BV as a dual space

BV as a Dual Space.

FABIO MACCHERONI(*) - WILLIAM H. RUCKLE(**)

ABSTRACT- LetCbe a field of subsets of a setI. It is well known that the spaceFA of all the finitely additive games of bounded variation onCis the norm dual of the space of all simple functions onC. In this paper we prove that the space BVof all the games of bounded variation onCis the norm dual of the space of all simple games onC. This result is equivalent to the compactness of the unit ball in BV with respect to the vague topology.

1. Introduction.

LetCbe a field of subsets of a setI. It is well known that the space FAof all the finitely additive games of bounded variation onC^{, equipped} with the total variation norm, is isometrically isomorphic to the norm du- al of the space of all simple functions onC, endowed with the sup norm (see [2] p. 258).

In this paper we establish a parallel result for the spaceBVof all the games of bounded variation onC. To this end, we first introduce simple games onC: the ones that assign non zero worth only to a finite number of elements ofC. Then we show that BV, equipped with the total varia- tion norm, is isometrically isomorphic to the norm dual of the space of all simple games, endowed with a suitable norm. A first proof of this result builds on the compactness of the unit ball in BV with respect to the vague topology, proved by Marinacci [4].

(*) Indirizzo dell’A.: Istituto di Metodi Quantitativi, Università Bocconi, via- le Isonzo 25, 20135 Milano, Italia. E-mail: fabio.maccheroniHuni-bocconi.it

(**) Indirizzo dell’A.: Clemson University, Clemson, South Carolina, USA.

E-mail: ruckleHexcite.com

We wish to thank Erio Castagnoli, Adriana Castaldo, an anonymous referee, and especially Massimo Marinacci for helpful discussions. Fabio Maccheroni gra- tefully acknowledges the financial support of MIUR and Università Bocconi.

Another contribution of this paper is showing that the duality of BV is indeed equivalent to the vague compactness of the unit ball. In order obtain this result, we provide a seconddirectproof of the dual nature of BV, and then observe that the vague topology can be viewed as the weak* topology induced by duality.

T h e p a p e r i s o r g a n i z e d a s f o l l o w s . A f t e r a br i e f s e c t i o n o f p r e l i m i - n a r i e s , i n S e c t i o n 3 we us e c o m p a c t n e s s t o o b t a i n d u a l i t y . I n S e c t i o n 4 , w e g o t h e o t h e r w a y a r o u n d : w e p r o v e d u a l i t y a n d o b t a i n c o m p a c t n e s s a s a co r o l l a r y . S e c t i o n 5 cl a r i f i e s t h e c o n n e c t i o n s b e t w e e n t h e t w o a p p r o a c h e s .

2. Preliminaries.

This section contains a few, well known, definitions and properties.

We follow the notation of cooperative game theory, instead of the more common one of measure theory, since the spaces of set functions we con- sider are widely used in that field (see, e.g. [1]).

Let C be a field of subsets of a non empty set I. A set function v: C_KR is a game if v(¯)40. A game on C ^{is monotone if v(A)}_G^v(B) whenever A’B.

A chain ]Si(ⁱn40 in C is a finite strictly increasing sequence

¯4S₀%S₁%R%S_n21%S_n4I of elements of C^{. BV} is the set of all games such that

VuV4sup

m

^!

n

A game inBVis said to beof bounded variation. The pair (BV,VQV) is a Banach space, generated by all monotone games (see [1] pp. 14, 26-28).

SinceBVis a (proper) subspace ofR^C, it inherits a topology from the product topology ofR^C. This is the weak topology generated by the pro- jection functionals

p_A: BV u

K O

R u(A)

whereAC^{. A net}_]^ua(converges touin this topology iffu_a(A)Ku(A) for all AC ^{(we write u}aK^C u). This topology is called vague topology (for the analogy with the vague topology on the set of probability measures).

3. From compactness to duality.

In this section, we start by observing that the unit ball inBVis com- pact in the vague topology, and we use this fact to show thatBVis a dual space.

THEOREM1 (MARINACCI). The unit ball U(BV)4 ]uBV: VuVG1( is compact w.r.t. the vague topology.

PROOF. See [4]. The proof is based on the Tychonoff Theo- rem. r

LetXbe the space of all the games which are non zero only on a finite number of elements of C^{, the} simple games. The pair (BV,X) is dual w.r.t. the following functional

D: BV3X (u,x)

K O

R

A

!

u(A)x(A) .

That is:

l D is bilinear,

l if D(u,x)40 for all u, then x40 , l if D(u,x)40 for all x, then u40.

Therefore, Xcan be interpreted as a total subspace of the algebraic dual of BV by identifying xX with the linear functional

D_x: BV u

K O

R D(u,x) .

For this reason, in the rest of this section, we will writeau,xbinstead of D(u,x). Some useful properties of Xare stated in the following lemma, whose easy proof is omitted.

LEMMA 2. X is a total subspace of the norm dual BV8 of BV, and VxV_BV84max

VuV41

g

^!

h

for all xX.

Moreover, the topology s(BV,X) coincides with the vague topology.

Given a gameuof bounded variation, letJ_u: XKRbe the functional defined by

J_u(x)4au,xb for all xX.

THEOREM 3. Let X8 be the norm dual of (X,VQV_BV8). The operator J: BV

u K O

X8 Ju

is an isometric isomorphism from BV onto X8. Moreover, J is a vague- weak* homeomorphism.

PROOF. The operator J is well defined, linear, injective and VJVG1.

LetU(BV) be the unit ball in BV,U(X8) be the unit ball inX8. For the Goldstine-Weston density Lemma J(U(BV) ) is weak* dense in U(X8) (see, e.g., [3] p. 126).

Consider the vague topology on BV and the weak* topology on X8. Let]u_a(be a net inBV. We have thatu_aK

C uiffu_aK

s(BV,X)

uiffau_a,xbK Kau,xbfor allxXiffJ_ua(x)KJ_u(x) for allxXiffJ_uaweak* converges to Ju (briefly Ju_aK^w*J_u). Thus J is vague-weak* continuous. Therefore, J(U(BV) ) is weak* compact in X8 and

J(U(BV) )4U(X8) .

Since VJVG1 , then VJuVGVuV, for all uBV. Suppose that there exists uBV such that VJ_uVEVuV. Then uc0 and 0EVJu/VuVV4rE1 , set w4_V^u_uV. SinceVJ_w/rV41 , it follows thatJ_w/rU(X8), but ^w

rU(BV) and Jis injective, which is absurd. ThusVJ_uV4VuV, for alluBVandJis an isometry.

If x8X8 2 ]0( then there exists uU(BV) s.t. J_u4_Vx^x8

8V , hence JVx8Vu4x8. That isJis surjective. Since, for every net]u_a(inBV,u_aK

C u iff J_uaK^w*J_u, then J is a vague-weak* homeomorphism. r

4. From duality to compactness.

In this section the opposite approach is adopted, we directly prove thatBVis a dual space and use this fact to provide an alternative proof that the unit ball inBVis compact w.r.t. the vague topology, thus obtain- ing the equivalence of the two results.

We define the game e_A:C_K^{R by} e_A(B)4

. /´ 1 0

if B4A otherwise for all AC_2]^¯₍^{, and e¯}₄^{0. Being x}₄

!

AC

x(A)e_A for all xX, we have X4ae_A:ACb.

For each chain V4 ]Si(ⁱn40 in C, define a seminorm on X by VxV_V4 max

0GkGn

N

^!

N

(1)

for allxX. LetX_V4ae_A:AVb. If xX_V, we say that x depends on the chain V.

For all xX, set

VxV4inf

!

l41 L

VxlV_V

l

where the inf is taken over all finite decompositionsx4

!

l41 L

x_lin whichx_l depends on the chain Vl and VQV_V

l is defined as in (1) for all l41 ,R,L.

LEMMA 4. The function VQV:XKR is a norm on X.

PROOF. It is easy to prove thatVQVis a seminorm. We just show that xc0 implies VxV c0. If xX_V and AV then

VxV_VF 1

2Nx(A)N. In fact,

Nx(A)N 4

N

^!

^!

N

N

^!

N

1

N

^!

N

Let xX2 ]0(. There exists AC ^{such that x(A)c0. If x}₄

!

l41 L

x_l,

x_lX_Vl for alll41 ,R,L, then AVlfor somel (otherwisex(A)40).

Hence,

Nx(A)N 4

N

^!

N

^!

and so

l

!

VxlV_V

lF

!

l:AV_l

Vx_lV_V

lF

!

l:AV_l

1

2Nx_l(A)N F1

2Nx(A)N, then VxVF¹₂Nx(A)ND0. r

Given a linear continuous functional f :XKR, define the game G_f as follows

Gf(A)4f(e_A) for all AC^.

THEOREM 5. Let X8 be the norm dual of (X,VQV). The operator G: X8

f K O

BV G_f

is an isometric isomorphism from X8onto BV. Moreover, G is a weak*- vague homeomorphism.

Notice that, together with the Alaoglu Theorem (see, e.g., [3] p. 70), the above result immediately yields Theorem 1, that is, the compactness of the unit ball U(BV) in the vague topology.

PROOF. We first show that if V4 ]Si(ⁱn40 is a chain in C ^then

k

!

NG_f(S_k)2G_f(S_k21)N GVfV, which implies that G_fBV and VG_fVGVfV.

Define xX_V by

x(S_n) x(S_n)1x(S_n21)

x(S_n)1x(S_n21)1R1x(S₁) x(S₀)

4 4

4 4

sgn (f(e_Sn)2f(e_Sn

21) ) , sgn (f(e_Sn

21)2f(e_Sn

22) ) , R

sgn (f(e_S1)2f(e_S0) ) , 0 ,

Obviously VxV_VG1 , so that VxVG1. Thus VfV F

4 4 4

f(x)4

!

j40 n

f(eS_j)x(Sj)4 f(e_S0)

!

k40 n

x(S_k)1

!

j41

n

k

^!

x(S_k)

l

j41

!

n

[ (f(e_Sj)2f(e_Sj

21) ) sgn (f(e_Sj)2f(e_Sj

21) ) ]4

j

!

Nf(e_Sj)2f(e_Sj21)N 4

!

j41 n

NG_f(S_j)2G_f(S_j21)N. Then G is well defined and obviously linear and injective.

Given uBV, we can define f_u on X by f_u(x)4

!

AC

u(A)x(A) , for all xX. Trivially, fu is linear.

If x depends on V4 ]Sj(^jn40, then f_u(x) 4

4 4 G G G

j

!

u(S_j)x(S_j)4 u(S₀)

!

k40 n

x(S_k)1

!

j41

n

k

^!

x(S_k)

l

j4

!

n

k

^!

x(S_k)

l

j

!

n

k

_N

^!

x(S_k)

N l

j

!

[Nu(S_j)2u(S_j21)NVxV_V]G VxV_V

!

j41 n

Nu(S_j)2u(S_j21)N GVxV_VVuV. If x4

!

l41 L

xl with xlX_Vl for all l41 , 2 ,R,L, then f_u(x)4

!

l41 L

f_u(x_l)G

!

l41 L

VuV Vx_lV_V

lGVuV

!

l41 L

Vx_lV_V

l, and so

f_u(x)Ginf

m

^!

^!

n

4VuV VxV.

We conclude thatf_uX8,G_(fu)4uandGis onto. For alluBV,f_u4G_u²¹ andVG_u²¹V4Vf_uVGVuV. Therefore, for allfX8,VfV4VG_(G²_f¹₎VGVG_fVand G is an isometry.

Finally, let]f^a( be a net inX8. We have thatf^aK^w*fifff^a(x)Kf(x) for all xX iff f^a(e_A)Kf(e_A) for all AC ^{iff G}f^a(A)KG_f(A) for all AC ^{iff G}f^aK^C G_f. Hence, G is a weak*-vague homeomorphism. r

We conclude the section by observing that Theorem 5 corrects Theo- rem 1.1 in [5]: it can be shown with a counterexample that the norm used there does not lead to an isometry.

5. Summing up.

In sections 3 and 4 we have given two independent proofs thatBVis the norm dual ofX, whenXis endowed with the normsVQV_BV8andVQV, re- spectively. The two approaches are indeed one the mirror image of the other. In fact:

PROPOSITION 6. The norm VQV_BV8 on X coincides with the norm VQV on X. Moreover, J4G²¹.

PROOF. Let (X8,VQV8) be the norm dual of (X,VQV). We have, for all x, VxV_BV84sup

m _N

^!

u(A)x(A)

N

n

4sup]Nf_u(x)N: uBV,VuV41( 4 4sup]Nf(x)N: fX8,VfV8 41( 4 4VxV.

In particular the norm dual of (X,VQV_BV8) and the norm dual of (X,VQV) are the same space X8. For all uBV and xX,

J_u(x)4au,xb4

!

AC

u(A)x(A)4f_u(x)4G_u²¹(x) , that is J_u4G_u²¹ for all uBV and J4G²¹. r

R E F E R E N C E S

[1] R. J. AUMANN- L. S. SHAPLEY,Values of Non-Atomic Games, Princeton Uni- versity Press, Princeton NJ, 1974.

[2] N. DUNFORD - J. T. SCHWARTZ, Linear Operators, Interscience, New York, 1958.

[3] R. B. HOLMES, Geometric Functional Analysis and its Applications, Springer-Verlag, New York, 1975.

[4] M. MARINACCI, Decomposition and Representation of Coalitional Games, Math. Oper. Res., 21 (1996), pp. 1000-1014.

[5] W. H. RUCKLE, Projection in Certain Spaces of Set Functions, Math. Oper.

Res., 7(1982), pp. 314-318.

Manoscritto pervenuto in redazione il 6 marzo 2001.