Journal of Mathematical Analysis and Applications 232, 1᎐19 Ž1999.
Article ID jmaa.1998.6215, available online at http:rrwww.idealibrary.com on
The Vector Measures Whose Range Is Strictly Convex
Stefano Bianchini
Scuola Internazionale Superiore di Studi A¨ anzati (S.I.S.S.A.), ¨ ia Beirut 2r4, 34013
Trieste, Italy
E-mail: [email protected]
and
C. Mariconda
Uni¨ ersita degli Studi di Pado¨ a, Dipartimento di Matematica Pura e Applicata, ¨ ia
Belzoni 7, 35131 Pado¨ a, Italy
E-mail: [email protected]
Submitted by Dorothy Maharam Stone
Received January 15, 1998
Let ␮ be a measure on a measure space Ž X, ⌳ . with values in ⺢ n and f be the
density of ␮ with respect to its total variation. We show that the range RŽ ␮ . s
␮ Ž E . : E g ⌳4 of ␮ is strictly convex if and only if the determinant
detw f Ž x 1 ., . . . , f Ž x n .x is nonzero a.e. on X n. We apply the result to a class of
measures containing those that are generated by Chebyshev systems. 䊚 1999
Academic Press
Key Words: Chebyshev measure; Chebyshev system; Exposed point; Strictly
convex; Lyapunov; Range of a vector measure.
1. INTRODUCTION
Let ␮ : Ž X, ⌳ . ª ⺢ n be a nonatomic vector measure. A theorem of
Lyapunov w15x states that its range RŽ ␮ . s ␮ Ž E . : E g ⌳4 is closed and
convex. In w5, 7x the authors, motivated from the study of some bang᎐bang
1991 Mathematics Subject Classification. Primary: 46G10, 52A20, 28B05. Secondary: 28A35,
41A50.
We warmly thank R. Cerf and the referee for having carefully read the manuscript and for
their useful remarks; C. M. wishes to thank G. Colombo for his helpful advices on
measurable multifunctions.
1
0022-247Xr99 $30.00
Copyright 䊚 1999 by Academic Press
All rights of reproduction in any form reserved.
2
BIANCHINI AND MARICONDA
control problems, were led to introduce a broad class of measures, Chebyshev measures, whose range is strictly convex. Their definition involves a
signed measure det ␮ defined on the product space Ž X n, ⌳mn . by the
relation
᭙A1 , . . . , A n g ⌳
det ␮ Ž A1 = ⭈⭈⭈ = A n . s det ␮ Ž A1 . , . . . , ␮ Ž A n . ,
where detw u1 , . . . , u n x denotes the determinant of u1 , . . . , u n .
In the simpler case when X s I s w0, 1x and ⌳ coincides with the set L
of its Lebesgue measurable subsets the measure ␮ is said to be Chebyshev
with respect to the Lebesgue measure ␭ in w0, 1x if the measure det ␮ is
strictly positive on the non ␭mn-negligible subsets of ⌫ s Ž x 1 , . . . , x n . g
⺢ n : 0 F x 1 F ⭈⭈⭈ F x n F 14 , ␭mn denoting the n-product measure of ␭. In
the case where ␮ is absolutely continuous with density g with respect to ␭
the above condition is equivalent to the fact that the determinant
detw g Ž x 1 ., . . . , g Ž x n .x is strictly positive ␭mn-a.e. in ⌫ i.e. that g is a
Chebyshev system Žor T system, following the terminology of w11x..
As it is shown in w7x the range RŽ ␮ . of such a measure is strictly convex
and contains the origin in its boundary. A peculiar property of a Chebyshev measure is that its range can be described through the values that the
measure assumes on the finite union of intervals. It is well known that a
compact, convex, centrally symmetric subset of ⺢ 2 containing the origin is
the range of a two dimensional measure, i.e. a bidimensional zonoid Žsee
w3x.. In w2x the authors show that every strictly convex, compact, centrally
symmetric subset of ⺢ 2 Žwith O in its boundary. is the range of a
Chebyshev measure. It is then natural to ask whether this result can be in
some way extended to greater dimensions and, more generally, to try to
characterize the measures whose ranges are strictly convex. The latter
question was asked during a workshop to R. Schneider who answered with
the following result.
THEOREM. w16x RŽ ␮ . is strictly con¨ ex if and only if for e¨ ery A such that
␮ Ž A. / O there exist A1 , . . . , A n in A such that ␮ Ž A1 ., . . . , ␮ Ž A n . are
linearly independent.
It seems difficult however to check whether or not a measure does
satisfy these conditions. One of the purposes of this paper is to show that
the range of a measure ␮ is strictly convex if and only if the density f of ␮
with respect to its total variation < ␮ < is such that detw f Ž x 1 ., . . . , f Ž x n .x is
nonzero a.e. on X n. The latter determinant being the density of det ␮ with
respect to the product measure < ␮ < mn it turns out that RŽ ␮ . is strictly
convex if and only if the total variation of det ␮ is equivalent to < ␮ < mn. The
main result is obtained via the study of the exposed faces of RŽ ␮ .; this
allows also to give an alternative simple proof of Schneider’s Theorem.
MEASURES WITH A CONVEX RANGE
3
In Section 4 we study some applications of this characterization to
Chebyshev measures. First we show that Žagain considering for simplicity
the case where X s I and ⌳ s L . ␮ is a Chebyshev measure with respect
to ␭ if and only if the measure det ␮ is positive and equivalent to < ␮ < mn on
⌫. We improve the main result of w2x showing that if the range of a
bidimensional measure ␮ is strictly convex and contains the origin in its
boundary then not only is RŽ ␮ . the range of a suitable Chebyshev
measure but ␮ is itself a Chebyshev measure. Finally we answer the initial
question: when n ) 2 there exist strictly convex zonoids Žwith the origin in
the boundary. that are not the range of a Chebyshev measure. Actually the
latter have a nonregular boundary.
2. EXTREME POINTS AND EXPOSED POINTS OF THE
RANGE OF A MEASURE
Notation. By ‘‘⭈’’ we denote the usual scalar product, 5 ⭈ 5 is the euclidean norm in ⺢ n and S ny1 s x g ⺢ n : 5 x 5 s 14 is the unit sphere in
⺢ n ; O is the zero vector in ⺢ n. In what follows X is a set and ⌳ is a ␴
algebra of subsets of X. If ␯ , ␯ 1 , . . . , ␯m are measures on Ž X, ⌳ . we denote
by ␯ 1 m ⭈⭈⭈ m ␯m Žresp. ␯ mm . the m-product measure of ␯ 1 , . . . , ␯m Žresp.
of ␯ . on Ž X m , ⌳mm ., where X m s X = ⭈⭈⭈ = X Ž m times. and ⌳mm is the
m-product ␴ algebra of ⌳. We set L1␯ Ž X, ⺢ n . to be the space of the
␯-integrable functions on X with values in ⺢ n.
In Sections 2 and 3 we assume that ␮ is a nonatomic vector measure on
Ž X, ⌳ . with values in ⺢ n ; we will denote by < ␮ < its total variation and by f
the density of ␮ with respect to < ␮ <; we recall that f belongs to L1< ␮ < Ž X, ⺢ n .
and that 5 f 5 s 1 almost everywhere Ža.e.. in X. The range RŽ ␮ . of ␮ is
the subset of ⺢ n defined by RŽ ␮ . s ␮ Ž E . : E g ⌳4 .
Unless the contrary is expressly stated, for A, B in X by A : B we
mean that B _ A is < ␮ < negligible and by A s B that A : B and B : A,
i.e., that < ␮ <Ž A^ B . s 0.
For a compact convex subset K of ⺢ n and p in S ny1 let hŽ K, p . s
max p ⭈ x : x g K 4 ; the supporting hyperplane H Ž K, p . with outer normal
¨ ector p is defined by
H Ž K , p. s x g ⺢ n : p ⭈ x s hŽ K , p. 4
and F Ž K, p . s H Ž K, p . l K is the exposed face with outer normal ¨ ector p.
We recall that a point x in K is said to be exposed if it coincides with an
exposed face, i.e., if there exists p in S ny 1 such that F Ž K, p . s x 4 ;
obviously each exposed point of K is an extreme point of K Žbut the
converse is not true, see for example w14x.. For p in S ny 1 we introduce the
4
BIANCHINI AND MARICONDA
following measurable subsets of X:
Dq Ž p . s x g X : p ⭈ f Ž x . ) 0 4 ,
Dy Ž p . s x g X : p ⭈ f Ž x . - 0 4 ,
D 0 Ž p . s x g X : p ⭈ f Ž x . s 04 .
Lyapunov’s Theorem Žsee for example w15x. states that RŽ ␮ . is closed and
convex. We describe here the exposed faces of RŽ ␮ ..
Assume that p belongs to S ny 1 ; then hŽ RŽ ␮ ., p . s p ⭈
PROPOSITION 2.1.
␮ Ž DqŽ p .. and
F Ž R Ž ␮ . , p . s ␮ Ž E . : E g ⌳ , Dq Ž p . : E : Dq Ž p . j D 0 Ž p . 4 .
Proof. For E in ⌳ we have
p ⭈ ␮Ž E. s
s
F
HE p ⭈ f Ž x . d < ␮ <
HElD
yŽ
p.
HElD
qŽ
p.
p ⭈ f Ž x. d< ␮< q
HElD
p ⭈ f Ž x. d< ␮< F
HD
qŽ
qŽ
p.
p.
p ⭈ f Ž x. d< ␮<
p ⭈ f Ž x . d < ␮ < s p ⭈ ␮ Ž Dq Ž p . . ,
proving the first part of the claim. Moreover the above inequalities show
that, for E in ⌳, the equality p ⭈ ␮ Ž E . s p ⭈ ␮ Ž DqŽ p .. holds if and only if
< ␮ <Ž E l DyŽ p .. s 0 and E l DqŽ p . s DqŽ p . or, equivalently, DqŽ p . :
E : Dq Ž p . j D 0 Ž p ..
COROLLARY 2.2. For p in S ny 1 the exposed face F Ž RŽ ␮ ., p . of RŽ ␮ .
with outer normal ¨ ector p is reduced to a point if and only if < ␮ <Ž D 0 Ž p .. s 0.
We recall that a compact convex subset of ⺢ n is strictly convex if and
only if each of its exposed faces reduces to a point. The above result yields
then directly a first characterization of the strict convexity of RŽ ␮ ..
PROPOSITION 2.3.
for each p in S ny 1.
RŽ ␮ . is strictly con¨ ex if and only if < ␮ <Ž D 0 Ž p .. s 0
As an application we give a short alternative proof to Schneider’s
characterization of the measures whose range are strictly convex.
If ␮␫ 4␫ g I is a set of vectors in ⺢ n we denote by ² u␫ :␫ g I the vector space
spanned by the vectors u␫ . The orthogonal space of a vector space L is
denoted by LH .
MEASURES WITH A CONVEX RANGE
5
THEOREM 2.4. w16x RŽ ␮ . is strictly con¨ ex if and only if for e¨ ery A such
that ␮ Ž A. / O there exist A1 , . . . , A n in A such that ␮ Ž A1 ., . . . , ␮ Ž A n . are
linearly independent.
Proof. Let A be such that ␮ Ž A. / O and assume that the vector space
L s ² ␮ Ž B . : B g ⌳ , B : A:
is at most Ž n y 1. dimensional. Then if p belongs to S ny 1 l LH we have
᭙B g ⌳ ,
B:A«
HB p ⭈ f Ž x . d < ␮ < s p ⭈ HB f Ž x . d < ␮ < s p ⭈ ␮ Ž B . s 0
so that A : D 0 Ž p . and, thus, < ␮ <Ž D 0 Ž p .. ) 0; Proposition 2.3 implies that
RŽ ␮ . is not strictly convex. Conversely if RŽ ␮ . is not strictly convex by
Proposition 2.3 there exists p in S ny 1 satisfying < ␮ <Ž D 0 Ž p .. ) 0: let
A : D 0 Ž p . be such that ␮ Ž A. / O. Then
᭙B g ⌳ ,
B : A « p ⭈ ␮Ž B. s
HB p ⭈ f Ž x . d < ␮ < s 0
and, thus, ² ␮ Ž B . : B g ⌳, B : A: : ² p :H / ⺢ n.
The next result is traditionally obtained from a celebrated Theorem of
Olech w12x; we prove it here in an elementary way.
PROPOSITION 2.5. Let E, F in ⌳ be such that ␮ Ž E . s ␮ Ž F . is an extreme
point of RŽ ␮ .. Then < ␮ <Ž E^ F . s 0.
Proof. Assume that < ␮ <Ž E _ F . ) 0 and let A : E _ F be such that
Ž
␮ A. / O. Set E1 s E _ A and E2 s F j A. Clearly we have ␮ Ž E1 . s
␮ Ž E . y ␮ Ž A. / ␮ Ž E ., ␮ Ž E2 . s ␮ Ž F . q ␮ Ž A. s ␮ Ž E . q ␮ Ž A. / ␮ Ž E .
and ␮ Ž E . s Ž1r2. ␮ Ž E1 . q Ž1r2. ␮ Ž E2 ., contradicting the extremality of
␮ Ž E ..
COROLLARY 2.6. Assume that the origin O is an extreme point of RŽ ␮ .
and let A in ⌳ be such that ␮ Ž A. s O. Then < ␮ <Ž A. s 0.
Proof. Since ␮ Ž A. s O s ␮ Ž⭋. is an extreme point of RŽ ␮ . then
Proposition 2.5 implies that < ␮ <Ž A. s < ␮ <Ž A^⭋. s 0.
As a consequence we obtain the following characterization of the
exposed points of RŽ ␮ ..
PROPOSITION 2.7. For E in ⌳ the point ␮ Ž E . is exposed in RŽ ␮ . if and
only if there exists p in S ny 1 such that E s Dq Ž p . and < ␮ <Ž D 0 Ž p .. s 0.
Proof. Assume that < ␮ <Ž D 0 Ž p .. s 0; then by Proposition 2.1 the exposed face F Ž RŽ ␮ ., p . coincides with ␮ Ž DqŽ p ..4 so that the latter is an
6
BIANCHINI AND MARICONDA
exposed point of RŽ ␮ .. Conversely, let E in ⌳ be such that F Ž RŽ ␮ ., p .
s ␮ Ž E .4 for some p in S ny 1. By Corollary 2.2 necessarily we have
< ␮ <Ž D 0 Ž p .. s 0 and therefore F Ž RŽ ␮ ., p . s ␮ Ž DqŽ p ..4 so that ␮ Ž E . s
␮ Ž DqŽ p ... Since ␮ Ž E . is an exposed Žand thus extreme. point of RŽ ␮ .
then Proposition 2.5 yields E s DqŽ p ..
COROLLARY 2.8. The origin is an exposed point of RŽ ␮ . if and only if
there exists p in S ny 1 such that p ⭈ f Ž x . - 0 a.e. on X.
Proof. Proposition 2.7 implies that O s ␮ Ž⭋. is an exposed point of
RŽ ␮ . if and only if there exists p in S ny 1 such that Dq Ž p . s ⭋ and
< ␮ <Ž D 0 Ž p .. s 0.
3. THE MEASURES WHOSE RANGE IS STRICTLY
CONVEX
The main result of this section stems from Corollary 2.3: it states that
the range of ␮ is strictly convex if and only if the vectors f Ž x 1 ., . . . , f Ž x n .
are linearly independent for a.e. Ž x 1 , . . . , x n . in X n. We introduce the
subset ⌺ of X n defined by
⌺ s Ž x 1 , . . . , x n . g X n : det f Ž x 1 . , . . . , f Ž x n . s 0 4 .
THEOREM 3.1.
RŽ ␮ . is strictly con¨ ex if and only if ⌺ is < ␮ < mn negligible.
Proof. If RŽ ␮ . is not strictly convex by Proposition 2.3 there exists p
in S ny 1 such that < ␮ <Ž D 0 Ž p .. ) 0; since Ž D 0 Ž p .. n : ⌺ then we obtain
< ␮ < mn Ž ⌺ . G w < ␮ <Ž D 0 Ž p ..x n ) 0.
We give two proofs of the opposite implication. For each subset S of X n
and Ž x 2 , . . . , x n . in X ny 1 let SŽ x 2 , . . . , x n . s x 1 g X : Ž x 1 , . . . , x n . g S 4
be the Ž x 2 , . . . , x n . section of S.
First proof. We first show that the set
B s Ž x1 , . . . , x n . g X n : f Ž x1 . g ² f Ž x 2 . , . . . , f Ž x n . :4
is measurable. For u1 , . . . , u m in ⺢ n Ž m F n. we denote by < u1 n ⭈⭈⭈ n u m <
their Gramian, i.e., the sum of the squares of the minors of order m of the
matrix Ž e i ⭈ u j . i, j Žwhere Ž e i . i is the standard basis in ⺢ n .; clearly u1 , . . . , u m
are linearly dependent if and only if their Gramian vanishes. For every
nonempty subset I s i1 , . . . , i k 4 of Ž2, . . . , n4 let BI be the measurable
subsets of B defined by
BI s Ž x 1 , . . . , x n . g X n : f Ž x 1 . n f Ž x i1 . n ⭈⭈⭈ n f Ž x i k . < s 0,
< f Ž x i . n ⭈⭈⭈ n f Ž x i . </0 4
1
k
7
MEASURES WITH A CONVEX RANGE
and set Z s Ž x 1 , . . . , x n . g X n : f Ž x 1 . s ⭈⭈⭈ s f Ž x n . s 04 . Let x s
Ž x 1 , . . . , x n . g B: then either x g Z or there exists a subset I of 2, . . . , n4
such that f Ž x i . : i g I 4 is a maximal subset of linearly independent vectors
among f Ž x i . : i g 2, . . . , n44 and f Ž x 1 . g ² f Ž x i . : i g I : or, equivalently,
x g BI . Thus, B s Z DŽDI : 2, . . . , n4 BI ., proving the claim.
Fubini’s theorem gives
< ␮ < mn Ž ⌺ . s
HX
ny1
½H
⌬Ž x2 , . . . , x n.
d < ␮ < Ž x 1 . d Ž < ␮ < Ž x 2 . m ⭈⭈⭈ m < ␮ < Ž x n . . .
5
Assume that RŽ ␮ . is strictly convex; then Proposition 2.3 yields
᭙ Ž x 2 , . . . , x n . g X ny 1
< ␮ < Ž x1 g X : f Ž x1 . g ² f Ž x 2 . , . . . , f Ž x n . :4 . s 0
so that if ⌺ 1 is the Žmeasurable . subset of ⌺ defined by
⌺1 s Ž x 1 , . . . , x n . g ⌺ : f Ž x 1 . f ² f Ž x 2 . , . . . , f Ž x n . : 4
from the above formula we obtain
< ␮ < mn Ž ⌺ . s
HX
ny1
½H
⌬ 1Ž x 2 , . . . , x n .
d < ␮ < Ž x 1 . d Ž < ␮ < Ž x 2 . m ⭈⭈⭈ m < ␮ < Ž x n . .
5
and thus Tonelli’s Theorem yields ⌺ s ⌺ 1 < ␮ < mn-a.e. Similarly if for i in
w2, . . . , n4 we put
⌺i s Ž x1 , . . . , x n . g ⌺ : f Ž x i .
f ² f Ž x 1 . , . . . , f Ž x iy1 . , f Ž x iq1 . , . . . , f Ž x n . : 4
the same arguments give ⌺ s ⌺ i < ␮ < mn-a.e. As a consequence
n
⌺s
F ⌺i
< ␮ < mn-a.e.
is1
n
Obviously the set Fis1
⌺ i is empty; the conclusion follows.
Second proof. Let g: X ny 1 = S ny 1 ª ⺢ be the map defined by
᭙ Ž y 1 , . . . , yny1 . g X ny 1
᭙ z g S ny 1
n
g Ž Ž y 1 , . . . , yny1 . , z . s
Ý Ž z ⭈ f Ž yi . .
is2
2
.
8
BIANCHINI AND MARICONDA
The function g is measurable in Ž y 1 , . . . , yny1 . and continuous in z:
Corollary 6.3 in w10x then implies that the set-valued map G: X ny 1 ª
P Ž S ny 1 . defined by
G Ž y 1 , . . . , yny1 . s z g S ny 1 : g Ž Ž y 1 , . . . , yny1 . , z . s 0 4
s ² f Ž y 1 . , . . . , f Ž yny1 . :H l S ny1
has a measurable graph: Theorem 5.7 in w10x then yields the existence of a
measurable selection p: X ny 1 ª S ny 1 of G, i.e., p is measurable and
pŽ y 1 , . . . , yny1 . g GŽ y 1 , . . . , yny1 . a.e. in X ny 1. For i in 1, . . . , n4 let A i
be the measurable subset of X n defined by
A i s Ž x 1 , . . . , x n . g X n : f Ž x i . ⭈ p Ž x 1 , . . . , x iy1 , x iq1 , . . . , x n . s 0 4 .
We claim that ⌺ s Di A i Žmodulo < ␮ < mn ..
In fact let x s Ž x 1 , . . . , x n . g ⌺: then detw f Ž x 1 ., . . . , f Ž x n .x s 0 so that
there exists i such that f Ž x i . g ² f Ž x 1 ., . . . , f Ž x iy1 ., f Ž x iq1 ., . . . , f Ž x n .:;
modulo a negligible set the latter vector space is contained in
² pŽ x 1 , . . . , x iy1 , x iq1 , . . . , x n .:H and, thus, x belongs to A i . Conversely let
Žfor instance . Ž x 1 , . . . , x n . g A1. Either f Ž x 2 ., . . . , f Ž x n . are linearly independent so that f Ž x 1 . g ² pŽ x 2 , . . . , x n .:H s ² f Ž x 2 ., . . . , f Ž x n .: or
f Ž x 2 ., . . . , f Ž x n . are linearly dependent: in both cases we obtain
detw f Ž x 1 ., . . . , f Ž x n .x s 0, proving the claim.
Assume that < ␮ < mn Ž ⌺ . ) 0: then there exists i such that a < ␮ < mn Ž A i . ) 0;
again without loss of generality we suppose that i s 1. Fubini’s Theorem
gives
< ␮ < mn Ž A1 . s
HX
ny1
½H
A 1Ž x 2 , . . . , x n .
d < ␮ < Ž x 1 . d Ž < ␮ < Ž x 2 . m ⭈⭈⭈ m < ␮ < Ž x n . .
5
so that there exists Ž x 2 , . . . , x n . in X ny 1 such that < ␮ <Ž A1Ž x 2 , . . . , x n .. ) 0.
Now we have A1Ž x 2 , . . . , x n . s D 0 Ž pŽ x 2 , . . . , x n ..: Proposition 2.3 implies
that RŽ ␮ . is not strictly convex.
The determinant measure det ␮ associated to ␮ s Ž ␮ 1 , . . . , ␮ n . was
introduced in w7x. It seems natural to use it here.
We shall denote by Sn the symmetric group of order n and, for ␴ in
Sn , by ⑀ Ž ␴ . its sign.
DEFINITION 3.2. The determinant measure of ␮ , denoted by det ␮ , is
the signed measure defined on Ž X n, ⌳mn . by
det ␮ s
Ý ⑀ Ž ␴ . ␮␴ Ž1. m ⭈⭈⭈ m ␮␴ Ž n. .
␴g Sn
MEASURES WITH A CONVEX RANGE
9
This is the only measure whose restriction to the product sets A1 = ⭈⭈⭈ =
A n satisfy det ␮ Ž A1 = ⭈⭈⭈ = A n . s detw ␮ Ž A1 ., . . . , ␮ Ž A n .x.
The next result appears in the proof of w7, Th. 3.4x but is not explicitly
stated.
PROPOSITION 3.3. The function det f defined on X n by
det f Ž x 1 , . . . , x n . s det f Ž x 1 . , . . . , f Ž x n .
is the density function of det ␮ with respect to < ␮ < mn.
Proof. Set f s Ž f 1 , . . . , f n .. For any measurable subset A of X n the
application of Fubini-Tonelli’s Theorem yields
det ␮ Ž A .
s
Ý ⑀ Ž ␴ . Ž ␮␴ Ž1. m ⭈⭈⭈ m ␮␴ Ž n. . Ž A .
␴g Sn
s
HA Ý
⑀ Ž ␴ . f␴ Ž1. Ž x 1 . ⭈⭈⭈ f␴ Ž n. Ž x n . d Ž < ␮ < Ž x 1 . m ⭈⭈⭈ m < ␮ < Ž x n . .
HAdet
f Ž x 1 . , . . . , f Ž x n . d < ␮ < mn Ž x 1 , . . . , x n . .
␴ g Sn
s
The measure det ␮ allows us to reformulate Theorem 3.1 in terms of the
behavior of < ␮ < mn with respect to det ␮. We recall that a vector measure ␶
is said to be absolutely continuous with respect to some other signed
measure ␰ Žboth defined in Ž X, ⌳ .., in symbols ␶ < ␰ , whenever for A in
⌳ the condition < ␰ <Ž A. s 0 implies ␶ Ž A. s O. Two positive measures ␶ , ␰
on X are said to be equivalent if they are absolutely continuous with
respect to each other. We will use the fact that if ␶ , ␰ are finite positive
measures and ␶ < ␰ then ␶ is equivalent to ␰ if and only if the density of
␶ with respect to ␰ is strictly positive a.e. on X.
THEOREM 3.4. The range of ␮ is strictly con¨ ex if and only if < ␮ < mn is
equi¨ alent to <det ␮ < Ž the total ¨ ariation of det ␮ ..
Proof. Proposition 3.3 together with w1, Ex. 26.10x imply that <det ␮ < is
absolutely continuous with density <det f < with respect to < ␮ < mn. Thus, the
condition that <det f < does not vanish in X n is equivalent to the absolute
continuity of < ␮ < mn with respect to <det ␮ <. Theorem 3.1 yields the conclusion.
10
BIANCHINI AND MARICONDA
4. SOME APPLICATIONS TO CHEBYSHEV MEASURES
As usual X is a set and ⌳ is a ␴ algebra of subsets of X. We consider
the following assumption.
ASSUMPTION Ž A.. Ž A1 .: M s Ž Mi . i g w0, 1x is an increasing family of
measurable sets Ži.e., Mi : M j if i - j . such that M0 s ⭋, M1 s X and ␯
is a positive nontrivial bounded measure on Ž X, ⌳ . such that the function
i ¬ ␯ Ž Mi . is continuous and strictly increasing.
Ž A 2 .: ␮ is a vector measure on Ž X, ⌳ . with values in ⺢ n and the
function i ¬ < ␮ <Ž Mi . is continuous.
Remark. The existence of such a family implies clearly that both ␯ and
␮ are nonatomic; conversely if these measures are nonatomic then Lyapunov’s Theorem applied to the vector measure Ž ␯ , < ␮ <. yields the existence of a family Ž Mi . i g w0, 1x such that ␯ Ž Mi . s i ␯ Ž X . and < ␮ <Ž Mi . s
i < ␮ <Ž X . for every i Žsee w8x. and, thus, satisfying the assumption Ž A..
The family M induces an order relation $M Žor simply $ when no
ambiguity may occur. on X defined by x $M y if there exists i in w0, 1x
such that x g Mi and y f Mi . By PM Žor P . we will denote the subset of
X n defined by PM s Ž x 1 , . . . , x n . g X n : x 1 $M ⭈⭈⭈ $M x n4 if n ) 1, PM s
X if n s 1. We will assume for simplicity that PM is measurable Žin the
general case one should replace PM with any of its measurable coverings..
EXAMPLE. If M s Žw0, i x. i g w0, 1x then PM s Ž x 1 , . . . , x n . g w0, 1x n : 0 F
x 1 - ⭈⭈⭈ - x n F 14 .
Chebyshev measures with respect to ␯ and M have been defined in w7x:
they are vector measures whose associated determinant measure is strictly
positive on the non ␯-negligible subsets of PM .
DEFINITION 4.1. The vector measure ␮ is a Chebyshe¨ measure with
respect to ␯ and to the family M Žor simply a T␯ measure or T measure
when ␯ s < ␮ <. if ␮ , ␯ , M satisfy assumption Ž A. and the measure det ␮
verifies
᭙A g ⌳mn l PM ,
␯ mn Ž A . ) 0 « det ␮ Ž A . ) 0.
When no ambiguity may occur we shall often omit to mention the
dependence with respect to M .
Remark. The measure det ␮ is absolutely continuous with respect to
< ␮ < mn ; it follows then directly from the definition that ␮ is a T < ␮ < measure
if and only if det ␮ is positive and equivalent to < ␮ < mn on PM .
Remark. When n s 1 then PM s X for every family of subsets M ;
moreover if ␮ is a signed measure then det ␮ s ␮. Therefore ␮ is a T␯
MEASURES WITH A CONVEX RANGE
11
measure whenever it assumes strictly positive values on the non ␯-negligible subsets of X. In particular a positive measure ␮ is a T␯ measure if and
only if ␯ < ␮. It may happen however that ␮ is not absolutely continuous
with respect to ␯ . Let, for instance, ␮ be the Lebesgue measure on w0, 1x,
E a measurable set such that 0 - ␮ Ž E l I . - ␮ Ž I . for every nontrivial
interval I, ␯ be the measure defined by ␯ Ž A. s ␮ Ž A l E . for every
measurable set A and set M s Žw0, i x. i g w0, 1x. Clearly, ␯ , ␮ , M verify assumption Ž A.; moreover ␯ is absolutely continuous with respect to ␮ so
that ␮ is a T␯ measure; however ␮ is not absolutely continuous with
respect to ␯ Ž ␯ Žw0, 1x _ E . s 0 whereas ␮ Žw0, 1x _ E . s 1 y ␮ Ž E . ) 0..
We will use the following result w7, Funny corollary 4.5x.
PROPOSITION 4.2. Let ␮ be a T␯ measure and A in ⌳ be such that
␮ Ž A. s O Ž the origin in ⺢ n .; then ␯ Ž A. s 0. In particular ␯ is absolutely
continuous with respect to < ␮ <.
Sketch of the proof. Assume that ␯ Ž A. ) 0. If ␮ is a T␯ measure with
respect to a family M s Ž Mi . i of subsets of X then the continuity of the
map i ¬ ␯ Ž Mi . allows to decompose A as a disjoint union of some non
␯-negligible sets A1 , . . . , A n such that their product A1 = ⭈⭈⭈ = A n is
contained in PM . It follows that detw ␮ Ž A1 ., . . . , ␮ Ž A n .x s det ␮ Ž A1
= ⭈⭈⭈ = A n . ) 0 so that the vectors ␮ Ž A1 ., . . . , ␮ Ž A n . are linearly independent. However O s ␮ Ž A. s ␮ Ž A1 . q ⭈⭈⭈ q␮Ž A n ., a contradiction.
It follows that if ␮ is a T␯-measure then the map i ¬ < ␮ <Ž Mi . is strictly
increasing. The term ‘‘Chebyshev’’ arises from the well-known concept of
T system Žwhere ‘‘T’’ stands for Tchebycheff., applied in approximation
theory and to moment problems in statistics, involving continuous functions defined on intervals of ⺢ Žsee w11x.. We recall here a slightly more
general definition.
DEFINITION 4.3. w7x Let ␯ , M satisfy Ž A1 .. A function g in L1␯ Ž X, ⺢ n . is
said to be a Chebyshe¨ system with respect to ␯ and M Ž or simply a T␯
system. if the determinant detw g Ž x 1 ., . . . , g Ž x n .x is positi¨ e ␯ mn-a.e. in PM .
Let g g L1␯ Ž X, ⺢ n . and ␮ be the measure with density function g with
respect to ␯ . The arguments involved in the proof of Proposition 3.3 show
that detw g Ž x 1 ., . . . , g Ž x n .x is the density function of det ␮ with respect to
␯ mn. As a consequence Chebyshev systems generate absolutely continuous
Chebyshev measures.
THEOREM 4.4. w7, Theorem 3.4x Let ␮ , ␯ , M satisfy Ž A. and let ␮ be
absolutely continuous with density g with respect to ␯ . Then ␮ is a T␯ measure
if and only if g is a T␯ system.
12
BIANCHINI AND MARICONDA
Chebyshev systems arise naturally from linear differential equations;
some of their applications to control theory and the calculus of variations
were given in w5x.
EXAMPLE. Let h g C ⬁Ž⺢. satisfy hŽ i. Ž0. s 0 for 0 F i F n y 2 and
h
0. s 1. There exists ␦ ) 0 such that the function f s
Ž hŽ ny1., hŽ ny2., . . . , h⬘, h. is a Chebyshev system on wy␦ , ␦ x with respect to
the Lebesgue measure and the family of intervals Mi s wy␦ , y␦ q 2 i ␦ x
Ž i g w0, 1x..
Ž ny1. Ž
We give now a remarkable example: a function with values in a half
plane of ⺢ 2 and whose inverse images of lines are negligible sets generates
a Chebyshev measure. For ␪ in ⺢ and u in ⺢ 2 _ O4 we denote by arg ␪ u
the argument of u in Ž ␪ y ␲ , ␪ q ␲ x.
PROPOSITION 4.5. Let Ž X, ⌳, ␯ . be a measure space Ž ␯ being nontri¨ ial .;
g in L1␯ Ž X, ⺢ 2 . be such that the set x g X : p ⭈ g Ž x . s 04 is ␯ negligible for
e¨ ery p in S 1 and q ⭈ g Ž x . ) 0 ␯-a.e. for some q in S 1. Then the two
dimensional measure ␮ defined by ␮ Ž A. s HA g d␯ for e¨ ery A in ⌳ is a
Chebyshe¨ measure Ž i.e., there exists a family M s Ž Mi . i g w0, 1x of subsets of X
with respect to which ␮ is a T␯ measure..
Proof. Let ␪ in ⺢ be such that q s e i ␪ : then if we set a s ␪ y ␲r2, b
s ␪ q ␲r2 the argument arg ␪ g Ž x . of g Ž x . in Ž ␪ y ␲ , ␪ q ␲ x belongs to
w a, b x for x in X _ Z, for some negligible set Z. For t in w a, b x let
Nt s x g X : arg ␪ g Ž x . F t 4
and ␾ : w a, b x ª ⺢q be the increasing map defined by ␾ Ž t . s ␯ Ž Nt . for
every t in w a, b x. Our assumption implies that for every t in ⺢ the sets
x g X : arg ␪ g Ž x . s t 4 are negligible. Clearly ␾ Ž a. s 0 and ␾ Ž b . s ␯ Ž X ..
Moreover the family of sets Ž Nt . t g w a, bx being increasing it follows that ␾ is
continuous and therefore ␾ Žw a, b x. s w0, ␯ Ž X .x. Let ␺ : w0, ␯ Ž X .x ª w a, b x
be a right inverse of ␾ and set, for every ␣ in w0, 1x, M␣ s N␺ Ž ␣␯ Ž X .. . The
definition of ␺ then implies that ␯ Ž M␣ . s ␾ Ž ␺ Ž ␣␯ Ž X ... s ␣␯ Ž X . so that
the map ␣ ¬ ␯ Ž M␣ . is continuous and strictly increasing. The absolute
continuity of ␮ with respect to ␯ yields the continuity of ␣ ¬ < ␮ <Ž M␣ . and
thus ␯ , ␮ , M fulfill assumption Ž A.. For x 1 , x 2 in X the relation x 1 $M x 2
here implies that there exists t in w a, b x such that arg ␪ g Ž x 1 . F t and
arg ␪ g Ž x 2 . ) t. Since b y a s ␲ it follows that if x 1 , x 2 are not in Z then
detw g Ž x 1 ., g Ž x 2 .x ) 0. Hence, this latter condition is fulfilled for ␯ m2 ᎏ
almost every couple Ž x 1 , x 2 . belonging to the set PM associated to M and
therefore g is a T␯ system. Theorem 4.4 yields the conclusion.
13
MEASURES WITH A CONVEX RANGE
The main result of w7x s tates that given a positive measure ␯ and a
prescribed increasing family of sets M s Ž Mi . i the range ␮ Ž E ., E g ⌳4 of
a n-dimensional T␯-measure ␮ with respect to M can be described through
the values that it assumes on the finite unions of Žat most n. sets of the
form M j _ Mi . Let ⌫ be the subset of ⺢ n defined by
⌫ s Ž ␥ 1 , . . . , ␥n . g ⺢ n : 0 F ␥ 1 F ⭈⭈⭈ F ␥n F 1 4 .
REPRESENTATION THEOREM 4.6. w7x Suppose ␮ is a T␯ measure with
respect to the family M s Ž Mi . i g w0, 1x and let ␳ be a measurable function such
that 0 F ␳ F 1 a.e. There exists ␣ s Ž ␣ 1 , . . . , ␣ n . in ⌫ satisfying
␮ Ž E␣ . s
HX␳ d ␮
where E␣ s
D Ž M␣
iq 1
_ M␣ i .
Ž ␣ nq1 s 1 . .
1FiFn
i odd
If 0 - ␳ - 1 on a ␯-nonnegligible set then ␣ is unique, it belongs to the
interior of ⌫ and ␮ Ž E␣ . lies in the interior of RŽ ␮ ..
Remark. We recall that Lyapunov’s Theorem w15x states that there
exists a set E in ⌳ such that HX ␳ d ␮ s ␮ Ž E .; the improvement here is
that the set E can be chosen among a family of ‘‘nice’’ sets. We refer to w6x
for some comments about this fact and to w5x for an application of this
result of the bang᎐bang principle in control theory.
Chebyshev measures are considered here in connection with Sections 2
and 3 because they represent a broad class of measures whose range is
strictly convex.
THEOREM 4.7. w7, Theorem 5.3x The range RŽ ␮ . of a T␯-measure ␮ is
strictly con¨ ex. The boundary points of RŽ ␮ . admit a unique representation
modulo < ␮ <; moreo¨ er a point ␮ Ž E . belongs to the boundary of RŽ ␮ . if and
only if there exists ␥ in the boundary of ⌫ such that < ␮ <Ž E^ E␥ . s 0. In
particular O s ␮ Ž⭋. s ␮ Ž EŽ1, . . . , 1. . belongs to the boundary of RŽ ␮ ..
Proof of strict con¨ exity. Let E, F in ⌫ be such that ␮ Ž E . / ␮ Ž F .: then
< ␮ <Ž E^ F . / 0; let for instance < ␮ <Ž E _ F . ) 0. Then for ␭ in Ž0, 1. the
function ␳ s ␭␹ E q Ž1 y ␭. ␹ F is such that 0 - ␳ - 1 a.e. on E _ F.
Theorem 4.6 implies that H␳ d ␮ s ␭␮ Ž E . q Ž1 y ␭. ␮ Ž F . belongs to the
interior of RŽ ␮ ..
In what follows we shall denote by ␮ a n-dimensional vector measure
and by f the density of ␮ with respect to its total variation < ␮ <.
Assume that ␮ is a T␯ measure with respect to a family M of subsets of
X. If ␮ is absolutely continuous with respect to ␯ then, trivially, ␮ is a T < ␮ <
measure; it follows from Theorem 4.4 that detw f Ž x 1 ., . . . , f Ž x n .x ) 0 < ␮ < mna.e. on PM . Otherwise, if ␮ is not absolutely continuous with respect to ␯ ,
14
BIANCHINI AND MARICONDA
it does not seem clear at all from Definition 4.1 whether the above
conclusion still holds. Certainly, by Proposition 4.2, ␯ is absolutely continuous with respect to < ␮ <; however one might think that there exists a
␯ mn-negligible Žbut not < ␮ < mn negligible. subset A in PM such that
det ␮ Ž A. F 0. The next result shows that this pathology does not occur; its
proof is based on the fact that a T␯ measure has a strictly convex range and
on our characterization of the measures having this property.
THEOREM 4.8. Let ␮ be a T␯ measure Ž with respect to a family M .. Then
␮ is a T < ␮ < measure Ž with respect to M ..
Proof. By Theorem 4.7 the range of ␮ is strictly convex; Theorem 3.1
then implies that
det f Ž x 1 . , . . . , f Ž x n . / 0
< ␮ < mn-a.e.
Let
Py
M s Ž x 1 , . . . , x n . g PM : det f Ž x 1 . , . . . , f Ž x n . - 0 4
.
and assume that < ␮ < mn Ž Py
M ) 0. Since, from Proposition 3.3
det ␮ Ž Py
M . s
HP
det f Ž x 1 . , . . . , f Ž x n . d < ␮ < mn ,
y
M
then by the continuity of < ␮ < with respect to the sets Mi , there exists a
Ž2 n.-tuple
Ž ␣ 11 , ␣ 12 , . . . , ␣ n1 , ␣ n2 . g ⺢ 2 n
such that
0 - ␣ 11 - ␣ 12 - ⭈⭈⭈ - ␣ n1 - ␣ n2 - 1
and
det ␮ Ž Py
M j Ž M␣ 12 _ M␣ 11 . = ⭈⭈⭈ = Ž M␣ n2 _ M␣ n1 . . - 0.
However, ␯ being a positive measure, we have
␯ mn Ž PMyj Ž M␣ 12 _ M␣ 11 . = ⭈⭈⭈ = Ž M␣ n2 _ M␣ n1 . .
G ␯ mn Ž Ž M␣ 12 _ M␣ 11 . = ⭈⭈⭈ = Ž M␣ n2 _ M␣ n1 . . ) 0,
a contradiction. Therefore f is a T system; Theorem 4.4 yields the
conclusion.
MEASURES WITH A CONVEX RANGE
15
Remark. The above result shows that, in Definition 4.1, the auxiliary
positive measure ␯ can be omitted: in dealing only with T measures, as we
do in the rest of the paper, there is undoubtedly a gain of clarity. However
in the applications Žw5x. it happens that ␯ and M are given a priori and
that ␮ is defined through a density function g in L1␯ Ž X, ⺢ n .: the easiest
way to see if ␮ is a Chebyshev measure is then to check whether g is a T␯
system.
Theorem 4.7 states that the range of a Chebyshev measure is strictly
convex and that the origin O belongs to the boundary of its range. It is
well known that each compact, convex, centrally symmetric subset of ⺢ 2 is
the range of a two dimensional measure w2, 3x, i.e., a zonoid. Conversely, in
w2, Theorem 2x the authors show that, in ⺢ 2 , each strictly convex zonoid
Žwith O in its boundary. is the range of a Chebyshev measure. The results
obtained in Sections 2 and 3 allow us to give a much stronger result: the
bidimensional Chebyshev measures are exactly those measures whose
range satisfies the above geometrical properties. Let ␮ : Ž X, ⌳ . ª ⺢ 2 be a
nonatomic bidimensional measure and, as usual, let f be the density of ␮
with respect to < ␮ <.
THEOREM 4.9. Assume that ␮ is a bidimensional measure whose range
RŽ ␮ . is strictly con¨ ex and contains the origin in its boundary. Then there
exists a family of sets with respect to which ␮ is a Chebyshe¨ measure.
Moreo¨ er there exists ␪ in ⺢ such that for e¨ ery measurable function ␳ with
¨ alues in w0, 1x there exist ␣ , ␤ in ⺢ satisfying
HX␳ Ž x . d ␮ s ␮ Ž x g X : ␣ F arg
␪
f Ž x. F ␤ 4..
Proof. By Corollary 2.8 there exists q in S 1 such that q ⭈ f Ž x . ) 0 a.e.
on X and, by Proposition 2.3, the sets x g X : p ⭈ f Ž x . s 04 are negligible
for every p in S 1. The application of Proposition 4.5 with Ž< ␮ <, f . instead
of Ž ␯ , g . yields the existence of a family M s Ž Mi . i of subsets of X with
respect to which ␮ is a Chebyshev measure. Furthermore, the proof of
Proposition 4.5 shows that there exists ␪ in ⺢ such that for every i we
have Mi s x g X : arg ␪ f Ž x . F ␰ i 4 for some ␰ i in Ž ␪ y ␲ , ␪ q ␲ x: the
conclusion follows from the representation theorem 4.6.
Let ␮ be a vector measure on Ž X, ⌳ ., E be in ⌳ and let ␮ E be the
vector measure defined by ␮ E Ž B . s ␮ Ž E _ B . y ␮ Ž E l B . for every B in
⌳. It is easy to verify Žsee w3, Lemma 1.3x. that the range RŽ ␮ E . of ␮ E is a
translate of the range of ␮ ; more precisely we have that RŽ ␮ E . s RŽ ␮ .
y ␮ Ž E . s x y ␮ Ž E . : x g RŽ ␮ .4 . The next characterization of the bidimensional strictly convex zonoids in ⺢ 2 follows then directly.
16
BIANCHINI AND MARICONDA
COROLLARY 4.10. Let ␮ be a measure on Ž X, ⌳ . with ¨ alues on ⺢ 2 . The
range of ␮ is strictly con¨ ex if and only if there exists a subset E in ⌳ such ␮ E
is a Chebyshe¨ measure.
Proof. If there exists E in ⌳ such that ␮ E is a Chebyshev measure
then by Theorem 4.7 the range of ␮ E is strictly convex. Thus, each of its
translates, in particular the range of ␮ , is strictly convex too. Conversely
assume that the range of ␮ is strictly convex. Let E be such that ␮ Ž E .
belongs to the boundary of RŽ ␮ .. Then the origin lies in the boundary of
the translate RŽ ␮ . y ␮ Ž E .. The latter set is the range of ␮ E and is
strictly convex: it follows from Theorem 4.9 that ␮ E is a T measure.
Our next result shows that, when n ) 2, the boundary of the range of a
n dimensional Chebyshev measures is not regular. In particular, Theorem
4.9 cannot be extended to greater dimensions: a measure whose range is a
ball Žit exists, see for instance w13x. is certainly not the range of some
Chebyshev measure.
PROPOSITION 4.11. Let ␮ s Ž ␮ 1 , . . . , ␮ n . be a T measure and n G 3.
Then the boundary of RŽ ␮ . is not a Ž n y 1.-dimensional T 1 manifold.
We need the following Lemma, whose proof is postponed at the end of
the paper.
LEMMA 4.12. Let ␮ be a T measure with respect to an increasing family
Ž M␣ .␣ g w0, 1x of subsets of X. Then there exists a ¨ ector measure ␮
˜ on the
Lebesgue ␴ algebra of w0, 1x such that ␮
˜ is a T measure with respect to the
inter¨ als Žw0, ␣ x.␣ g w0, 1x and RŽ ␮ . s RŽ ␮
˜ ..
Proof of Proposition 4.11. By Lemma 4.12 it is not restrictive to suppose
that ␮ is a Chebyshev measure on w0, 1x with respect to the intervals
Žw0, x x. x g w0, 1x. Fix x in w0, 1. and let ␭ x : w0, 1 y x x ª ⺢ n be the curve
defined by
᭙ t g w 0, 1 y x x :
␭ x Ž t . s ␮ Ž w x, x q t x . .
Since n G 3 and ␹ w x, xqt x has at most two jumps Theorem 4.7 implies that
the set ⌫x s ␭ x Žw0, 1 y x x. is entirely contained in the boundary ⭸ RŽ ␮ . of
RŽ ␮ .. Remark further that the origin O s ␭ x Ž0. belongs to ⌫x . Assume
that ⭸ RŽ ␮ . is a manifold of cl ass C 1 and let p be a unit normal vector to
the tangent plane ⌸ of ⭸ RŽ ␮ . at the origin. For every point q of RŽ ␮ .,
the distance of q from ⌸ is the absolute value of q ⭈ p; therefore
lim q ª 0, q g ⭸ R Ž ␮ .Ž q ⭈ p .r5 q 5. s 0. Since, by Proposition 4.2, ␮ Ž I . / O for
every nontrivial interval I it follows that lim t ª 0 Ž ␮ Žw x, x q t x. ⭈
p .rŽ5 ␮ Žw x, x q t x.5. s 0 and therefore, recalling that 5 ␮ Ž A.5 F < ␮ <Ž A.
for every measurable set A, lim t ª 0 Ž ␮ Žw x, x q t x..rŽ< ␮ <Žw x, x q t x.. ⭈ p s 0.
17
MEASURES WITH A CONVEX RANGE
Since, by w17x, lim t ª 0 Ž ␮ Žw x, x q t x..rŽ< ␮ <Žw x, x q t x.. s f Ž x . a.e. in w0, 1x
we obtain that D 0 Ž p . s x g w0, 1x : f Ž x . ⭈ p s 04 s w0, 1x Ž< ␮ <-a.e... However the set RŽ ␮ . is strictly convex: Proposition 2.3 then implies that
< ␮ <Ž D 0 Ž p .. s 0, a contradiction.
Proof of Lemma 4.12. Let ␮ s Ž ␮ 1 , . . . , ␮ n . and, for each i in 1, . . . , n4 ,
y
w x
let ␮ i s ␮q
i y ␮ i be the Jordan decomposition of ␮ i . Let g i : 0, 1 ª ⺢ be
the bounded variation function defined by
y
g i Ž ␣ . s ␮q
i Ž M␣ . y ␮ i Ž M␣ .
and let ␮
˜ i be the Lebesgue-Stieltjes measure generated by g i . Clearly ␮
˜ is
regular and if we set ␮
˜sŽ␮
˜ 1, . . . , ␮
˜ n . the continuity of the functions g i
yields
᭙␣ , ␤ g w 0, 1 x , ␣ F ␤
␮
˜ Ž w ␣ , ␤ x. s ␮
˜ Ž Ž ␣ , ␤ x. s ␮
˜ Ž w ␣ , ␤. . s ␮
˜ Ž Ž ␣ , ␤ . . s ␮ Ž M␤ _ M␣ . .
We shall show now that ␮
˜ is a T measure with respect to M˜s Žw0, i x. i g w0, 1x.
Notice first that, ␮ being a T < ␮ < measure, by Proposition 4.2 for
every ␣ - ␤ in w0, 1x we have ␮ Ž M␤ _ M␣ . / O. Therefore < ␮
˜ <Žw ␣ , ␤ x. G
5␮
Žw
x.5
5
Ž
.5
˜ ␣ , ␤ s ␮ M␤ _ M␣ ) 0. Moreover, it is clear that < ␮
˜ < F < ␮ <. It
follows that ␮
˜, < ␮
˜ <, M˜ satisfy assumption Ž A.. We recall that in this case
the set PM˜ associated to M˜ is given by PM˜ s Ž x 1 , . . . , x n . g w0, 1x n : 0 F x 1
- ⭈⭈⭈ - x n F 14 .
Let A : PM˜ be such that < ␮
˜ <mn Ž A. ) 0. The measure < ␮
˜ <mn being
n
regular and PM˜ being open in w0, 1x there exists a G␦ subset E of PM˜ such
that A : E and < ␮
˜ <mn Ž E _ A. s 0. We may assume that
Es
⬁
F Vm ,
ms1
where Ž Vm . m g ⺞ are open subsets of PM˜ and V1 = ⭈⭈⭈ = Vm = Vmq1 = ⭈⭈⭈ .
Moreover we can choose the sets Vm such that for each m in ⺞
Vm s
⬁
D Imk , 1 = ⭈⭈⭈ = Imk , n ,
ks1
where the
k
Im,
i
k
k
are subintervals of w0, 1x satisfying sup Im,
i F inf Im, iq1 and
Ž Imk , 1 = ⭈⭈⭈ = Imk , n . F Ž Iml , 1 = ⭈⭈⭈ = Iml , n . s ⭋
If
set
k
␣ m,
j
s inf
k
Im,
j,
k
␤m,
j
s sup
Gs
k
Im,
j,
⬁
we define
k
Jm,
j
s M␤ mk , j _ M␣ mk , j and we
⬁
F D Jmk , 1 = ⭈⭈⭈ = Jmk , n
ms1
ž
ks1
if k / l.
/
.
18
BIANCHINI AND MARICONDA
k
Clearly G is a subset of PM . Moreover, by definition of Jm,
i,
⬁
< ␮ < mn Ž G . s lim
mª⬁
Ý < ␮ < Ž Jmk , 1 . ⭈⭈⭈ < ␮ < Ž Jmk , n .
ž
žÝ
ks1
⬁
s lim
mª⬁
s<␮
˜ <mn
<␮
˜ < Ž Imk , 1 . ⭈⭈⭈ < ␮
˜ < Ž Imk , n
ks1
⬁
⬁
ž ž
ks1
/
./
F D Imk , 1 = ⭈⭈⭈ = Imk , n
ms1
//
s<␮
˜ <mn Ž E . s < ␮
˜ <mn Ž A .
so that < ␮ < mn Ž G . ) 0: the measure ␮ being T < ␮ < we deduce that det ␮ Ž G .
) 0. Let ␴ s Ž ␴ 1 , . . . , ␴n . be a permutation of 1, . . . , n. We have
Ž ␮␴
m ⭈⭈⭈ m ␮␴ n . Ž G . s lim
1
mª⬁
s lim
mª⬁
⬁
Ý ␮␴ Ž Jmk , 1 . ⭈⭈⭈ ␮␴ Ž Jmk , n .
ž
žÝ
n
1
ks1
⬁
␮
˜␴1Ž Imk , 1 . ⭈⭈⭈ ␮
˜␴ nŽ Imk , n
ks1
/
./
sŽ␮
˜␴1 m ⭈⭈⭈ m ␮
˜␴ n . Ž E . ;
moreover < ␮
˜␴1 m ⭈⭈⭈ m ␮
˜␴ n < F < ␮
˜ <mn and, thus
Ž ␮˜␴
1
m ⭈⭈⭈ m ␮
˜␴ n . Ž E . s Ž ␮
˜␴ 1 m ⭈⭈⭈ m ␮
˜␴ n . Ž A . .
As a consequence
Ž ␮␴
1
m ⭈⭈⭈ m ␮␴ n . Ž G . s Ž ␮
˜␴ 1 m ⭈⭈⭈ m ␮
˜␴ n . Ž A .
so that
det ␮
˜ Ž A . s det ␮ Ž G . ) 0
and thus ␮
˜ is a Chebyshev measure.
Finally Theorem 4.6 applied to the T-measures ␮ and ␮
˜ gives
RŽ ␮ . s
s
½ž
½ž
␮
␮
˜
D Ž M␣
iq 1
_ M␣ i . : Ž ␣ 1 , . . . , ␣ n . g ⌫
/
1FiFn
i odd
D Ž ␣ i , ␣ iq1
1FiFn
i odd
/
5
5
Ž ␣ nq1 s 1 .
: Ž ␣1 , . . . , ␣n . g ⌫ s RŽ ␮
˜. .
MEASURES WITH A CONVEX RANGE
19
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