Let C be a field of subsets of a set I. It is well known that the space FA of all the finitely additive games of bounded variation on C is the norm dual of the space of all simple functions on C. In this paper we prove that the space BV of all the games of bounded variation on C is the norm dual of the space of all simple games on C. This result is equivalent to the compactness of the unit ball in BV with respect to the vague topology.Set functions; duality; compactness; coalitional games

Fabio Maccheroni

William H. Ruckle

English

Research Papers in Economics

INTERNATIONAL CENTRE FOR ECONOMIC RESEARCH
WORKING PAPER SERIES
Fabio Maccheroni and William H. Ruckle
BV AS A DUAL SPACE
Working Paper no. 29/2001
October 2001
APPLIED MATHEMATICS
WORKING PAPER SERIESBV as a Dual Space∗
Fabio Maccheroni†
Universit` a Bocconi, Milano, Italia and
ICER, Torino, Italia
William H. Ruckle
Clemson University, Clemson, South Carolina, USA
October 2001
Abstract
Let C be a ﬁeld of subsets of a set I. It is well known that the space FA of all the ﬁnitely
additive games of bounded variation on C is the norm dual of the space of all simple functions
on C. In this paper we prove that the space BV of all the games of bounded variation on
C is the norm dual of the space of all simple games on C. This result is equivalent to the
compactness of the unit ball in BV with respect to the vague topology.
Key words: Set functions, duality, compactness, coalitional games.
1 Introduction
Let C be a ﬁeld of subsets of a set I. It is well known that the space FA of all the ﬁnitely additive
games of bounded variation on C, equipped with the total variation norm, is isometrically
isomorphic to the norm dual of the space of all simple functions on C, endowed with the sup
norm (see [2] p. 258).
In this paper we establish a parallel result for the space BV of all the games of bounded
variation on C. To this end, we ﬁrst introduce simple games on C: the ones that assign non
zero worth only to a ﬁnite number of elements of C. Then we show that BV , equipped with
the total variation 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].
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 second direct
∗We wish to thank Erio Castagnoli, Adriana Castaldo, an anonymous referee, and especially Massimo Mari-
nacci for helpful discussions. Fabio Maccheroni gratefully acknowledges the ﬁnancial support of MIUR and
Universit` a Bocconi.
†Corresponding Author: Istituto di Metodi Quantitativi, Universit` a Bocconi, viale Isonzo 25, 20135 Milano,
Italy. Phone: ++39 +02 5836 5606. Fax: ++39 +02 5836 5634. Email: fabio.maccheroni@uni-bocconi.it
1proof of the dual nature of BV , and then observe that the vague topology can be viewed as the
weak* topology induced by duality.
The paper is organized as follows. After a brief section of preliminaries, in Section 3 we use
compactness to obtain duality. In Section 4, we go the other way around: we prove duality
and obtain compactness as a corollary. Section 5 clariﬁes the connections between the two
approaches.
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 consider 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 → R is a game if
v (∅) = 0. A game on C is monotone if v (A) ≤ v (B) whenever A ⊆ B.
A chain {Si}
n
i=0 in C is a ﬁnite strictly increasing sequence
∅ = S0 ⊂ S1 ⊂ ... ⊂ Sn−1 ⊂ Sn = I
of elements of C. BV is the set of all games such that
kuk = sup
(
n X
i=1
|u(Si) − u(Si−1)| : {Si}
n
i=0 is a chain in C
)
< ∞.
A game in BV is said to be of bounded variation. The pair (BV,k·k) is a Banach space,
generated by all monotone games (see [1] pp. 14, 26-28).
Since BV is a (proper) subspace of RC, it inherits a topology from the product topology of
RC. This is the weak topology generated by the projection functionals
pA : BV → R
u 7→ u(A)
where A ∈ C. A net {ua} converges to u in this topology iﬀ ua (A) → u(A) for all A ∈ C (we
write ua
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 in BV is compact in the vague topology,
and we use this fact to show that BV is a dual space.
Theorem 1 (Marinacci) The unit ball U (BV ) = {u ∈ BV : kuk ≤ 1} is compact w.r.t. the
vague topology.
2Proof. See [4]. The proof is based on the Tychonoﬀ Theorem. 
Let X be 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 : BV × X → R
(u,x) 7→
P
A∈C
u(A)x(A) .
That is:
• D is bilinear,
• if D(u,x) = 0 for all u, then x = 0,
• if D(u,x) = 0 for all x, then u = 0.
Therefore, X can be interpreted as a total subspace of the algebraic dual of BV by identifying
x ∈ X with the linear functional
Dx : BV → R
u 7→ D(u,x).
For this reason, in the rest of this section, we will write hu,xi instead of D(u,x). Some useful
properties of X are stated in the following lemma, whose easy proof is omitted.
Lemma 2 X is a total subspace of the norm dual BV 0 of BV , and
kxkBV 0 =max
kuk=1
 
X
A∈C
u(A)x(A)
!
for all x ∈ X.
Moreover, the topology σ (BV,X) coincides with the vague topology.
Given a game u of bounded variation, let Ju : X → R be the functional deﬁned by
Ju (x) = hu,xi
for all x ∈ X.
Theorem 3 Let X0 be the norm dual of (X,k·kBV 0). The operator
J : BV → X0
u 7→ Ju
is an isometric isomorphism from BV onto X0. Moreover, J is a vague-weak* homeomorphism.
3Proof. The operator J is well deﬁned, linear, injective and kJk ≤ 1. Let U (BV ) be the unit
ball in BV , U (X0) be the unit ball in X0. For the Goldstine-Weston density Lemma J (U (BV ))
is weak* dense in U (X0) (see, e.g., [3] p. 126).
Consider the vague topology on BV and the weak* topology on X0. Let {ua} be a net in BV .
We have that ua
C → u iﬀ ua
σ(BV,X)
→ u iﬀ hua,xi → hu,xi for all x ∈ X iﬀ Jua (x) → Ju (x) for
all x ∈ X iﬀ Jua weak* converges to Ju (brieﬂy Jua
w∗
→ Ju). Thus J is vague-weak* continuous.
Therefore, J (U (BV )) is weak* compact in X0 and
J (U (BV )) = U
 
X0
.
Since kJk ≤ 1, then kJuk ≤ kuk, for all u ∈ BV . Suppose that there exists u ∈ BV such that
kJuk < kuk. Then u 6= 0 and 0 <

 


J u
kuk

 


 = ρ < 1, set w = u
kuk. Since

 


J w
ρ

 


 = 1, it follows
that J w
ρ ∈ U (X0), but w
ρ / ∈ U (BV ) and J is injective, which is absurd. Thus kJuk = kuk, for
all u ∈ BV and J is an isometry.
If x0 ∈ X0 − {0} then there exists u ∈ U (BV ) s.t. Ju = x0
kx0k, hence Jkx0ku = x0. That is J
is surjective. Since, for every net {ua} in BV , ua
C → u iﬀ Jua
w∗
→ Ju, then J is a vague-weak*
homeomorphism. 
4 From duality to compactness
In this section the opposite approach is adopted, we directly prove that BV is a dual space
and use this fact to provide an alternative proof that the unit ball in BV is compact w.r.t. the
vague topology, thus obtaining the equivalence of the two results.
We deﬁne the game eA : C → R by
eA (B) =
(
1 if B = A
0 otherwise
for all A ∈ C−{∅}, and e∅ = 0. Being x =
P
A∈C
x(A)eA for all x ∈ X, we have X = heA : A ∈ Ci.
For each chain Ω = {Si}
n
i=0 in C, deﬁne a seminorm on X by
kxkΩ = max
0≤k≤n
 
 

n X
i=k
x(Si)
 
 

(1)
for all x ∈ X. Let XΩ = heA : A ∈ Ωi. If x ∈ XΩ, we say that x depends on the chain Ω.
For all x ∈ X, set
kxk = inf
L X
l=1
kxlkΩl
where the inf is taken over all ﬁnite decompositions x =
L P
l=1
xl in which xl depends on the chain
Ωl and k·kΩl is deﬁned as in (1) for all l = 1,...,L.
Lemma 4 The function k·k : X → R is a norm on X.
4Proof. It is easy to prove that k·k is a seminorm. We just show that x 6= 0 implies kxk 6= 0.
If x ∈ XΩ and A ∈ Ω then
kxkΩ ≥
1
2
|x(A)|.
In fact,
|x(A)| =
 
 
 
X
i:Si⊇A
x(Si) −
X
i:Si⊃A
x(Si)
 
 
 
≤
 
 
 
X
i:Si⊇A
x(Si)
 
 
 
+
 
 
 
X
i:Si⊃A
x(Si)
 
 
 
≤ 2kxkΩ
Let x ∈ X − {0}. There exists A ∈ C such that x(A) 6= 0. If x =
L P
l=1
xl, xl ∈ XΩl for all
l = 1,...,L, then A ∈ Ωl for some l (otherwise x(A) = 0). Hence,
|x(A)| =
 
 
 
X
l:A∈Ωl
xl (A)
 
 
 
≤
X
l:A∈Ωl
|xl (A)|,
and so
L X
l=1
kxlkΩl ≥
X
l:A∈Ωl
kxlkΩl ≥
X
l:A∈Ωl
1
2
|xl (A)| ≥
1
2
|x(A)|,
then kxk ≥ 1
2 |x(A)| > 0. 
Given a linear continuous functional f : X → R, deﬁne the game Gf as follows
Gf (A) = f (eA)
for all A ∈ C.
Theorem 5 Let X0 be the norm dual of (X,k·k). The operator
G : X0 → BV
f 7→ Gf
is an isometric isomorphism from X0 onto 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 Ω = {Si}
n
i=0 is a chain in C then
n X
k=1
|Gf (Sk) − Gf (Sk−1)| ≤ kfk,
which implies that Gf ∈ BV and kGfk ≤ kfk.
5Deﬁne x ∈ XΩ by
x(Sn) = sgn
 
f (eSn) − f
 
eSn−1

,
x(Sn) + x(Sn−1) = sgn
 
f
 
eSn−1

− f
 
eSn−2

,
...
x(Sn) + x(Sn−1) + ... + x(S1) = sgn(f (eS1) − f (eS0)),
x(S0) = 0,
Obviously kxkΩ ≤ 1, so that kxk ≤ 1. Thus
kfk ≥ f (x) =
n X
j=0
f
 
eSj

x(Sj) =
= f (eS0)
n X
k=0
x(Sk) +
n X
j=1


 
f
 
eSj

− f
 
eSj−1

n X
k=j
x(Sk)

 =
=
n X
j=1
 
f
 
eSj

− f
 
eSj−1

sgn
 
f
 
eSj

− f
 
eSj−1

=
=
n X
j=1

f
 
eSj

− f
 
eSj−1

 =
n X
j=1
|Gf (Sj) − Gf (Sj−1)|.
Then G is well deﬁned and obviously linear and injective.
Given u ∈ BV , we can deﬁne fu on X by
fu (x) =
X
A∈C
u(A)x(A),
for all x ∈ X. Trivially, fu is linear.
If x depends on Ω = {Sj}
n
j=0, then
fu (x) =
n X
j=0
u(Sj)x(Sj) =
= u(S0)
n X
k=0
x(Sk) +
n X
j=1

(u(Sj) − u(Sj−1))
n X
k=j
x(Sk)

 =
=
n X
j=1

(u(Sj) − u(Sj−1))
n X
k=j
x(Sk)

 ≤
≤
n X
j=1

|u(Sj) − u(Sj−1)|
 
 
 
n X
k=j
x(Sk)
 
 
 

 ≤
≤
n X
j=1
[|u(Sj) − u(Sj−1)|kxkΩ] ≤
≤ kxkΩ
n X
j=1
|u(Sj) − u(Sj−1)| ≤ kxkΩ kuk.
6If x =
L P
l=1
xl with xl ∈ XΩl for all l = 1,2,...,L, then
fu (x) =
L X
l=1
fu (xl) ≤
L X
l=1
kukkxlkΩl ≤ kuk
L X
l=1
kxlkΩl ,
and so
fu (x) ≤ inf
(
kuk
L X
l=1
kxlkΩl : x =
L X
l=1
xl, xl ∈ XΩl
)
=
= kukkxk.
We conclude that fu ∈ X0, G(fu) = u and G is onto. For all u ∈ BV , fu = G−1
u and



G−1
u



 = kfuk ≤ kuk. Therefore, for all f ∈ X0, kfk =

 


 
G−1
(Gf)

 


 
 ≤ kGfk and G is an isometry.
Finally, let {fa} be a net in X0. We have that fa w∗
→ f iﬀ fa (x) → f (x) for all x ∈ X iﬀ
fa (eA) → f (eA) for all A ∈ C iﬀ Gfa (A) → Gf (A) for all A ∈ C iﬀ Gfa
C → Gf. Hence, G is a
weak*-vague homeomorphism. 
We conclude the section by observing that Theorem 5 corrects Theorem 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 that BV is the norm dual of X,
endowed with the norms k·kBV 0 and k·k, respectively. The two approaches are indeed one the
mirror image of the other. In fact:
Proposition 6 The norm k·kBV 0 on X coincides with the norm k·k on X. Moreover, J = G−1.
Proof. Let
 
X0,k·k
0
be the norm dual of (X,k·k). We have, for all x,
kxkBV 0 = sup
( 
 

X
A∈C
u(A)x(A)
 
 

: u ∈ BV,kuk = 1
)
=
= sup{|fu (x)| : u ∈ BV,kuk = 1} =
= sup

|f (x)| : f ∈ X0,kfk
0 = 1
	
=
= kxk.
In particular the norm dual of (X,k·kBV 0) and the norm dual of (X,k·k) are the same space
X0. For all u ∈ BV and x ∈ X,
Ju (x) = hu,xi =
X
A∈C
u(A)x(A) = fu (x) = G−1
u (x),
that is Ju = G−1
u for all u ∈ BV and J = G−1. 
7References
[1] R.J. Aumann and L.S. Shapley, “Values of Non-Atomic Games”, Princeton University Press,
Princeton NJ, 1974.
[2] N. Dunford and 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), 1000-1014.
[5] W.H. Ruckle, Projection in Certain Spaces of Set Functions, Math. Oper. Res. 7 (1982),
314-318.
8Copies of the working papers can be downloaded from ICER￿s web site www.icer.it
INTERNATIONAL CENTRE FOR ECONOMIC RESEARCH
APPLIED MATHEMATICS WORKING PAPER SERIES
1.  Luigi Montrucchio and Fabio Privileggi, ￿On Fragility of Bubbles in Equilibrium
Asset Pricing Models of Lucas-Type,￿ Journal of Economic Theory, forthcoming
(ICER WP 2001/5).
2.  Massimo Marinacci, ￿Probabilistic Sophistication and Multiple Priors,￿
Econometrica, forthcoming (ICER WP 2001/8).
3.  Massimo Marinacci and Luigi Montrucchio, ￿Subcalculus for Set Functions and
Cores of TU Games,￿ April 2001 (ICER WP 2001/9).
4.  Juan Dubra, Fabio Maccheroni, and Efe Ok, ￿Expected Utility Theory without the
Completeness Axiom,￿ April 2001 (ICER WP 2001/11).
5.  Adriana Castaldo and Massimo Marinacci, ￿Random Correspondences as Bundles
of Random Variables,￿ April 2001 (ICER WP 2001/12).
6.  Paolo Ghirardato, Fabio Maccheroni, Massimo Marinacci, and Marciano
Siniscalchi, ￿A Subjective Spin on Roulette Wheels,￿ July 2001 (ICER WP
2001/17).
7.  Domenico Menicucci, ￿Optimal Two-Object Auctions with Synergies,￿ July 2001
(ICER WP 2001/18).
8.  Paolo Ghirardato and Massimo Marinacci, ￿Risk, Ambiguity, and the Separation
of Tastes and Beliefs,￿ Mathematics of Operations Research, forthcoming (ICER
WP 2001/21).
9.  Andrea Roncoroni, ￿Change of Numeraire for Affine Arbitrage Pricing Models
Driven By Multifactor Market Point Processes,￿ September 2001 (ICER WP
2001/22).
10. Maitreesh Ghatak, Massimo Morelli, and Tomas Sjostr￿m, ￿Credit rationing,
wealth inequality, and allocation of talent￿, September 2001 (ICER WP 2001/23).
11. Fabio Maccheroni and William H. Ruckle, ￿BV as a Dual Space,￿ Rendiconti del
Seminario Matematico dell’Universit￿ di Padova, forthcoming (ICER WP
2001/29).
12. Fabio Maccheroni, ￿Yaari Dual Theory without the Completeness Axiom,￿
October 2001 (ICER WP 2001/30).

BV as a dual space.

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BV as a dual space.

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