Radon-Nikodym theorems îor set-valued measures whose values are convex and closed

ANNALES SOC1ETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X X (1978)

A. C

and E . P

B

(Caen, Paris)

Radon-Nikodym theorems îor set-valued measures whose values are convex and closed

The present work was motivated by the study of two articles : Debreu- Sclimeidler [ 6 ] and Artstein [ 1 ]. These deal with the existence of B adon- Nikodym derivatives for convex set-valued measures in finite dimension.

These works remain within the framework of strict set-valued measures, but in the course of their construction their authors implicitly use the notion of weak set-valued measures. Our first contribution is to give a, new method of construction of a Badon-Nikodym derivative for weak set-valued measures using Ioneseu-Tulcea’s lifting theorems. The ela­

boration of this construction made us realize th a t this new method led to interesting results in the infinite dimensional case. These are the results obtained in the first p art of this paper. They were announced in Coste- Pallu de la Barrière [5].

On the other hand, it seemed to us th a t certain difficulties encount­

ered in proceeding to the case of strict set-valued measures were insuffi­

ciently brought out in Artstein [1] to the extent of leading to certain errors. This led us to deepen the problem of the Badon-Nikodym density for a finite dimensional strict set-valued measure. These results are pre­

sented in the second part.

0. Notations and definitions. Throughout this paper (E , F) denotes a dual pair of real vector spaces. F is endowed with the a ( E , F) topology and F with the Mackey topology r ( F , E) .

Let / be a convex function from F to ] — oo, +oo]. We call domain of f the set dom/ = {y e F/ f ( y) < + oo}. We recall th a t in case F is finite dimensional / is continuous on intdom / (“in t” stands for interior), and in the general case the same is true if and only if / is majorized on a non­

empty open subset of F.

Let G be a closed convex subset of E, we call support function of G the function <5*(-/C) from F to ] — oo, +oo] defined by

ô*(y!G) = sup «ж, y>/® eO}.

4 — Roczniki PTM Prace Mat. XX.2

284 A. Coste and R. P a llu de la Barrière

This support function is sublinear (i.e., convex and positively homogeneous) and lower semicontinuous on F. If y e F we call face of G in the direction y the set

C'y = {x<=CKa>,y> — ô*{ylC)}.

If / : _F->] — oo, -f oo] is a sublinear map, we denote by p / the closed convex set {x e E j\fy e F , <a?, у} </(?/)}. We recall th a t whenever p / is non-empty Ô* ( • /p /) is equal to the lower semicontinuous regularization / of /.

Let A and В be two closed convex subsets of E such th a t A n B Ф 0 . Then the support function ô* (• /Аг\В) is equal to the lower semicon­

tinuous regularization of the (so-called inf convolution) map:

y e F-+ inf {ô* (yx IA) + d* ( yJB)l yx + y 2 = y , y x., y 2 e F } .

All these basic results in convex analysis can be found in Ekeland [ 8 ] or Moreau [11].

We denote by Ж{Е, F) the set of closed convex locally compact non-empty subsets of E which contain no straight line.

Let (T, ST) be a measure space (i.e., ЗГ is a c-algebra of subsets of T) we call weak set-valued measure a map M defined on T whose values are closed convex non-empty subsets of E such th a t for every y e F the map A->ô* [yfM(A)) from У to ] — oo, -f oo] is a (cr-additive) measure.

If the values of M belong to Ж { Е , F) we say th a t M is a weak Ж( Е, F)-valued measure.

We call selector of a weak set valued measure M, a vector measure from to E such th a t m(A) e M( A) for every A e ZT. We denote by Sf M the set of selectors of M.

R e m a rk . In case E is finite dimensional, a weak Ж{ Е, P)-valued measure M is simply a weak set-valued measure such th a t every M( A) i contains no line. In this case we merely say th a t “I f contains no line”.

1 . Complements on liftings. We consider a complete measure space ( T , у), where у is a positive <r-finite measure.

We denote by g a lifting: th a t is a map о from £C°°(T, F , y) to if°°(T, y) such th a t:

( 1 ) e(f ) = f (a.e.),

( 2 ) f = g (a.e.) =>Q(f) = Q(9),

(3) Q( 1 ) = 1 ,

(4) f > o (a.e.) =>e(f)> o,

(5) g is linear,

( 6 ) Q(f-g) == e ( f ) ’Q(9) V/,

Bade п-Ж ilcodym th eorems 285

Such a lifting does exist (Ionescu Tulcea [9]). A function / e is said to be g-consistent iff g(f) = /.

We recall the following theorem ([9], p. 52):

T

1.1. Let E be a completly regular topological space, and Мд be the set of all weakly measurable maps f from T to E (i.e., <po f is meas­

urable for every continuous <p: E->R) such that <p{T) is relatively compact.

On Мд we consider the following equivalence: / = g iff u o f — u o g (a.e.) for every continuous map и from E to R. Then there exists one and only one mapping q from Мд to Мд such that:

(1) 8(f) ^ f ,

(2) f * * g » e ( f ) = 8(3), .

(3) g(uof ) = uo g(f) for every f e Мд and every continuous map и from E to R .

C

1.1 . Let К be a compact metrizable topological space, and let У д be the set of all measurable (x) maps from T to K. Then there exists one and only one mapping g from У°к to У°к such that:

(x ) C ( f ) = f (a.e),

<2) f = g (a.e) =>6(f)=Q(g),

(3) Q(uof) = uoç(f), y f e y ° K, V u e V ( K ) .

P ro o f. The weak measurability used in Theorem 1.1 is in fact the measurability with respect to the cr-algebra T on T, and the Baire cr-al- gebra on E (i.e., the smallest cr-algebra with respect to which every real continuous map is measurable). If К is compact metrizable, the Baire

« 7 -algebra and the Borel or-algebra coincide. Hence, we have У д = Мд.

Moreover, К being metrizable and compact, r£(K) is separable and the equivalence on М д mentioned in Theorem 1.1 is the equality a.e.

L

1 .1 . In the hypothesis of Theorem 1 .1 , a necessary and suffi­

cient condition for q (/) —f is that for every u e t ë ( E ) , g (uof) — uof.

P ro o f. The necessary p art comes from part (3) of Corollary 1.1.

On the other hand, if this condition is satisfied we deduce from p art (3) of Corollary 1 . 1 th a t for every и e r€ (K) uoQ(f) = g (uof) = uof . Thus 6(f) = f .

D

. If f e У д and / = g(f), we say th a t / is £-consistent.

The mapping q is called lifting on У°к associated to the lifting g on У°°.

i1) The measurability is understood with respeet to the cr-algebra У on T and

the Borel cr-algebra on E.

286 A. Coste and E. P a llu de la Barrière

P roposition 1.1. Let K x and K 2 be metrizable compact topological spaces, gx and

be the liftings on and associated to the lifting о on ££°°. Let (p be a continuous map from K x to K 2. Then for every f e we have

<P°Qxif) = M<P°f)-

P ro o f, (a) Suppose first th a t f = gx{f). Then for every и e ^ { K 2) we have

(ucxp)of = Q[(ucxp)of\ since uo<p e ^ ( K x).

Hence uo(<pof) = gluo((pof)]. Thus q>of = g%{<pof).

(b) We deduce from (a) th a t for every / e ^°Ki

<P°Qi(f) = QsfyoQiif)']-

Moreover, r

<P°Qiif) =<pof a.e.

Hence

<P°Qi(f) r= Q*(<P°f)-

C orollary 1.2. In the hypothesis of Proposition 1.1, (a) i f cp is an isomorphism from K x onto K 2 we havè

Qi(f) = <P~l [Qi(<pof)Vi (b) i f li 1 <= К 2, then q 2 is an extension of gx.

A pplication . Let В denote [ — 00 , + <*>] = 22 и { — 00} и { + °°}.

The lifting q on associated to q on £P°° can be defined by g(f)

= ta n (g (A rc ta n o /)); and g is an extension of q . A ^-consistent function / 6 i f 00 is still ^-consistent when considered as an element of

P roposition 1.2. Let K x and K 2 be metrizable compact topological spaces and К = K x x K 2. Let qx , q 21 q be the liftings on -&к2>

associated to the lifting q on J§?°°. Then, for every f e â?°K we have 5(f) = (61 ( P ) , e A f % №here m =

P ro o f. Setting g(f) as above we obviously have

{ ? ( / ) = / a.e. and f = g a.e. =><?(/)= 0 ( 0 ).

I t remains to show th a t for every и e # ( K) u o g ( f ) = g{uof).

Let us suppose first th a t и e ^ ( K x) 0 ^ ( K 2), i.e.,

и = ^ V {0Wi, V i E ^ ( K x), W{ e & ( K 2).

Radon-Nikodym theorems 287

Then n

UOQ(f) = £ v {o дх(Р)' Щ°9г(Р)

г -1

n г =1

п

= Q ( ^ { v i of)-{wiof)) = g(uof).

г— 1

Let и е ^ ( К ) and e ^ { Ef ) <S>^(K2) such th a t un->u for the topology of uniform convergence. Then uno f - > u o f uniformly. Hence, Q(unof)-+

g(uof) uniformly. On the other hand unoQ(fn) converges to и о <?(/) uni­

formly. From the relation unog{f) = g(unof) we deduce th a t uo g ( f )

= o(uof).

C

1.3. Let K x, K 2, К , &<? metrizable compact topological spaces and let gx, g2, Q be the liftings on J£°Ki, У к 2, associated to a liftingr

q on Ж°°. Let p be a continuous map from K x x K 2 to K . Then for every f x e J£°Ki and f 2 e we have

г М Л ( •)>/»( -))1 = ? > o » (/i)(-),5 (/.)(-)].

Finally, we have the following theorem.

T

1.2. Denote by g the lifting on У 0 - or on jSf+ = associated to a lifting g on j 5?°°. {We recall that these liftings are extensions of one another.)

Then

(i) Q(f+g) = Q(f) + Q(g), /е=^°°, 9 (ii) Qif+g) = 6(f) + Q(g), f e & + \ g e ^ + , (iii) g(A-f) = Ag(f), f e & +, A > 0 ,

(iv) / < y a-e. ^Q(f)<Q(g), f e ^ + , g e£?+.

P ro o f. To show (i) and (ii) we respectively apply the previous corollary to the following two cases:

(i) K x compact in R such th a t f (T) c K x\ K 2 = R) p( x , y ) = x-{- y . (ii) K x = K 2 = R +; p ( x , y ) = x + y.

As for (iii) we apply Proposition 1.1 with K x = K 2 = R + and p{x)

= Ax, (iv) is obvious.

Bern a rk . We can also prove (ii) for f and g bounded below. Indeed if / > a, g ^ b , take K x = a + R + , K 2 — b + R +, К = a -f & -j-B+ .

2. Properties of convex locally compact sets. Troughout this paper (E, F) denotes a dual pair of real vector spaces. E is endowed with the topology a ( E , F ) and F with the Mackey topology r ( F, E) .

We recall th a t Ж( Е, F) denotes the set of closed convex locally

288 A. C oste and R. P a lin de la Barrière

compact non-empty subsets of E which contain no straight line. For further notations of convex analysis see Section 0.

L emma 2.1. Let G e Ж( Е, F). I f As(0) = 0, then G is compact (where As (0) denotes the asymptotic cone of G).

P ro o f. Suppose 0 e 0. Let p be a continuous seminorm on E such th a t VnG is compact, where F = {же Ejp(x) < 1 }. Let Й1 be an ultrafilter on C and suppose th a t lim p(x) = + °o. Then --- e V nG for every x. x

x ® p

So lim --- = £ exists and p(£) = 1 . Moreover, £ e As(0): indeed for

2 >(®) x x

every A > 0, A£ = lim — — •x a n d ---x e G whenever p(x) > A. This v Р(а>) p(a>)

contradicts the assumption As(O) = 0 . So th a t lim p (a?) < +oo. There­

sa"

fore, there exists 1c > 0 and U such th a t U <= Gn f k - V) c Jc(GnV) which is compact. Thus is convergent, and so О is compact.

T heorem 2.1. Let C be a closed convex non-empty subset of E. Then 1° i f y 0 e F, the following properties are equivalent :

(a) <5*( • /С) is finite and continuous at y 0 for the topology r ( F , E ) . ((H) For every a e R, the set {x e O /O , y 0y ^ a} is compact.

2° In order that there exists an element y 0 e F satisfying (a) and ((3) above it is necessary and sufficient that G e Ж( Е, F).

P ro o f. 1° follows from Moreau [11].

2° is a theorem of Joly [10]. For the sake of completeness we give the proof of the non-trivial p art which is th a t G e Ж (E, F) implies th a t there exists y 0 e F satisfying ((H). First suppose th a t О is a cone.

Let F be a neighbourhood of 0 such th a t V r\G is compact. By Bour­

baki [3] there exists y 0 e F such th a t the sets {ж/0> y 0} < a } n 7 n O ; a > 0 is a base of neighbourhoods of 0 in F nG (use the fact th a t 0 is extremal in VnG). So there exists a 0 > 0 such th a t {x/(x, y üy < a0}n n V n G a ^ V n G . This implies th a t {#/<#, y0> < a0}nG c VnG. Indeed suppose this is not true: so there exists x e C with О» y f) < a0 and x $ V.

B ut there is A e [ 0 , 1 ] such th a t Ax e F \ | F . Hence Ax g { x / ( x } y 0y < a0} n n V n G and Ax $ |F , which leads to a contradiction. So {ж/О? y f) < a0}nO is compact and it is easy to see th a t this remains true if we replace a 0 by any a g В. Now observe th a t if G g Ж( Е , F ), then As (0) e Ж (E, F).

So there is y0 e F such th a t {x e As(0)/<®, y0> < a) is compact for every a > 0 . Let 1 = (же 0/O> yf) < a}. We have

As (A) = As(0)nAs{a? e E / ( x , y 0} < a}

= A s(G)n{x e E K x , y 0> < 0} = {0},

so A is compact by Lemma 2.1.

Radon-Nikodym theorems 289 R e m a rk 1 . If y 0 satisfies (a) and ((3), then the face Cy of C in the direction y 0 is compact and non-empty.

T heorem 2.2. Let A and В be closed convex non-empty subsets of E such that closure { A+B) belongs to Ж( Е , Е ) . Then,

1 ° closure (A -\-B) = A -\-B.

2° I f У о is a point of continuity of ô* (• /А-\-В), it is a point of con­

tinuity of ô*('IA) and ô*(‘/B).

3° A and В belongs to Ж{Е, F).

4° (A AB)'V = Ay +B'y for every y

F.

P ro o f. A + B is closed by a result of Dieudonné [7]. Furthermore we have <3*( • /А +B) = <5*( • I A) + <5*( • /В), whence ô* (у /А) < ô* { y j AA + Б ) — <X, y}, where x 0 is an arbitrary point of B. Hence if <3*( • /А +B) is continuous at one point y0, this is also true for <5*( • /А), which proves 2 °, hence 3° by Theorem 2.1 and 4° is readily checked.

Let A e Ж { Е , F) and y 0 g F, where d*( ■ /А) is finite and continuous (Theorem 2.1). Let В be a closed convex subset of E such th a t 4 п Б is compact and non-empty.

Then the support function <5*(-/AnB) is equal to the map у-+Ш{0*{уг1А) + 0*(у21В)1у1 + уь = y ,

F, y %e F ) .

Indeed the latter is majorized by d*{‘ /A). Therefore it is continuous and finite a t y 0. Its semicontinuous regularization <5*( • / A n B ) (see Section 0) is everywhere finite. Hence its domain is an everywhere dense convex cone with non-empty interior which therefore is the whole F.

So it is everywhere continuous and coincides with ô* (• I A n B ) . As a con­

sequence we have the following result. (See Wegmann [14] for analoguous results.)

L emma 2.2. Let A e Ж {E, F) and y 0

F, where ô*( • I A) is finite and continuous. Let B = {x g E j ( x , yfy = ô* {y0/A)}. Then we have

ô*(ylAnB) = I n f { ô * ( y - K y 0IA) + ZÔ*(y0IA)!l.GB}-, V g F.

T heorem 2.3. Let M be a weak Ж( Е, F)-valued measure. Let y0

in td o md* jM{T)). Then the map M yQ: A->[M{A)f VQ is a weak convex compact set-valued measure. {We recall from Section 0 that [M(A)fyQ denotes the face of M( A) in the direction y 0.)

P ro o f. Let us set r 0 (A) = ô\(y0IM(A)), 4 e . f ; and H(A) = {x

E l К®,УоУ = v 0 (A)}, i e l We have М Уо{А) = М(А)Г\Н(А). By Theo­

rem 2.2, MyQ is finitely additive. Let us prove the a-additivity of each map A-+ô (ylMyo(A)),

we have (Lemma 2.2)

d ' l y i K M ) ) = In l { ^ { y - y j m ^ + ^ y x X i A ) ) ]

y e F

= Inf {<5*{ y - Xy J M( A) ) + b ll(A)\-

290 A. Coste and R. P a llu de la Barrière

Since ô*(y I M'yo{A)) is finite for every y e F and every 4 e f , there exist 1 such th a t S*(y — l y 0/M(T)) < +oo. Thus ô*(y — l y 0/M( • )) is a finite measure. Let A n be a decreasing sequence whose intersection is empty.

We have

(f (y IM'V0 (A )) < à* ( y - k o / м (A ) ) + A (A ) •

Since the second side of this inequality converges to 0 for + oo, we have:

lim sup ô*(yIМ'щ{Ап)) < 0 .

n

— >00

From the relation ô*(у1М'щ(А))+ й*( — ylM'V(i{A)) > 0 we deduce th a t lim inf d*(y I M ' ( A n)) > 0.

n-»oo

Hence lim ô*(yIМ'Уо{Ап)\ = 0 . Thus 6*(yjM'4 {‘)) is (r-additive.

ft— >oo

C onsequence . I f M is a weak , F)-valmd measure, then M has a selector.

P ro o f. I t suffices to take a selector of M'y with у e intdom<5*( • IM{T))

#(see Pallu de la Barrière [12] or Coste [4]).

We now tu rn to proving the following property and lemmas which will be used in the next paragraph.

P roposition 2.1. l e t Q e Ж{Е,Ж). Then Q is weakly complete.

P ro o f. We may assume th a t 0 eQ. We must prove th a t the bipolar ф 00 of Q in F is equal to Q. Let 2 е ф 00. Then, for every у e F , <£, y>

<; 6*{ylQ). Thus since ô* ( • IQ) is finite and continuous at one point of F for the topology r { F , E ) (Theorem 2.1), this is also the case for <z , •>.

And so z e E.

The following lemma is a strengthening of lemma communicated to us by M. Yaladier.

L emma 2.3. Let {E, F) be a dual pair, F being endowed with the Mackey topology x (F, E). Let f be a convex function from F to ] —- oo, + ° ° ] which is finite and continuous at one point y 0 e F. Let D be a dense subset of F.

Then we have

I n f { f(y)l y e F } = In f {f{y)l y e H n in td o m /} .

P ro o f. We have in td o m / Ф 0; / is continuous on intdom / and B n in td o m / is dense in intdom /.

Hence inf {f{y)ly edom /} = inf {f(y)/y eD n d o m /} .

Let Ух e d o m /\in td o m /. The restriction of / to [у0,Ух] is convex, finite on [y 0 ) 2 /i[? whence

/ Ы > Inf {f(y)ly e [y0, ух[}.

So th a t f ( y x) > Inf {f(y)/y e intdom /}, completing the proof.

Radon-Nikodym theorems 291 C orollary 1. I f h is a sublinear function on F finite, continuous at one point of F, then for every dense subset of F, we have

Vh — {x e F [{со, уУ < h (y) , Vy e D }.

P ro o f. If for every y e B (je, y} < h{y), we have Inf {h(y) - <x , уУ1у

JD} > 0.

Hence Inf {h (y) — (x , y y/y e F} > 0. But this infimnm cannot take values other than 0 and — oo, so th a t it is equal to 0, whence x e Vh. The con­

verse is obvious.

C orollary 2. Let A

Ж {E, F). Then i f I) is a dense subset of F we have

A = {x e E / ( x , yy < ô*(yfA), Vy e D).

L emma 2.4. Let Q

Ж ( E , F) and К be a convex compact subset of F.

Then Q A K E 3 i f { E , F ) .

P ro o f. Q + K is convex closed and we have:

д*(-1Я+Щ = ô*(-IQ) + Ô*(-IK).

Since Q

{F, F),

IQ) is finite and continuous for r ( F , E ) at one point of F. Hence <5*(• IQ A-К) is finite and continuous a t this same point.

This implies th a t Q + ül e Ж ( E , F). I t is also possible to deduce this lemma from a more general unpublished result by Saint-Pierre on the local compacity of the sum of a compact and a locally compact.

3. Radon-Nikodym theorem for weak Ж ( Е , F)-valued measures. In the following (E , F ) is a dual pair, where E is endowed with o{ E, F) and F with r ( F, E ) . We consider a complete measure space (T, 2Г, p), where p is positive, q denotes a lifting of JS?°°(T, Ж , и), and q denotes the lifting on associated to q according to Section 1.

T heorem 3.1. Let M be a weak Ж (E , Ffvalued measure defined on Ж and such that AI{A) — {0} whenever p{A) = 0 . Assume that either

(a) E is finite dimensional.

(b) E is suslin and there exists ( ^ е Ж ( Е , Е ) such that AI(A)

c А е Ж.

Then there exists a set-valued function taking its values in Ж ( Е , Е ) , scalarly measurable (i.e., the map t-><5*[у/Г (/)) is measurable for every

У e

F) such that the negative part of each function t->ô*(y/-T(i)), y

is fi-integrable and that

0*<ylM(A)) = j A e T .

292 A. Coste and R. P a llu de la Barrière

P ro o f. 1° First, we assume th a t 0 e M( A) for every l e i Then for every y e F, b*{yjM{‘)) is a measure defined on T with values in [0, + oo] absolutely continuous with respect to pi. For every y x, y 2 e F we have

<% , + y% m ( • » < <5* ( y J M ( • )) + Й* ( y J M( ■ )).

Now, consider the map h from F x T to [0, + ° ° ] such th a t h( y , •) is the ^-consistent version of b*{y/M( • )). Then h(-, t) is sublinear

dpi for each t e T, and by definition we have

0*(ylM{A)) = f h(y, t)pi{dt); А е Г , y e F .

2 ° For every t e T , we set _T(<) = Vh{-,t), i.e., r(t) — { x e E j ^ x , y }

< h{y, t), У y e F}. Then we have b*(-/r(t)) — h(-,t), where the latter denotes the lower semi-continuous regularization of Ji(-,t).

3° Note th a t 0 g Г(<). Let us show th a t F{t) is almost everywhere in Ж{Е, F): First assume th a t E is finite dimensional. Let D be a count­

able dense subset of F and let us set D 0 = D nintdom d* ( • fM{T)).

For every y e D0, we have

b*(y/M(T)) = j h { y , t)pi(dt)-,

. t

Hence h ( y , t ) < -f oo except if t belongs to a null set depending on y.

Since H 0 is countable, we have h { y , t ) < + oo except if t belongs to a null set independent of y e D 0. Since E is finite dimensional the convex hull of JD0 is intdom b* (• /M{T)) and so

dom /Ц -, t) => intdom <5* (• /M(T)) a.e.

Hence h( - , t ) is finite and continuous at one point, and consequently r(t) e Ж { E, F) a.e. (Theorem 2.1). Now, under assumption (b) we have for every y e F

I t follows from Theorem 1.2 (iv) th a t h{y, t) < b*(y/Q), t e T , y e F. Thus r(t) <= Q for every t e T , and so Г{Ь) e Ж {E, F ). Moreover, the inequality h(y, t) < Ô* (y IQ) shows th a t for every t e T , Ji(-,t) is finite and con­

tinuous on intdom ô*(• IQ).

4° The graph of the set-valued function Г belongs to Ж where

& is the borelian <r-algebra of E. Indeed this graph is equal to C\{{t,x) e T xE\<oc,yy ^ h { y , t ) } ,

y e D

Radon- Nikodym theorems 293 where D is a countable dense subset of F (apply Corollary 1 of Lemma 2.2).

Since E is suslin, and is complete, this implies th a t Г is scalarly meas­

urable (Sainte-Beuve [13]).

5° Now we show th a t h ( y , t ) = h{y, t) except if t belongs to a null subset N y depending on y : if not, there would exist y 0 e F and A e&~

with /л (А) > 0 such th a t

h{y0, t) < HVoi t) for every t e A .

Hence y0 belongs to the boundary of dom h(-, t) for every t e A . Consider one point y x e int d o m h ( -, t) for every t e A . (Under assumption (a), take y e int dom <5* ( • /M(T)) and under assumption (b) take y x e int dom ô*( • IQ); see p art 3 of the proof.) For every y e [y0, yx] let us set

H(y) = f h ( y , t) y(dt), and H(y) = f Ji{y, t) y(dt).

A A

Taking into account the positivity of h and h, the maps H and H are well defined and convex. We have H(y) — <5* (y/31(A)) and H is lower semi-continuous, whence continuous on [yx, y0] since H{yx) < + oo.

Moreover, since [yu y0[ int dom h(-, t) we have for every t e T and У e lyx, y0[, h(y, t) = h_(y, t). Therefore H{y) = H{y) Vy e [y0, y x[. Any­

way, we always have H(y) < H(y). According to the properties of convex functions on intervals we have

H( y 0) ^ limsup S ( y ), since H( yt)<i -f-oo.

V-Vo

By continuity of H on [у0, у г], we get

H ( y 0) = lim H(y) = l i m % ) < # ( y 0),

V-*Vo V-+V0

which leads to a contradiction. The proof is complete in case 0 e l ( i ) , VA евГ,

6° We recall (Proposition 2.1) th a t Q is weakly complete. Let В be a countable dense subset of F and H be the vector subspace of F generated by D. Let us show th a t Q e j f { E , H ) and th a t on Q the topolo­

gies induced by a{E, F) and a{E, H) coincide. First note th a t Q is closed for a ( E , H) since, by Lemma 2.2, Q = П ty e U/<a?, y> < à*{уIQ)}.

y e D

Now, let F be a neighbourhood in Q for the topology induced by o(E, H) of a point x 0 in Q. Let y be a point in H such th a t {x e Ql ( x , y> •

> a} is compact for every a (this condition being satisfied for every

У e int dom<S*( • /Q) and H being dense in F for r ( F , E ) , it is sufficient

to take y e JT n int dom <5* ( • IQ) which is not empty). Let a e B such th a t

(fioi УУ > ot-J-1.

294 A. Coste and R. P a lin de la Barrière

Then Vr\ {x eQI(oo, уУ> a +1} is a neighbourhood W of x 0 in Q endowed with a{E, E ), and we have W <= {x e Qj ( x, y} > a +1} which is a compact neighbourhood of x 0 in Q endowed with cr(E, F). Therefore W is also a neighbourhood of x 0 in Q endowed with a(E, F). This implies th a t a (E, F) and a ( E , E ) coincide on Q. Therefore Q е Ж ( Е , H ), and Q is a polish space when endowed with a(E, E ) since it is complete for a{E, E) and th a t the topology a { E , E ) is metrizable.

7° Denote by E* the algebraic dual of i / ; th at is the weak conrple- tion of E for the tojmlogy a(E, E). Let m be a selector of M and / be a weak density with values in E* ; th at is a scalarly measurable and in­

tégrable (for the pairing (H *, H)) such th a t

<m{A), y} = f <f{t), У> V y e E . л

In fact / is a measurable mapping from T endowed with the <r-algebra to E* endowed with the borelian (7-algebra associated to a{E*, E). More­

over, since Q e Ж(Е*, E), it is the intersection of a sequence of closed hyperplanes. Therefore f(t) e Q for almost every t. Since the two topo­

logies a ( E , E ) and a ( E , F ) coincide on Q, f is a measurable mapping from T endowed with the u-algebra ,T to E endowed with the borelian cr-algebra associated to a{ E, F), and therefore / is scalarly measurable for the pairing (E , F ) (i.e., < /(•), уУ is measurable for all y eF).

We end this point by showing th a t f is scalarly integrable. Let {Tn}

be a ^"-partition of T such th a t for every n, f { T n) is relatively compact in Q. Then the restriction of / to T n is scalarly integrable for every n.

For A c T n let m' (A) be the element of Q defined by <m ' ( A ) , y } =

= f (.f(t), уУ p(dt), у e F . From the definition o f /, we have у >

A

= (m{A), y>, Vy e E , and since E separates the points of Q, this equality remains true for all y e F. Therefore we have

(m{A), У> = f уУр{М), у e F , A a T n.

A

Now, let A e X . We have

<m{ AnTn), y} = f (f(t)j уУ fz{dt), V y e F .

Ас лТп

The series ( m { A n T n), yy is convergent and its sum is <m(A), yy. Hence the series / </(<), y> fi{dt) is convergent; this implies th a t < /(•), yy is integrable and th a t

<m(A), yy = /< /(< ), уУ fi(dt), V i / e T , V l e J .

A

Thus, / i s a weak density of m.

295 f

8° Let us set M X{A) = M( A) — m{A), VJL Then M x is a weak Ж( Е, F)-valued measure such th a t 0 e M X(A). Consider again a «^"-par­

tition {Tn} of T such th a t f { T n) is relatively compact in Q.

Let {Kn} be a sequence of compact convex subsets of Q such th at f{t) e K n, Vt e Tn. For all A a Tn, we then have M X(A) a y {A) (Q —K n).

Since Q —K n

Ж ( E , F) (Lemma 2.2), we may apply to the restriction of Mx to Tn the results obtained in part 6° above : there exists a set-valued function Гп defined on Tn whose values are in Ж { Е , F), scalarly meas­

urable such th a t:

«*(ÿlitx(A)) = J V (yi rn(t))n(dt), A = Tn.

A

Setting r{t) — r n(t) +/(#) if t e Tn, we define a set-valued function whose values are in Ж { Е , F), which satisfies the conditions of the theorem.

9° If E is finite dimensional, parts 7° and 8° become simpler: We consider a selector m of M and a density / of m with respect to y. The weak Ж{Е, F)-valued measure defined by M X(A) = M( A) — m(A), A e^~

is such th a t 0 e M X(A), A e 2 T . We consider а Ж( Е, F )-valued density Гх of M x and we set F(t) = r x(t) -{-fit). The set-valued function Г sat­

isfies the conditions of the theorem. The proof is complete in either case.

D

3.1. Let M be a weak Ж (E, F)-valued measure. We call weak density of M a scalarly measurable set-valued function (i.e., V y e i \ <->г*(у/Г( 0 j is measurable) whose values are closed convex non-empty subsets of E such th a t for every y e F the negative part of the map t-^ô* [y /r{t)) is -intégrable and th a t for every A e&~,

f d*{9 i m ) f H d t ) = дЦу / ЩА) ) . A

We end tljiis paragraph by proving the uniqueness of the weak density of a weak Ж (E, F )-valued measure.

T

3 .2 . We assume that E endowed with a{ E, F) is suslin.

Let M x and M 2 be two weak Ж (E , F)-valued measures absolutely contin­

uous with respect to у {i.e., M X{A) = M 2{A) = 0 whenever у {A) = 0), and Гх, Г 2 some weak densities of M x and M 2. We assume for every А еЖ that M X{A) c M 2(A) and that Г 2 is Ж {E, F)-valued. Then we have al­

most everywhere Г х{1) c r 2(t).

P ro o f. We have f ô*( yi r x{t)) y(dt) < / б* (уЦ\{Щ y{dt) for all A & T

a a

and all у e F . Hence ô*(yirx{t)) < ô*(y/r2{t)) except if t belongs to a null set dependent on y. Let D be a countable dense subset of F. We then have for every y e D

'à*\yH\{t))< ô*{yir2(t)),

Radon-Nikodym theorems

296 A. Coste and R. P a llu de la Barrière

except if t belongs to a null set N independent on y . If t $ N, then we have

A W = | * E Î K * , J > < S*(y/A(«)),Vy е £ ) , i.e., r x(t) c r 2(t) (Lemma 2.2).

C

1. I f M is a weak Ж( Е, Ffvalued measure such that Щ А ) c p(A)-Q, 'iA g ,T 1 where Q is an element of Ж( Е, F), then every weak density Г of M satisfies r(t) <= Q a.e.

P ro o f. Indeed the set valued function A-+p(A) Q is a weak Ж (E, F)- valued measure of which a weak density is given by the constant mapping equal to Q.

C

2. Let M be a weak Ж (E , Ffvalued measure such that M( A) = 0 whenever у (A) = 0 and Г х, Г 2 be two weak densities of M.

We assume that Г 2 is Ж{Е, Ffvalued. Then we have r x{t) = r 2{t) a.e.

P ro o f. Indeed we have r x(t) a r 2{t) a.e. Therefore Гх is almost everywhere Ж{Е, F) valued. By interchanging Fx and Г 2 we get

r 2(t) c r x(t) a.e.

This completes the proof.

4. An extension in case of finite dimension. The aim of this paragraph is to show th a t in finite dimension it is possible to obtain a generalization of the Badon-Nikodym theorem to the case of weak set-valued measures whose values may eventually contain straight lines. Throughout this paragraph we assume th a t E is finite dimensional. If G is a closed convex non-empty subset of E, and x 0 e G, we set

5 Л?(С) — {h e E l%0-\-Xh g G, V/l g В}.

I t is easy to see th a t J£{C) is a vector subspace of E which is the intersec­

tion of the asymptotic cone of C with its symétrie. In order th a t G be without straight line, it is necessary and sufficient th a t JF{G) == {0}.

If Gx c C2, we have 3?{Сг) c= <?(C2). And if a e E we have S£ (C + <*) = JSP (0).

T

4.1. Let M be a weak set-valued measure such that M( A)

= {0} whenever /л (A) = 0 . Then there exists a countable F -partition {T{}

of T such that : 1 ° p(Tt) > 0 ,

2° S£(AI(A)) — (Jf(f\-)) whenever A c z Ti and p(A) > 0.

P ro o f. 1° First let us show th a t there exists T 0 e T with y ( T 0) > 0 such th a t ( if (A)) = J?(M{T0)) whenever A A cz T 0, /л(А) > 0.

Indeed let p be the minimum value of dim ( i f (A)j for A g /л(А) > 0,

and let T 0 be such th a t dim (AI (T0)( = p. For each A cz T Q with p(A) > 0,

we have M{A) cz M ( T 0) ~ a with a e M ( T 0\ A ) , so th a t &( M( A) \

a S£ ( i f (T Q)). By the minimum assumption on p, this implies th a t jSf ( if (A))

Radon-Nikodym theorems 297 2° Consider the set consisting of all non-empty families {TJ of sub­

sets of T such th a t

Т, е ^ Г, p( Tt) > 0 , the X) are pairwise disjoint, and

Se[M{*A)) = ^ ( Ж ( Т г-)) V A e .T , A œ T{, y(A) > 0.

Such families exist by 1°. Choose a maximal family (according to Zorn theorem). This family is necessarily countable. Let us show th a t T = is ii-mill. If not, by considering the restriction of M to T,

i

we could claim by 1°, th a t there exists T 0 с T with y ( T 0) > 0 such that

£f[M(A)) = S£[M(T0)), VA a T 0, у (A) > 0. The set T 0 could therefore be joined to the family {T{} which would no longer be maximal. Thus changing one of the sets T t by we get a family satisfying the con­

ditions of the theorem.

T

4.2 . Let M be a weak set-valued measure such that 31(A)

= {0} whenever /л(A) = 0 . (We recall that E is finite dimensional.) Then M has a weak density (Definition 3.1).

P ro o f. 1 ° Assume first th a t jУ?[М(А)) = <g[M(T)), for every i e . f with у (A) > 0. Let us set H = £?[M(T)). Let E x be a complement space of H and M X(A) = M( A ) r \ E x for l e J , We have M( A) = M X( A ) +H and M X(A) contains no straight line, so th a t M x is a weak set-valued measure containing no line. Let Fx be a weak density of M x and let us set r(t) == r x(t)+H. I t is readily verified th a t Г is a weak density of H.

2° We

turn to the general case. Let {TJ be a «^-partition of T satisfying the conditions of Theorem 4.1. For each i, the restriction of 31 to Ti has a weak density Fi on T i with respect to the restriction y T.

of у to T t . Setting F(t) = F{(t) whenever t e Т {, луе define а луеак density of M.

5. Strict set-valued measures. Throughout this paragraph and the follo>\7ing one, E denotes a finite dimensional vector space and F — E ' . We consider a fixed complete measure space (T , ^ , y ) , where у is po­

sitive cr-finite. As лее make use of several results stated in the appendix, we repeat here some of the notations from there: луе say th a t a set-valued function Г from T to E is measurable if its graph belongs to ЗГ ® ^, where ââ is the borelian ^-algebra of E. For every measurable set-valued func­

tion denotes the set of integrable selections of F, and r i / T denotes the set valued function tf-^ri.T(£) (i.e., relative interior of F(t)).

We recall th a t E being finite dimensional, a given weak set-valued measure M is a weak Ж( Е, F )-valued measure if an only if M( T) con­

tains no line (and consequently every M( A) contains no line). In this

case we merely say th a t “M contains no line”.

298 A. Coste and R. Fallu de la Barrière

Some of the following results possess a generalization to the infinite dimension case. However, some others are proved by specific finite di­

mension arguments, so th a t the results which we have in infinite dimen­

sion are too partial to be presented here.

Let {TFJ be a finite or countable family of subsets of E. We denote by ZW< the set of elements where {xf\ is a summable selection of

i i

the family {Жг}, i.e., is a summable family of E such th a t xt e W{.

D

. Let (T, . T) be a measurable space. We call a strict set­

valued measure defined on IT whose values are in E, a map M from .T to the set of non-empty subsets of E such th a t for every A e - T and every countable ^"-partition {JLJ of A, we have

Щ Л ) = ] ? М ( А , ) .

We call selector of the strict set-valued measure M, a measure m defined on ST with values in E such th a t m(A) e M( A) for every l e f , and we denote by the set of selectors of M.

R e m a rk . If the property M( A) — (Af) is only valid for finite

г

partitions {J.J of A, we say th a t M is additive (“additive set-valued set function”).

T

5.1. Let M be a strict set-valued measure whose values are convex. Then the map A->closure M{A) is a weak set-valued measure (see Definition in Section 0).

P ro o f. Since the family {M y A)} has a summable selection (due to M( A) Ф 0), the relation

b*(y!M(A)) = 2 ’ й *( 2 //Ж (Л )), i

comes from Proposition A.5 of Appendix.

P

5.1. Let AI be a strict set-valued measure whose values are convex and y 0 e F . I f the face of M(T) in the direction y 0 is not empty, then the map A- >[M (A))^o from ST into E is a strict set-valued measure which we denote by М'Уо.

P ro o f. If Cx and (7a are two convex subsets of E f we have (Сх+<32)y

= {Сх)у + {Сг)'у , y

F. Hence the condition (M(T))yQ Ф 0 implies (M{A))'yQ Ф 0 V A e3~. Let А Е.Ф, {A y be a countable ^ -p a rtitio n of A and x e (M (A ))'yfj. There exists a summable selection {.rj of (Ж(Аг-)} such th a t x = y^x^ . We then have

i

<æ, yf> = £ <®i,

(Xi, Уо> < à*(y9j M { A {))\

Radon-N ikodym theorems 299

and

£ <*<,?/„> = »*\Уо1М(А)) = £ д * ( у 01 ЩА {)).

i i

This implies <^,2/0> = ô*(y0/3I(A)), i.e., х{ е(31(А))'щ. Conversely let {a?t} be a summable selection of the family |(Ж (А г-))уо|. We have

<®<, Уо> = д*(Уо1М(А{)), so th at setting x = x{, we get

г

<®,У„> = 2 à*{yJM(At)) = g>{y,IM(A)), i

i.e., æ е[М{А))'щ.

The following theorem is fundamental in the study of strict set­

valued measures. Its proof requires E to be finite dimensional. We do not know if it is possible to generalize it to infinite dimension.

T

5.2. Let ( T, $~, p) be a complete measure space, and M be a strict set-valued measure whose values are convex subsets of E (dim(E)

< +oo) such that closure (M(T)) contains no straight line. We suppose that M( A) = 0 whenever p(A) — 0. Then for every A we have

31(A) = {m(A)lm e S^M}.

P ro o f. I t is sufficient to prove it in case A = T. Assume th a t dimen­

sion of M( T) = Tc and let us suppose the result true whenever dimension of 3I( T)<l c. Let |е Ж ( Т ) . First suppose th a t Ç $ri(M(T)) (see nota­

tions in Appendix). Then there exists у й е Е such th a t £ e M'y^(T) (the face of M( T) in the direction y 0). We may assume <•, y 0) to be not con­

stant on the affine variety generated by M(T), so th a t dimension of Myo(T) < dimension of M(T). Since M'y is a strict set-valued measure (Proposition 5.1), there exists by assumption m e such th a t m(T) = |, consequently m&SLM. Let us suppose now th a t £ еп (Ж (Т )). Let Ж be the weak set-valued measure defined by 31(A) — closure 31 (A), l e J , and Г be a weak density of 31 (Theorem 3.1).

By considering the density / of a selector of Ж which does exist by the consequence of Theorem 2.3, we obtain an integrable selection of Г (Theorem 3.2). So th a t S£\ïr ф 0 by Proposition A .6 of Appendix.

By Proposition A.5 of Appendix, the set {/ fp If e i ^ r ) bas the same

T

support function than M(T), whence the former contains ri M(T). There­

fore, there exists / e 5£\ir such th a t | = f f /л. Let us show th a t f f /л e ri Ж (A)

t a _

for every A e . If not there would exist A e such th a t / f p ф ri 31(A),

A _

hence also y 0 e F such th a t - <3 *( —у0/М(А)) < < ///* , Уо> = à* (у 0/Ж (А)),

А 5 — Roczniki PTM Prace Mat. ХХ.2

300 A. Coste and E. P a llu de la Barrière

Le., - / ô * ( - y j r ( t )) p(dt) < J </(<), у0У = / à*(y0jr(t)) p(dt). But

A A A

we have — ô* ( —y 0ir(t)) < </(£), y0> < à*(y0/r{t))f Wt e T. There would therefore exist A ' with /л{A') > 0 contained in A such th a t Vtf e A', - à * ( - y 0ir{t))< <f{t),yQy = ô*(y0ir(t)), which implies f(t) £ri(r(*)) for every t e A', which leads to a contradiction. Now set m = fp. We have m(T) = I and, for all A

e ri M ( A ) = ri M(A). Therefore m e S?M.

Thus the theorem is proved in case dimension M(T) — к if it is true whenever dimension M ( T ) < k. Since it is true for к — 0, it is also true for all k.

The aim of the following proposition is to show th a t in finite dimen­

sion; the study of weak set valued measures containing no line can he reduced to the study of set-valued measures whose values are subsets of J?” . Let a base (ef) of E be given, we call positive orthant, the set of elements of E whose coordinates with respect to this base are non-negative.

P

5.2. Let M be a weak set-valued measure containing no line. Then there exists a base {et} of E and a E-valued measure such that

M( A) c p ( A ) + E +, where E + is the positive orthant.

P ro o f. We have intdom<5* ( • jM(T)j Ф 0 . This set is an open cone of F. Let {e'i} b e 'a base of F such th a t — e\ e intdom<5*(- jM(T)) for every i. Let {e% } be the conjugate base of E, and E + the positive orthant of E. We have for every i, inf {<#, ef) jx e M(T)} > — 00 . For every A e Ф, we have

Ô*(y/M(T)) = 0*{ylM(A)) + 0*(yIM(T\A)),

which implies 6*[y jM(A)) < +00 whenever 6*{y jM(T)) < + 00 . If we define for A P(A) by

<p(A), <•> = inf {<a>, efyjx e M{ A) } ,

(the second side being well defined according to the previous remark), then p is a F-valued measure and we have

M( A) cz р ( А ) ф Е +.

T

5.3. Let Ж be a weak set-valued measure. For every A e ^ ,

let us set M( A) = {m(A)jm e S?M}. Then M is an additive map from F to the subsets of E. I f , moreover, M contains no line, then M is a strict set­

valued measure.

P ro o f. The additivity of M is readily seen. Suppose th a t M has

its values included in . Let' {AJ be a countable «^"-partition of A . If

x e M(A), there exists m such th a t m(A) = x , so th a t x = f? x {

Radqn-jyikodym theorems 301 with оо{ е М ( А {), and so х{ е М ( А {). Therefore M( A) cz £ M ( A {). Now

i

let {хг) be a summable selection of the family [М{А{)}. For every i, there exists e SfM such th a t œt = т{(А{). Since the measures m4 are positive, there exists a positive measure m whose restriction to each A { coincides with the restriction of m{. Moreover, we may suppose th a t the restriction of m to T \ A coincides with the restriction of a selector of M arbitrarily chosen. Then we have oo{ = т ( А {). Let us show th a t m e SfM. Let

We have m (£) = £ m ( B n A i)-{-m(Bn(T\A)' ) with m( Br \ A{) — т{( В п C\A{) e M ( B n A {) for every i and m ( B n ( T \ A ) )

M ( B n ( T \ A ) ) , whence m{B) g M(B). Now assume th a t AI contains no line. Let p and É + defined by Proposition 5.2, and let us set M X(A) = M ( A ) —p(A), А е У . Then M X(A) a E + and we have = £? m ~ V • Hence

M( A) = {m(A)lm е У щ } + р(А), i.e., M{A) = M 1 (A)+p{A).

But according to the first part of this proof, M x is strict set-valued measure. I t is easy to see th a t this implies th a t M is also a strict set­

valued measure. This completes the proof.

R e m a rk . If M contains no line and has a weak density Г with respect to p, then M defined by Theorem 5.3 may also be written:

M( A) = { j f r l f e S ?]■},

A

i.e., M( A) = J <£lr p.

A

This follows at once from the following proposition:

P

5.3. I f M is a weak set valued-measure, containing no ine and having a weak density Г with respect to p, then we have

SlM ^ { f p l f e ^ r }.

P ro o f. Let f e < £ lr . We have f p e SfM. Indeed, for all y

F we have

</(*)> 2 /> < ô*(y/r{t)), whence

< J 7 i» ,ÿ > < è‘ {ylM(A)), А е Г .

A

Hence f f p

M(A).

A

Conversely if m e S ? M, there exists /

such th a t m = fp. For every A we have

/< /( * ) ,» > /» ( * ) < / л*(у/Л 0 ) /»(*)•

A A

Hence </(*), y> < <5* (;y /r(t)) except if t belongs to a /i-null set N y depend­

ing on y. If i) is a countable dense subset of F, we have </(<), y> <

302 A. C oste and R. P a llu de la Barrière

< ô* (y IT(t)) for every y e D and every t not belonging to а /л-null set N independant of y. This, by Corollarv 2 of Lemma 2.1, implies th a t m e m , if t t N .

In case AI satisfies M(A) = 0 whenever /л(A) = 0 , Theorem 5.3 may be strenghtened to

T

5.4. Let AI be a weak set-valued measure containing no line such that M( A) = 0 whenever y (A) = 0. Let M be the strict set-valued measure defined by M( A) = {m(A)/m e ^ M}. Then for every strict set- valued measure N satisfying N {A) c M(A), for every i e . f , we have Af (A) cz M(A). Otherwise stated M. is uthe greatest” strict set-valued measure majorized by M. Moreover, M( A) is dense in M( A) for all A.

P ro o f. Let N be a strict set-valued measure such th a t N (A) cz M (A) for every A е!Г. By Theorem 5.2 луе have Af(A) = {m(A)jm e SfN}. Since S?N cz SfM, we then have AT (A) cz AI(A), which proves the first part.

On the other hand, M (A) = { f f y If e £?lr ), due to Г being a weak density A

of M. Hence M{A) is dense in AI(A) according to Proposition A.5 of Appendix.

6. Radon-Nikodym derivative of strict set-valued measures.

P

6.1. Let Г be a measurable set-valued function whose values are convex subsets of L — E n. I f Ф 0 , the set-valued mapping M from F to E defined by

M( A) e&b}

A

is additive and we have

(<*■) »*{ylM(A)) = J дЦу1т ) / л ( Ш ) , A elT, y e E ' .

A

I f we set M( A) — closure AI(A) for every A e.T, then M is a weak set valued measure of which a weak density is the set-valued function

Г: t-*closure r(t).

If, moreover, Г has its values included in , then M is a strict set-valued measure.

P ro o f. The additivity of AI is easily checked. Relation (a) follows from Proposition A.5 of Appendix. Since ô*(yir{t)) — ô*(yir(t)), t e T, y е й , we do have ô* {у/Ж(A)) = j ô*[yjr{t)) y{dt), l e f , which shows

A

th a t P is a weak density of M. Assume now th a t Г has its values included

in В \ . Let i e f and {AJ be a countable ^"-partition of A, and let {XJ

be a summable selection of { M( Ai)}. We have = f f {y with f i e£?lr .

Since fi takes its values in B \ , the map / equal to f t on each A { and equal i

Badon-N ïkodym theorems 303 to an arbitrary integrable selector of Г on T \ A is an integrable selector of Г and we have

X i = j f p .

^i Hence = / f p e M ( A ) .

i A

Therefore M{ A i) c M(A). The reverse inclusion being obvious, i

the proof is complete.

ISTow we consider a given strict set-valued measure and we proceed to investigate whether or not there exists a measurable set-valued mapping Г whose values are closed convex and such th at

M U ) = { j f r l f e ±?}•}.

A

When this holds we say th a t Г is a strong density of M with respect to p. The following theorem gives a necessary and sufficient condition for the existence of a strong density and also gives the method for construct­

ing it when it exists.

T

6 .1 . Let 31 be a strict set-valued measure such that M( A) = 0 whenever p(A) = 0 , and sueh that M( T) contains no line. Then a necessary and sufficient condition in order that 31 possesses a strong density with respect to p is that M( A) = {m(A) fm e S?—}, i.e., 31 is the greatest strict set-valued measure majorized by M. In this case the strong density Г is almost everywhere equal to the weak density of M.

P ro o f. Suppose th a t M( A) = {m(A)/m e Let Г be a weak density of M. According to Proposition 5.3, we also have

M U ) = { J f r l f e & ' r } , A

i.e., Г is a strong density of M.

Conversely suppose th a t M possesses Г as strong density. We have M(A) = { J f p / f e S £ \\. By Proposition 6.1, Г is a weak density of M.

A

According to Proposition 5.3 we also have

^ 31(A) = {т(А){т e £f—} .

Hence 31 is the greatest strict set-valued measure majorized by 31.

The following theorem is another version of the previous one:

T

6 .2 . Let 31 be a weak set-valued measure containing no line

such that 31(A) — {0} whenever p(A) = 0 . Among the strict set-valued

measures N such that N (A) a 31 (A) and closure X (A) = 31 (A) for every A ,

there is one and only one possessing a strong density : that is the set valued

304 A. Coste and K. F a llu de la Barrière measure 31 defined by

M( A) = {m(A)!m e M) '

7. A counter example. Suppose th a t T is countable, th a t .T is the set of all subsets of T and th a t у is the measure defined by y({n}) = 1, n e T. Then every strict set-valued measure M defined on X is of the form

A ^ M ( A ) =

ne A

where Г(п) = M({n}).

Now if wre are given a set-valued function Г defined on T whose values are convex subsets of E such th a t the set llr consisting of all sum- mable selections of {Г{п)} is not empty, then according to Theorem 6.1, the map M defined on J by 31(A) = У. Г(п) is an additive set-valued

ne A

set function. But, as we shall see, M is not necessarily a strict set-valued measure. However, the set-valued mapping 31: A ->closure 31(A) is a weak set valued measure possessing Г : closure Г(п) as weak density.

If Г is closed convex-valued, we have a bijection between If— and If, namely m~>{m({n})\, and we have 31(A) = {m(A)jm e-.^—}. We set ourselves to studying the following example: E = P 2; T = N + kj N_ , where N + = { 1 , 2 , 3 , . . . } and N_ = {—1, —2, —3 ,...} . For n e N + we set

Г(п) = {(x , у) e Б,21ж ^ 0, у > 0, п х ф п гу < 1},

and for n e JV_, Г(п) — {(x , y) e B 2 jx ^ 0, y > 0, п х фп^ у < 1}. We do have If Ф 0. Let us set

P + = {(x , y) e B 2 /y ^ 0} and P_ = {(x , y) e B 2 ly < 0}.

Let us show first th a t M is <т-additive. Setting A n — {n, —n ) ; n e N + and 1 = (J 1 й = f , we have M ( A n) = Г(п) +Г( —n) = P + (closme of

П

P +) and 31 (An) = P + . On the other hand, 31(A) does not contain

n e N.|_

{ix iy)l y = 0 } : indeed, if X n = (xn, yn) e P(n) is an element of If we cannot have yn = 0 for every n, since then \xn\ > 1 [n and £ \xn\ = + oo.

П

In fact, since 31(A) is dense in 31 (An), we have 31(A) = P + ./ j e t us П

show now th a t the set-valued set fimction 31 x : 31X(A) = int E(n) is

ne A

a strict set-valued measure (i.e., is c-additive). In this connection let us observe th a t if $ <= P + is such th a t <5* ((0, —1) /$) = 0, then P ++ S = P +:

indeed we have P + + 8 = {x+ P +lx e 8} = U {P+ + (0, y)l(cc, y) e S}. Let

then (A J be a partition of a subset A of T. First suppose th a t there exists

i 0 such that A iQC\N+ Ф 0 and l j on I _ Ф 0 . Let p e A i()n N +J q e i i()n