a measurable function p:
PROOF
all (/,
begin by proving an elementary
fact.
^0
for almost
all
40
FACT
6.1
Ibea
Let
:
HausdorfT linear topological space,
C' =
having an interior point w and
C.
Then
r
•
u <
PROOF
any
for
x e X,
x <
X*: p
for
V
=
0, a
a closed
x e C) f
all
convex cone of
{0} the dual
X
cone of
eC*.
r
± Xx e V and
i.e., r
e.
Suppose by way of contradiction that
:
symmetric neighborhood
have that
[p
C
Vc
of zero with u +
consequently
±
Xr
•
Therefore,
contradiction.
C.
for
some
x e X, then
If
x = r(u ± Ax) <
r
•
u <
e
r
for
any
for
C\
€
u =0.
r
Pick a
some A e R A >
we
,
Hence,
0.
r
C\
x =
r
and
this
for each
completes the
proof of the Fact.
We now proceed
that of
Theorem
F((,p) =
Y
is
a separable
the arguments used in the proof of
fies all the properties
GF e
F(t
,
/^'(A)
r(x)
p{t
))
=
of
Theorem
A
for almost all
e
A
in
t
Banach space,
Theorem
e
f(/,p"(/))nC f
(6.6)
sup
yec
then
r
•
all
t
y <b <
z
e<(t,p)).
A
is
a
now
3,8, p.
compact metric space. Adopting
T
has to use
Lemma
measurable function p
:
x
2 A satis-
A—
show
3.9 to
T— A
*•
that
such that
,
all
/
p(t
))
such that q
z
<
for almost
all
/
in T.
u
in T.
in a non-null subset
inf
r
r
A of T, $(t,p(t))n C =
e Y* /{0} and b e R such
0.
By
the
that
x.
*e?(t,p(t))
and
Notice that b >
A
one
exists a
$ (t
for almost
separating hyperplane theorem there exist
£
all
by
i.e.,
there exists z
Suppose otherwise, then for
r
for
2A
that (6.4) implies that
(6.5)
if
A—
one can easily see that F:
6.3
6.2 (of course,
T,
x
same with
essentially the
is
weak* compact [Jameson (1970, Theorem
is
^'(A)). Hence, there
(6.4) for all q
We show
T
>0
e A: q -z
by Alaoglu's theorem
Moreover, since
123)].
{q
whose idea
6.4,
Define the correspondence F:
6.3.
First notice that
Theorem
with the proof of
G
r
int
e C*
C
.
Without
implies
loss of generality
(recall
Fact
6.1
)
we may assume
that
that r
r-u <0 and we
a A. In
can
fact,
replace
41
r
by
——
-
-r
follows from (6.6) that
It
.
r
•
z >
for
all
z
e
f (f
,
p (/
for all
))
€A
t
Hence,
iction to (6.4)
REMARK
6.2
:
(6.5)
holds and this completes the proof of
As we noted
earlier
Theorem
6.4
may be
Theorem
arguments adopted for the proof of Theorem
above deterministic equilibrium
that a version of
Theorem
Yannelis (1985) with the
THEOREM 6.4'
(ii')
:
£(•,•)
is
{(t,p)e
T
Suppose that conditions
Theorem
seen as a generalization of the
6.4
By Theorem
t
)
Replace assumption
measurable,
(i), (iii)
and
(iv)
W(
3.1 in
Yannelis (1985,
n C #0. Therefore, W(t
•
in
Theorem
every
f 0} belongs
of
Theorem
in
by
closed
to
subset
V
of
Y,
the
set
r® £ W *(A).
Then
the conclusion of
)
p.
)
597) for each fixed
for
?
all
e
t
f
•
(
,
Appeal now
),
all
t
i.e.,
$•
(/
,
p (/
))
n C
Bibliographical Notes:
deterministic results on
the
^0
for almost
all
/
e T there
x A:
f
(assumption
to the
ensure the existence of a measurable function p:
F,
t
exists
p e
t
A
such that
Observe that
T.
{(t,p)eT
e\V(t)) =
from the measurability of
has a measurable graph.
by
eA: ^(/,p)nC^).
to
in
6.4
6.4 are satisfied.
-» 2 K
T
Define the correspondence W:
:
follows, at once
i.e..
(ii)
for
i.e.,
r(/M)n V
G w ={(t,p)eT xA:p
It
wish however to indicate
measurable selection theorem as follows:
W(t) = {p
p
We do
prove the
to
can be easily obtained by combining the deterministic result
Aumann
x A:
above authors.
results of the
way
provide an alternative
6.4
Moreover, our
6.4 holds.
PROOF
,
a contrad-
6.4.
deterministic equilibrium results of Florenzano (1983) and Yannelis (1985).
f (/
,
w
•
T
Aumann
—>
A
(r,/>)nC ^0}.
(ii'))
that
Gw e
r
® P w *(A),
measurable selection theorem
such that
p~(t
)
e W(t
)
for almost
in T.
All the results in this section are new.
existence of maximal
They
generalize the
elements of Sonnenschein (1971), and
42
Yannelis-Prabhakar (1983) as well as the excess demand equilibrium existence theorems of
Debreu (1959), Aliprantis-Brown (1983), Florenzano (1983) and Yannelis (1985), among
ers.
oth-
43
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J.
Brown, 1983, "Equilibria
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S.,
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N. C. Yannelis, 1988,
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Balder, E.
M
Bhattacharya, R. N. and
Theory, 6, 37-67.
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F.,
C,
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1979,
Languedoc, No.
lexistence
"Sur
5, p. 14.
Castaing, C. and
M Valadier,
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New
Debreu, G., 1959, Theory of Value, John Wiley and Sons,
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Debreu, G., 1967, "Integration of Correspondences," Proc. Fifth Berkeley Symposium on
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Dugundji,
1966, Topology, Allyn and Bacon, Boston.
J.,
Fan, K., 1952, "Fixed Point and Minimax Theorems
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Fan, K., 1969, "Extensions of
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