PROCEEDINGS
OF THE
THE
PROCEEDINGS OF
AMERICAN
AMERICAN MATHEMATICAL
MATHEMATICAL
SOCIETY
SOCIETY
Volume
Number 2, February
Volume 99,
1987
99, Number
February 1987
A GENERALIZATION
GENERALIZATION OF
LYAPOUNOV'S
CONVEXITY THEOREM
THEOREM
LYAPOUNOV'S CONVEXITY
MEASURES WITH ATOMS
TO MEASURES
JOHN ELTON AND
AND THEODORE
THEODORE P. HILL
ABSTRACT. The distance
distance from the convex
convex hull of the range of an n-dimenn-dimenmeasure is no more than
sional
measure to the range of that
sional vector-valued
vector-valued measure
that measure
than
a is the largest
o.n/2,
an/2, where 0.
atoms of the mealargest (one-dimensional)
(one-dimensional) mass of the atoms
= 0 yields
sure. The case 0.a =
yields Lyapounov's
Lyapounov's Convexity
Convexity Theorem; applications
applications
are given to the bisection
bisection problem
problem and to the bang-bang
bang-bang principle
principle of optimal
optimal
control theory.
control
1. Introduction.
Introduction. The celebrated Convexity Theorem of Lyapounov
Lyapounov [8]
[8] states
that the range
range of a nonatomic finite-dimensional vector-valued measure
measure is compact
and convex (where
(where throughout this paper "measure"
"measure" means "countably-additive
"countably-additive
nonnegative finite measure").
measure"). The range
range may not be convex if the measure has
atoms (see for
for example, Figure 1) but, as is the main purpose of this paper to
prove (Theorem 1.2), a fairly
far from
from convex
prove
fairly sharp bound can be given on how far
the range
range can be, as a function of the mass of the largest atom. Intuitively, if the
atoms all have very small mass, the range is very close to being convex.
Throughout this paper, (X,l)
(X, i) will denote a measurable
measurable space;
space; M
JMn
n is the set of
measures on (X,l)
is a measure on
n-dimensional measures
... , J-Ln):
(X, .) (Le.,
(i.e., M
Mn
On): J-Li
,ui is
n = {(J-Ll,
{(iu....,.
= (J.tl,
for all i <
(X,l)
(X,5) for
~ n}; and R(;)
R(,u) is the range
range of ;,u =
(1 ... , J-Ln)
Mn.
n) E M
n . The
first theorem, a result of Lyapounov
states that the range
first
Lyapounov [8],
[8], states
range of every
every vector
measure (nonatomic or not) is always compact; this conclusion is classically proved
proved
in conjunction with the Convexity Theorem (see for
for example Diestel and Uhl [3],
[3],
Halmos [4],
[4], Lindenstrauss [7],
[7], or Lyapounov
Lyapounov [8]).
[8]). However,
However, only the conclusion in
= 1 will be used in the proof of Theorem 1.2, and this case is fairly
the case n =
fairly easy
to establish directly without convexity (cf.
(cf. Halmos [5,
[5, Problem 4, p. 174]).
174]).
a
1.1 (LYAPOUNOV
THEOREM1.1
THEOREM
(LYAPOUNOV[8]). 1/;
If ,uE
M
R(;)
Mn,
then R(
compact.
i) is compact.
n , then
To state the next theorem, the main result of this paper, some additional notation
are needed:
and definitions are
needed: coCA)
co(A) is the convex hull of A eRn;
c Rn; Ilxll
llxll is the Euclidean
norm of x ERn;
y; and d(x,A)
E Rn; d(x,y)
d(x, y) = Ilx
d(x, A) =
lIx- yll
yII is the distance between x and y;
from x to the set A.
inf{
d( x, y): YEA}
inf{d(x,
y E A} is the distance from
DEFINITION. For
D(A) = sup{d(x,
sup{d(x,A):
For A eRn,
DEFINITION.
A): x E co (A)}. (For
c Rn, D(A)
(For a set A,
D(A)
"dent size"
size" of A;
D(A) represents
represents the maximum "dent
A; see Figure 1.)
Received by the editors
July 26, 1985 and, in retised
editors July
re-rised form, December
December 16, 1985. Presented
Presented to
Received
the International
Mathematicians in Berkeley, California,
California, on August
August 4, 1986.
International Congress
Congress of Mathematicians
1980 Mathematics
Mathematics Subject
Revision). Primary
Primary 28B05,
28B05, 46G10,
46G10, 60A10; SecSubject Classification
CZasficafion (1985 Revisin).
ondary 49E15,
49E15, 93C15.
93C15.
phrases. Range
Range of a vector
measure, convexity
measures with
with
convexity theorem,
theorem, vector
vector measures
Key words
words and phrases.
vector measure,
atoms,
zonohedron.
atoms, dent-size
dent-size of a nonconvex
nonconvex set, zonotope,
zonotope, zonohedron.
Partially supported
NSF Grants
Grants DMS-83-16627
DMS-83-16627 (Elton)
(Elton) and DMS-84-01614
DMS-84-01614 (Hill).
(Hill).
Partially
supported by NSF
@1987
Mathematical
Society
Mathematical
American
(?1987 American
Society
+ $.25
$.25 per
$1.00 +
0002-9939/87
$1.00
per page
page
0002-9939/87
297
297
298
298
JOHN ELTON
ELTON AND
AND T.
T. P.
P. HILL
HILL
JOHN
2
1
1
2"
1
!
A
~
~~
~
~~~
~ ~
~--------.-----------.:--+
1
11
~
~l
P-i
FIGURE 1
The set A is the range of the vector Borel measure (J-Ll,
J-L2)
(/11, /2)
= A and
= 3; it
on [0,1] defined by:
by: ,1({0})
J-L({O}) =
J-L2({0}) =!;
J-Ll =,x
= ,U2({0})
A on (~,
= !,x,
= ,X
= !,x
'A on (0,
J-L2
~]; and J-Ll
/2 =
'A, J-L2
/2 =
ql =
(2,1]1] (where
(0,2];
,X
A = Lebesgue measure).
{J7
for all i ~
DEFINITION. Pn(a)
< n and all J-Li-atoms
< a for
P f(a)= {,t E M
n : J-Li(A)
/ii-atoms A}.
Mn:
ii(A) ~
coordi(
(So P
a)
is
the
collection
of
n-dimensional
vector
measures
none
of
whose
coordiPn(a)
n
nate measures
have atoms of mass greater
greater than a.)
measures have
J7
< an/2.
then D(R(J7))
an/2.
THEOREM
D(R(,u)) ~
1.2. If ,u E Pn(a),
THEOREM1.2.
P,(a), then
then R(,ut) is
are nonatomic, thenR(J7)
THEOREM
.... ,J-Ln
If /11, ...
1.3 (LYAPOUNOV
(LYAPOUNOV [8]). IfJ-Ll,
THEOREM 1.3
i,,u are
convex.
convex.
from Theofollows from
1.3 follows
The proof of Theorem 1.2
Theorem 1.3
1.2 will be given
?3; Theorem
given in §3;
rems
lemma.
1.2 and
and the following
following easy lemma.
1.1 and
rems 1.1
and 1.2
0.
LEMMA
and only
is convex
convexif and
B C R' is
A closed
set BeRn
only if D(B) = o.
1.4. A
closedset
LEMMA1.4.
some
is to prove
2. Purely
prove some
measures. The purpose
atomic measures.
purpose of this section is
Purely atomic
is purely
preliminary
each J-Li
where each
case where
purely atomic with
results corresponding
corresponding to the case
preliminary results
,ii is
of (not
finite set
set of
is aa finite
only
V is
atoms. Throughout
(not
number of atoms.
section, V
finite number
Throughout this section,
only aa finite
< n}.
for all
all ii ~
> 00 for
in R+.
... , rrn):
necessarily
R+ = {(rl,
{(ri,....,
distinct) points in
necessarily distinct)
ri ~
ri EE R, ri
n ): ri
DEFINITION
2. 1.
DEFINITION 2.1.
E(V) =
ZE
ixi: bi = 0 or 1};
C(V) =
{
Z tixi: ti E [0,1]}.
GENERALIZATION
LYAPOUNOV'S CONVEXITY
THEOREM
GENERALIZATION OF LYAPOUNOV'S
CONVEXITY THEOREM
C({a,b,c})
//
04/
a
o
299
b
+
C(a, c})/
b
/
0~~~~
/
/
/
o~~~~~~~~~~~~~
~
0o
FIGURE 2
FIGURE
(So ~(V)
RI and C(V)
Y(V) is a finite set of points in R+.
C(V) contains ~(V);
3(V); see Figure 2.
C(V)
zonotope [1],
zonohedron [2],
C(V) is a zonotope
[1], or zonohedron
[2], and most of the results of this section
may be rephrased
rephrased using that terminology.)
= C(V).
LEMMA
2.2. co(~(V))
LEMMA 2.2.
co(E(V)) ==
C(V).
PROOF.
PROOF. Routine. 0Ol
The next lemma states that C(V) can be expressed as the union of translates of
form C(V)
subsets of the form
~ n. Its proof is similar to an
C(V) (see Figure 2) where IVI
IVI <
argument
argument of Caratheodory (see [11, p. 35]).
= U{I(V\V)
LEMMA2.3.
VC
2.3. C(V) ==
c V,
V,IVI
LEMMA
U{~(V\V) + C(V): V
IVI <~ n}.
fix x ==
so fix
V cc V,IVI
Clearly
U{~(V\V) +
~ n},
D U{I(V\V)
V, IVI <
=
+ C(V):
Clearly C(V)
C(V) :J
C(V): V
n}, so
n, the conclusion
{ti}~l C [0,1] with
is trivial, so suppose m > n. It will be shown that there exist {t,}=
= 0 or 1 for
= Zi
for some jj <
tj ==
~ n so that x ==
L~l tixi,
tixi, and the conclusion will then
follow
follow by induction.
= tii
for all i <
Assume further
further that 0 <
< ti <
< 1 for
~ m (for
(for otherwise taking ti ==
= 0.
not all zero,
m > n, there exist constants {ai}IT
suffices).
{ai}~l not
zero, so L~l
suffices). Since m
aixi ==
Em aiXi
Let
= min{ti/lail,
= 1,
.. .,, m,
b ==
O},
--&
ai #
1, ...
m, ai
O},
min{ti/lai 1,(1
(1 - ti)/Iail:
ti)/Iai1 ii ==
00.
b
and observe that 0 <
<
< < 00.
= tk/lakl
< m.
for some
some 1 <
k~
m. Let
Let
CASE 1. b ==
CASE
for
~ k
tk/Iakl
PROOF.
PROOF.
L~l
E C(V), where {Xi}~l C
C [0,1]. If m
~
CV
Z>mltixi
V and {til
m<
tixi E C(V),
{ti} C [0,1].
{x}xil
°
°
°
M
i
ti=ti-(bsgnak)ai'
i==I,
,...m
... ,m.
ii = ti -(b sgn ak)ai,
= 0, Eim=V
Note that tk ==
L~l tixi == EmZ1
L~l tixi,
# 0,
tixi, and if ai #&
ti
(1 -- ti)
= 00 and
and ti:::;
ii 2'>t?tii < titi + ( lail
ail =
ti
: ti -ti-~Iail
lail = 1,
jaiIti)jail
jail
= 0, ti ==
= ti).
for each i (since if ai ==
so ti E [0,1] for
ti).
for some 1 <
k<
CASE 2. b ==(1-tk)/Iakl
CASE
(1- tk)/Iakl for
~ k
~ m. Let
Ni
ti==ti+(bsgnak)ai,
= ti+ (bsgnak)an,ic
i=I, ... ,m.
at{..1
= 1,
= j:m
Note tk
1 ,,m=
check as before that {til
L~l tixi
L~l1 tixi,
{ti} C [0,1].
tk ==
ixi ==
tixi, and check
[0, 1].
001
HILL
JOHN ELTON AND T. P. HILL
300
= n, then
any point
the distance
distancefrom
2.4. If V
and IVI
|V| =
then the
LEMMA
from any
point in
V C
LEMMA 2.4.
C [O,I]n
[0, 1]n and
< n/2.
vertexis ~
nearest vertex
0(V) to
to the
the nearest
the
parallelepiped C(V)
the parallelepiped
PROOF.
for all x, y E Rn
Rn and all t E [0,1],
[0,1],
PROOF. First it will be shown that, for
(1)
(1)
min{llx + (1
ty112} < IIxI12 + 11y112/4.
t)y112, IxI-
-
> (t
To see (1), first
y) ~
t)2)llyI12. Then
(t22 - (1(1- t)2)11y112.
first consider the case 2(x,
2(x,y)
2+
= IIxI1
2t(x, y)
11X112
+ t 211yl12 -- 2t(x,
Ilx - tyl12 =
ix
t211y112
ty112
2
< IIxW12
::;
IIxl1 2 +
t 211yl12 -_ t(t
+ t211y112
t(t2 -_ (1(1 - t)2)llyI12
t)2)11y112
2
2
< IIxII2
+ t(l+ 11y112/4.
=
=
IIxl1
t)llyl12 ~
IIxl1 +
IlyI12/4.
t(l - t)11y112
11xI12 +
< (t
<
yielding Ilx
+ (1 - t)yll2
t)2) 11y112
is similar,
similar, yielding
(t22 -- (1 - t)2)llyI12
The
2(x, y) ::;
The case
case where
where 2(x,
t)y112 ~
IIx+
2
IIxl1
+ IlyI12/4.
IIyII2/4.
IIX112
Next, let V =
C(V). Applying (1) n times
= {Xl,
= E~l
{Xli ... . ,,x
xnn } and fix xX =
En tixi EE C(V).
?
1}
satisfying
of
{0, I}
implies the existence
{Di}
i:: 1 E {O,
{&I}LE
~2
n
Zixii-x
n.
=
ZE(i
-ti)xi
<
n -n/4=n2/4,
proof. 0O
which completes the proof.
< o:n/2.
PROPOSITION
then D(~(V))
2.5. If V
VC
C [0,
D(E(V)) ~
an/2.
PROPOSITION2.5.
[0, o:]n,
a]n, then
PROOF.
X E
Y(V)): x
PROOF. By Lemma 2.2 and the definition of D, D(~(V))
D(E(V)) = sup{d(x,
sup{d(x, ~(V)):
< max{D(~(V)):
C(V)},
c
D(Y(V)) ~
max{D(E(V)): V C
C(V)}, which together with Lemma 2.3 implies D(~(V))
< n}.
? o:n/2
n}. Then Lemma 2.4 (and rescaling)
for all such
V,
rescaling) implies D(~(V))
an/2 for
V, IVI
D(Yf(V)) ~
VI~
V.
V.O
0
The next example shows that the bound in Lemma 2.4 (and hence in Proposition 2.5 and Theorem 1.2) is of the correct
n; in fact the best possible
correct order
order in n;
bound (which is not known to the authors) is at least n/8
n/8 for
for general n and at least
n/4
n/4 if n is a power of 2.
1
k 1
= 2
2kk ~
let m =
2k+1,
and let
let {Wi}
be
< n <
EXAMPLE 2.6.
Fix n, let
EXAMPLE
+ , and
be the
the m -- 1
2.6. Fix
< 2
{wit}lm1
f~nctions on m poin~
wi..lWj
l}m, wilwj
mean-zero Walsh
Walshfunctions
{-1, l}m,
points (see [12]). Then Wi
wi E {-I,
::1:
$ j,j, and wil
for i =1=
wi..l11 for
for each i, where 1
for
and m == 4)
(so kk =
= 2 and
W1
-1, -1),
-1),
= (1,1,
(1, -1,
W =
=
= (1,1,
... ,1).
,1).
(1, 1, ...
W2
= (1, -1,1,
w2 =
-1,1, -1),
-1),
and
For example, when n =
For
= 44
W3
-1 1).
= (1, -1,
-1, -1,
W3=
- 1, so xi
..., ,m
for i =
m -1,
Let xi
Xi = (Wi
= 1, ...
Xi E
{O, l}m
l}mcC [O,
[0,1]m,
l]m, and
E{0,
1)/2 for
+ 1)/2
(wi +
(Xi,Xj)
(Wi +
1 , wj + 1)/4
+ l,wj+
1))4
(xi, xj) =
=(wi
= ((Wi,Wj)
((w%,wj) +
1) + (1, wj) + (1, 1))/4
+ (Wi,
(wi, 1)+(I,wj)+(I,
=
1))/4
if
i
j
j,
_
m/4
=
4 if i =1= j,
m/2
if ii=j.
= j.
|m/2
{m
= {Xl,
... . , xm
center
the distance
distance from the
now be
shown that
Let V =
Xm-11};
}; it will
be shown
that the
the center
will now
{xi, ...
of C(V) to the nearest vertex is at least m/4.
m/4.
GENERALIZATION
GENERALIZATION OF LYAPOUNOV'S
LYAPOUNOV'S CONVEXITY
CONVEXITY THEOREM
301
= 1= ±1
= 0Oor 1 for
for ii=== 1,
1,......, ,m,
1- 2Si,
?1 for
Let 8Sii ==
m, and let Ei
Ci ==
28i, so Ei
Ci ==
for each i. Then
Since m
m - 1 is odd,
o~
so
(~1 €i)
~=1
> -(in-2).
E
-(m seje ~
Z Ei;:j:.j
Zii CiCj
4~~
1|Egx|.. 1
!4 11~1
L..J C~X~
(t €j)
-1 =
J=l
I:~€i€j + (m - 2),
~;:j:.J
2). Thus
2
(m(rn - 1)
1) _
_ (m(m
(m(rn-2)))
> ~! (m(m
- 2))) ===(in)2
(m)2
-
4
2
4
4'
i=l
1
1
> m/4.
m/4.
d(Z=?21 Xi/2,
En=j
6jixi)~
x,/2, E::1
so d(E::1
8iXi)
k 1
Since m <
~ n, one may consider Xi
+
2k+1
e [0,
[0, l]n for
for each i, and since 2m == 2
xi E
n/8.
the best possible upper bound in Lemma 2.4 is greater than n/8.
-
>
> n,
of Theorem
1.2.
Proof of
Theorem 1.2.
3. Proof
LEMMA3.1.
3.1. For
For each
each ;,
each Ce
LEMMA
,u and each
there exists a measurable
measurable partition
> 0, there
{Bi},N of X satisfying
{Bi}~l
(2) for
eachB
BeEE 1,
., ,N}
n 3JJ c{1,
||
(B)-ii(U3e
(2)
for each
C {I, ....
N} 931117(B)
- 17 (UjEJ Bj)
Bj) II||< < e€.
{x1,..... ,, xXj}
of
that
PROOF.Since
SinceR(;')
is bounded,
bounded,there
thereisisIan
an c-net
-net {Xl,
PROOF.
R(;'); that
R(,u) is
m } of
R(,X);
{Xi,. . ..
. .,Xm}
and for
eachXx EER(
there isisan
i5, {Xl,
CR(/(i)
() there
||x - x~i
is,
, Xm } C
R(J1;) and
for each
R(J1;)
an ii<~ mwith
m with Ilx
xiii <
.1 satisfy ;'(A
e 1
e. Let {Ai}~l
e 1
= xi,
= 1,
1,...
c.
{Ai}m1 E
,(Ai)i ) =
Xi, i =
... ,m,
, m, and let {.Bi}~l
{.Bi}ff=1 E
{B}N
BN)
=
,Am).
be disjoint
disjoint with u(B
,
...
,BN)
==
u(A
,
...
,A
).
It
is
easily
seen
that
{Bi}~l
(7(Bi,...
(7(Ai,...
1
1
m
satisfies (2). 00I
each B
B EE ".1, ::l] a measurable
measurablepartition
LEMMA
3.2. If;'
If ,lb Ee Pn(o:),
then for
LEMMA 3.2.
Pn(at), then
for each
partition
K k.
each ij ~
n and i ~
k.
btj(Bi) <
a for each
< nand
{B.}k= of B such that J-Lj(Bi)
{Bi}f=l
~ 0:
a. Let
PROOF. Assume J-L1(B)
bti(B) > 0:.
PROOF.
A ==
= inf{J-L1(E):
c B, E E
e 1,
~, and J-L1(E)
Al(E) ~
a/2}.
inf{,u1(E): E C
> 0:/2}.
.1 with E1
E11 E
e 1
B and
1.1, A is attained, that is, there exists E
By Theorem 1.1,
E 1 c Band
J-L1(E
= A~
> 0:/2.
a/2. Next it will be shown
shown that
bti(El)1) ==
(3)
(3)
AK
A
~ Q.
<o.
odd , E
a, then
E11 is not an atom of J-L1 (since J-L1 E
e P
P1(a)),
If A > 0:,
1(0:)), so there is a
a/2
measurable subset C of E
E11 with 0 <Aut(C)
measurable
J-L1 (C) < J-L1tl(El).
(E1). But then either /11(C)
J-L1 (C) > 0:/2
JOHN ELTON AND T. P. HILL
HILL
302
> a/2
or p1a(El\C)
J..Ll(E 1 \C) >
0./2 since iut(E1)
J..Ll(E 1 ) > a.
0.. This contradicts the definition of "\,
A, since
A and
and also
also Al
A.
J..Ll(C)
<,.\
J..L1(E
<"\.
(El\C)
li(C) <
1 \C) <
for B\E1
argument for
Repeat this argument
B\E1 in place of B, etc., and then for
for J..L2,
t2 J..L3,·
U3,..X.. ,J..Ln
,An
arrive at the desired partition {Bi}f=l.
to arrive
0
O
{Bi}k_
REMARK. Notice Theorem 1.1 is used here only in the case n = 1.
REMARK.
OF THEOREM
Fix it E Pn(a).
PROOF OF
Pn(o.). If;
If ,t is purely atomic with only
PROOF
THEOREM1.2.
1.2. Fix;
number of atoms, then R(;)
a finite number
R(pi) = E(V),
Ei(V), where V = {;(A):
{,u(A): A is an atom
of ;}
follows by Proposition 2.5. For
c [0,
[0, o.]n,
a]n, and the conclusion follows
For general
general J.,t,.L,
AI} C
Lemmas
Lemmas 3.1 and 3.2 will be used to approximate this finite-atom case.
0. It must be shown that
First assume a0. > o.
(4)
d(x, R(;))
for all x E co(R(;)).
d(x,
~ o.n/2
an/2 for
co(R(,u)).
R(,t)) <
Fix E
0. By Lemma 3.1 and repeated application of Lemma 3.2, there is a
c >
> O.
measurable partition {Bi}f::1
< a
of X satisfying both (2) and J..Lj(Bi)
0. for
~ n
measurable
for all jj <
{Bi},1
uj (Bi) ~
and i <
~ N. Let io be the restriction of ;iu to l1(B
,BN ); then ito is purely
o(Bi,...
1 , .. . ,BN);
Pn(o.), so by Proposition 2.5
atomic with only a finite number
number of atoms, and ito E Pn(a),
;0
(5)
(5)
;0
;0
D(R(o))
D(RCllo))
(E (I}, Jl(Bi)) ) ::;
= D (
=
AB)))
(N
2n.
Q
2
= 2::::1
C R(;)
> 0,
for some {Xi}
co(R(,u)); then x ==
Fix x E co(R(;));
- tixi
R(_s) and some {til
{xi} c
{ti} ~
t,xi for
=
{1,.
.
.
1.
..
there
exist
,N}
(2)
2::::1
ti
=
1.
By
(2)
there
exist
J1,
...
,
J
C
{I,
...
satisfying
Ji,
.,
Jm
C
satisfying
m
Ei=1ti
(6)
Xi xi<-
;
(,UU Bj)
B
< ec
foralli=1,...,
m,
for
all i == 1, ... ,m,
JEJi
so letting y = Eiz=1t, IUjEj, Bj) E co(R(Ao))
m
i=l
(
m
m
~ L:tic = c.
jGJj
i=l
i=l
Next observe
observe that
(8)
(8)
d(x,
R(;))
~ d(x,
y) +
R(;))
R
+ d(y,
d(x, y)
d(x,R(,))
d(, R())
< c+
+ an/2,
?<
+ d(y, R(;o))
R(,o)) ?~
~ c+
o.n/2,
where the second inequality in (8) follows
follows from
from (7)
C R(;),
(7) and the fact that R(;o)
R(1AO)c
R(A),
and the third inequalilty from
Since
c
was
arbitrary,
this
E
from (5) since y E co (R(;o)).
arbitrary,
(R(Ao)).
= 0 follows
completes the proof of (4); the case 0.
a=
follows easily by continuity. 0
4. Applications.
Applications.
Lyapounov's Convexity Theorem has been applied to a vadiverse areas
riety of problems in such diverse
areas as Banach space theory,
theory, optimal stopping
and
statistical
decision
theory
(see
Diestel and Uhl [3]);
statistical
theory, control theory,
theory,
[3]); the
purpose of this section is to mention two similar applications of Theorem 1.2.
. . .,,J..Ln
BISECTIONPROBLEM.
BISECTION
J..L1, ...
PROBLEM.If ,1,
are nonatomic probability measures on the
,Unare
same measurable space (X,1),
(X, i), then there is always
always a measurable subset A of X
=
with J..Li(A)
for
a
=
1/2
for
all
i
(this
is
special
case
of a theorem of Neyman [9]).
(A)
1/2
[9]). The
ui
following
following theorem generalizes this result.
GENERALIZATION
OF
CONVEXITY
THEOREM
GENERALIZATION
OF LYAPOUNOV'S
LYAPOUNOV'S
CONVEXITY
THEOREM
303
303
... .,, A7n
THEOREM4. 1. If J.,tl,
are probability
measures and pi
THEOREM
J.,tn are
probability measures
J.,ti E P
Pil (a)
(c) for all i,
li, •••
then
measurable subset A of X satisfying
then there
there is a measurable
!
alli=
< can/2
lJ.,ti(A)
an/2 for all
i == 1,2,
... , n.
1,2,...,n.
upi(A) - I1I ~
.
~
1
1
1
~
E co(R(J.,t)).
PROOF. SInce
Since (0,0,
(O,O,...
....,O)
(1, 1,......,1)
co(R(-I))
E R(-),
PROOF.
,0) and (1,1,
,1) E
R(J.,t), (2'2""'2)
(2 2 *
) E
Apply Theorem 1.2. 0c1
ApplY'
BANG-BANG
BANG-BANG PRINCIPLE.
PRINCIPLE. Lyapounov's Convexity Theorem was used by
LaSalle [6]
LaSalle
[6] to establish a principle in control theory which says that if an admissible
steering function (in an absolutely continuous problem)
problem) can bring the system from
from
one state to another in time t, then there is a "bang-bang"
"bang-bang" steering function that
can do the same thing in the same time. A generalization of a particular form
form of
this principle is given by the next theorem, which essentially says that in a system
with point masses (or
(or discontinuities or jumps in the process), given an arbitrary
arbitrary
steering function there is always a bang-bang steering function which will bring the
system within distance an
cn of the state arrived
arrived at by the given steering. (In the
following
following theorem, the set M is viewed as the collection of all admissible steering
functions, and MO
MO as the set of bang-bang steering functions-see
functions-see LaSalle [6].)
[6].)
THEOREM
THEOREM4.
4.2.2 . Let -;;
be a finite n-dimensional
n-dimensional vector-valued
measure
,i be
vector-valued Borel measure
< n and all t E
< a
with J.,ti({t})
ca for all i ~
be the
the set of
on [0,1]
[0, 1] with
[0, 1]. Let M be
E [0,1].
ui({t}) ~
< 1 for all
all real-valued
real-valued Borel measurable
measurable functions f on [0,1]
[0, 1] satisfying If(t)1
If(t)I ~
= 1 for all t E
be the
tE
MO be
the subset of M with
with If(t)1
[0, 1], and let MO
E [0,1],
[0, 1]. Define
E [0,1].
If(t)I ==
{!
{!
EM}
K=
K = {ffd,u:
fdj7: fEM}
f
and
KO
f dj7 : f E MO
MO} ·
K? = {ffdiI:
= K and D(KO) ~
Then
KO is compact,
an.
< cmn.
Then KO
compact, co(KO)
co(K?) ==
PROOF.
R(-;;) is compact and D(R(p,))
D(R(-;;)) ?~ an/2.
PROOF. By Theorems 1.1 and 1.2, R(1,)
can/2.
= 2IE(t)
- 1 for
= 2R(-;;)Letting fE(t)
fE(t) ==
KO ==
for each subset E of [0,
2IE(t) -1
[0, 1],
1], it is clear that K?
2R(,u) < cn.
U([O, 1]), so
SO KO
-;;([0,1]),
KO is compact and D(K?)
D(KO) ~
an.
It is clear that K and KO
KO are
are unaffected if we consider (via the usual equivalence
classesof I-;;I-a.e.
classes
M and
MO to be
be subsets
Loo(I-;;I),
where 1-;;1
and MO
subsets of Lo(I,u
equal functions)
functions)M
), where
I,uI
1a1-a.e.equal
is the total variation of -;;.
-closure
Krein-Milman theorem M is the weak*
weak*-closure
,t. By the Krein-Milman
= J
for z ==
of the convex hull of MO,
MO, and it follows
d-;; E
follows easily that, for
E K, z is in
f ffd,u
KO is compact, so (by [10, Theorem 3.25]) co(KO)
the closure of co(KO).
co(K0). But K?
co(K0) is
also compact, which implies z E
co(K0). The opposite inclusion co(KO)
co(K0) C
c K is
E co(KO).
trivial. 001
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INSTITUTE OF TECHNOLOGY,
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