Random Correspondes as Bundles of Random Variables

RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM
VARIABLES
ADRIANA CASTALDO, FABIO MACCHERONI, AND MASSIMO MARINACCI
Abstract. We relate the distributions induced by random sets with the distributions induced by their measurable selections, thus providing a probabilistic
foundation for viewing random sets as \sets" of random variables. In so doing
we revisit and extend some results of Artstein, Hart, Hess, and Kohlberg, and we
obtain a \change of variable formula" for the Aumann integral of a random set.
1.
Introduction
Since the seminal works of Debreu [6], Dempster [8] and [9], Hildenbrand [15],
Kendall [18], and Matheron [20], random sets have been widely used as a generalization of standard random variables. Given a probability space (S, , P) and a
topological space X , while random variables associate to elements of S single elements of X , random sets relax this assumption by associating nonempty subsets of
X to elements of S . This added exibility turned out to be useful in several areas
{ e.g., statistics, mathematical economics, stochastic geometry, optimization { and
we refer the interested reader to the original works, as well as to the surveys in
Klein and Thompson [19], Stoyan et al. [26] and Barndor-Nielsen et al. [4]. For
example, building on Dempster's works, random correspondences have been recently
used in Bayesian decision theory to model unforeseen contingencies and preference
for exibility by Mukerji [21], Nehring [22], Ghirardato [12], and Ozdenoren [24].
A suitably measurable random set F : S ⇒ X induces an upper distribution νF
and lower distribution ν F on X as follows:
νF (B )
=
ν F (B ) =
P ({s ∈ S : F (s) ∩ B 6= ∅}) ,
P ({s ∈ S : F (s) ⊆ B}) ,
for all Borel subsets B of X . In the special case of a random variable f : S →
X , we have νf (B ) = ν f (B ) for all B and νf reduces to the standard probability
distribution Pf induced by f .
Date : June 2004.
We wish to thank Anna Battauz, Sergiu Hart, Giuliana Regoli, Roberta Renzini, Marco Scarsini,
an anonymous referee, and especially Claude Dellacherie for helpful comments. The nancial
support of Ministero dell'Istruzione, dell'Universita e della Ricerca is gratefully acknowledged.
1
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
2
The distributions νF and ν F therefore generalize the usual probability distributions induced by random variables. The purpose of our work is to study these
distributions and in particular their relationships with the standard probability distributions induced by the measurable selections of the random set F .
Specically, let Sel F be the set of all measurable selections of the random set F ,
that is, the set of all random variables f : S → X such that f (s) ∈ F (s) for all
s ∈ S . Each selection f ∈ Sel F induces a probability distribution Pf on X dened
by Pf (B ) = P ({s ∈ S : f (s) ∈ B}) for all Borel subsets B of X . Our purpose is
to relate the distributions νF and ν F with the set {Pf : f ∈ Sel F } of the standard
probability distributions induced by the selections of F . In our probabilistic setting
we thus follow Aumann [3]'s lead, who showed that a fruitful way to look at multifunctions is as \bundles" of their selections, a standpoint that makes it possible to
relate multifunctions with the more familiar single-valued functions.
We have two main results. Consider a bounded Borel function u : X → R dened
on the space X , which in applications will be, in general, the space of interest { e.g.,
a space of outcomes. Since
νF and ν F are nonadditive set functions, we consider
R
R
their Choquet integrals udνF Rand udν F ,R dened in the next section. Our rst
result, Theorem 3.2, shows that udνF and udν F are,
respectively, theupper and
R
lower envelopes of the sets of the standard integrals udPf : f ∈ Sel F . That is,
we prove that
Z
udPf : f ∈ Sel F ,
Z
inf
udPf : f ∈ Sel F ,
udνF
= sup
udν F
=
Z
Z
provided X is Polish and F compact-valued, conditions often satised in applications.
Our second main result, Corollary 3.4, considers the set Pν of all probability
measures P on X such that P (B ) ≤ νF (B ) for all Borel subsets B of X . This
set is often associated with the distribution νF (see, e.g., Dempster [8], Huber and
Strassen [17], Artstein [1], Philippe et al. [25]). Corollary 3.4 shows that Pν is
nothing but the weak* closed convex hull of the set {Pf : f ∈ Sel F } of induced
probability distributions, that is,
F
F
(1.1)
PνF
= co {Pf : f ∈ Sel F } .
Our results show that there exists a tight connection between the generalized
distributions νF and ν F and the standard probability distributions {Pf : f ∈ Sel F }
that are naturally associated with them. In this way, we can relate these generalized
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
3
notions with more familiar standard notions and oer a perspective on random sets
as sets of random variables.
Relation (1.1) extends some results of Hart and Kohlberg [13] and Artstein and
Hart [2] to random sets dened on general probability spaces, not required to have
a nonatomic underlying probability measure. This is discussed in Section 3, as well
as the relations of (1.1) with the results of Artstein [1] and Hess [14].
The paper is organized as follows. Section 2 contains notation and some preliminary results. Section 3, the heart of the paper, states our main results. Section 4
gives couple of additional results. The paper ends with two Appendices: Appendix
A provides some tools useful in proving the results of the paper, while Appendix B
contains all proofs.
2.
Preliminaries
Throughout the paper, (S, , P) is a probability space and X a Polish space.
2.1. Polish Spaces. Let B be the Borel σ-algebra of the Polish space X . Denote
by B (X ) the space of all bounded Borel functions u : X → R, endowed with the
supnorm topology, and by M the set of all Borel probability measures P : B → R on
X , endowed with the weak* topology induced by the bounded continuous functions
on X (sometimes called the vague topology).
On the family K of all nonempty compact subsets of X , we consider the Vietoris
topology, generated by the sets of the form {K ∈ K : K ⊆ G} or {K ∈ K : K ∩ G 6= ∅},
where G is an open subset of X . This topology is Polish, and for each B ∈ B the
set {K ∈ K : K ∩ B 6= ∅} is universally measurable in K (see, e.g., [25]).1
2.2. Capacities. A (Choquet) capacity on X is a set function ν : B → R such that
(c.1) ν (∅) = 0, ν (X ) = 1,
(c.2) ν (A) ≤ ν (B ) for all Borel sets A ⊆ B ,
(c.3) ν (Bn ) ↑ ν (B ) for all sequences of Borel sets Bn ↑ B ,
(c.4) ν (Cn ) ↓ ν (C ) for all sequences of closed sets Cn ↓ C .
A capacity ν is alternating of order n if
ν
n
\
i=1
!
Bi
≤
X
∅6=I⊆{1,...,n}
(−1)|I|+1 ν
!
[
Bi
∀B1 , ..., Bn ∈ B ;
i∈I
it is innitely alternating if it is alternating of order n for each n ≥ 2. For example,
ν is alternating of order 2 if
ν (A ∪ B ) + ν (A ∩ B ) ≤ ν (A) + ν (B ) ∀A, B ∈ B,
That is, it belongs to the completion of the Borel σ -algebra of K with respect to any (Borel)
probability measure on K.
1
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
4
that is, if it is submodular.
The Choquet integral of a bounded Borel function u with respect to a set function
ν satisfying (c.1) and (c.2) is dened by
Z
Z
+∞
udν ≡
0
=
Z
0
+∞
ν ({u ≥ t}) dt +
Z
ν ({u > t}) dt +
Z
0
−∞
0
−∞
[ν ({u ≥ t}) − 1] dt
[ν ({u > t}) − 1] dt.
The integrals on the right are Riemann integrals, which are well dened since
ν ({u ≥ t}) and ν ({u > t}) are monotone functions in t (and they coincide where
the rst one is continuous). The Choquet
integral is positively
homogeneous, monoR
R
tone, and translation invariant (i.e., (u + c) dν = udν + c if c is constant). It
reduces to the standard Lebesgue integral when ν is a probability measure.
Given a capacity ν , set
Pν ≡ {P ∈ M : P (B ) ≤ ν (B )
∀B ∈ B} ,
that is, Pν is the set of all probability measures set-wise dominated by ν . The set
Pν is a weak* compact and convex subset of M.2
The conjugate ν : B → R of a set function ν is dened by ν (B ) ≡ ν (X ) − ν (B c )
for all B ∈ B. In general, the properties of ν have dual versions for its conjugate.
For example, ν is an innitely alternating capacity if and only if ν satises (c.1),
(c.2),
(c.3') ν (Bn ) ↓ ν (B ) for all sequences of Borel sets Bn ↓ B ,
(c.4') ν (G
n ) ↑ ν (G) for all sequences of open sets Gn ↑ G, and
P
Sn
T
(c.5') ν ( i=1 Bi ) ≥ ∅6=I⊆{1,...,n} (−1)|I|+1 ν i∈I Bi for each nite collection B1 , ..., Bn
of Borel sets.
In the words of Philippe et al. [25], this means that ν is an innitely alternating
capacity if and only if ν is an innitely monotone cocapacity.3
Lemma 2.1 (Philippe-Debs-Jaray). Let X be a Polish space. Then a set function
ν : B → R is an innitely alternating capacity on X if and only if there exists a
probability measure π on K such that, for each B ∈ B,
ν (B ) = π ({K ∈ K : K ∩ B 6= ∅}) ,
where π still denotes the (unique) extension of π to the class of universally measurable subsets of K. Moreover, such π is unique.
See [17, Lemma 2.2] and Lemma A.2 in Appendix A.
See Lemma A.3 in Appendix A.
2
3
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
5
We denote by νπ the innitely alternating capacity on X induced by a probability
measure π on K.
2.3. Random Sets and their Distributions. A multifunction (or correspondence ) F : S ⇒ X is a map with domain S and whose values are nonempty subsets
of X . Its graph is the set gr F ≡ {(s, x) ∈ S × X : x ∈ F (s)} . For any A ⊆ X we
put
F −1 (A) ≡ {s ∈ S : F (s) ∩ A 6= ∅} ,
F−1 (A) ≡ {s ∈ S : F (s) ⊆ A} .
Notice that
A ⊆ X.
F −1 (A)
= prS (gr F ∩ (S × A)) and
F −1 (A)
=
S − F−1 (Ac )
for any
A multifunction F : S ⇒ X is measurable if F −1 (C ) ∈ for every closed subset C
of X . A measurable and compact-valued multifunction is called random (compact)
set. Random sets reduce to standard random variables when they are functions.
As to counter-images of Borel subsets of X , while for random variables they belong
to , for random sets all we can say is that they belong to its completion P ; that
is, F −1 (B ) ∈ P for all B ∈ B (see [19, Chapter 13]).
A measurable selection (resp., measurable a.s. selection ) of a multifunction F :
S ⇒ X is a random variable f : S → X such that f (s) ∈ F (s) for all (resp.,
for P-almost all) s ∈ S . The set of all measurable selections (resp., measurable
a.s. selections) of F is denoted by Sel F (resp., Sela.s. F ). The Selection Theorem of
Kuratowski and Ryll-Nardzewski (see [16, p. 60]) guarantees that Sel F is nonempty
when F is a random set.
A random variable f : S → X induces a probability distribution Pf on X dened
by
Pf (B ) ≡ P ({f ∈ B})
for all B ∈ B. Analogously, a random set F : S ⇒ X induces an upper distribution
νF and a lower distribution (or belief function ) ν F on X dened, for all B ∈ B , by
νF (B ) ≡ P F −1 (B ) ,
ν F (B ) ≡ P (F−1 (B )) ,
where P still denotes the (unique) extension of P to P . The lower distribution ν F is
easily seen to be the conjugate of νF , thus explaining our notation. Moreover, when
regarded as a random variable from S to K, the random set F induces a probability
distribution PF on K given by PF (E ) ≡ P ({F ∈ E}) for all Borel subsets E of K.
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
6
Capacities and upper distributions are related as follows (see [5, Section 26.8] and
[23, p. 534]).
Lemma 2.2. Let X be a Polish space.
(i) If (S, , P) is a probability space and F : S ⇒ X is a random set, then νF is
an innitely alternating capacity on X , and νF = νPF .4
(ii) If ν is an innitely alternating capacity on X , then there exists a random set
F : [0, 1] ⇒ X such that ν = νF , where [0, 1] is endowed with the Lebesgue
σ -algebra and the Lebesgue measure.
3.
Main Results
Before stating the results we introduce a class of functions.
Denition 3.1. A Borel function u : X → R is upper (resp., lower) Weierstrass if
it attains its supremum (resp., inmum) on all nonempty compact subsets of X .
The class of upper Weierstrass functions is broad and includes:
(i) all upper semicontinuous functions u : X → R;
(ii) all simple Borel functions u : X → R.
Continuous functions and simple Borel functions are both lower and upper Weierstrass.
We can now state our main result.
Theorem 3.2. Let (S, , P) be a probability space, X a Polish space, and F : S ⇒ X
a random set. Then, for each u ∈ B (X ),
(3.1)
Z
udνF
(3.2)
=
=
sup
Z
udPf
f ∈Sel F
Z
sup u (x) dPF (K ) .
K x∈K
If, in addition, u is upper Weierstrass, the suprema in both (3.1) and (3.2) are
attained.
Remark. A dual version of Theorem 3.2 holds, where ν , sup, and upper Weierstrass
are replaced with ν , inf, and lower Weierstrass, respectively. For example,
νF (B ) =
max
f ∈Sel F
P f (B )
and ν F (B ) = f min
Pf (B )
∈Sel F
for each B ∈ B.
The proof of Theorem 3.2 builds on the following lemma of independent interest.
The capacity νPF is dened right after Lemma 2.1.
4
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
7
Lemma 3.3. Let (S, , P) be a probability space, X a Polish space, and F : S ⇒ X
a random set. Then, for each upper Weierstrass u, there exists f ∈ Sel F such that
νF ({u ≥ t}) = Pf ({u ≥ t}) and νF ({u > t}) = Pf ({u > t})
∀t ∈ R.
Inter alia, Lemma 3.3 shows that, given any nite chain B1 ⊇ ... ⊇ Bn of Borel
sets, there exists f ∈ Sel F such that νF (Bi ) = Pf (Bi ) for each i = 1, ...,Pn. In fact,
it is enough to apply the lemma to the upper Weierstrass function u = ni=1 1B .
Consider the set
DF ≡ {Pf : f ∈ Sel F }
of the probability distributions induced by the measurable selections of F . From a
probabilistic standpoint, DF is a very important subset of Pν since it has a direct
connection with the random correspondence F . It would be therefore desirable that
DF were also a mathematically important subset of Pν . In general, the set DF is
neither closed nor convex (see [13]) and so in general Pν 6= DF .
However, the next result { based on Theorem 3.2 { shows that DF is still an
important subset of Pν . In fact, Pν is nothing but the weak* closed convex hull
of DF .
i
F
F
F
F
F
Corollary 3.4. Let (S, , P) be a probability space, X a Polish space, and F : S ⇒
X a random set. Then,
(3.3)
PνF
= co {Pf : f ∈ Sel F } .
If, in addition, P is nonatomic, then
(3.4)
PνF
= {Pf : f ∈ Sel F }.
The second part of Corollary 3.4 builds on the results of Hart and Kohlberg [13],
Artstein and Hart [2], and Artstein [1], while the rst one extends them to the non
nonatomic case.
In [13] and [2] it is shown that, if Fi : Si ⇒ X are random sets on nonatomic
probability spaces (Si , i , Pi ), with i = 1, 2, then5
νF1 = νF2 ⇐⇒ DF1 = DF2 .
Under nonatomicity, equally distributed random sets thus induce the same, up to
closure, sets of probability distributions. Eq. (3.3) of Corollary 3.4 completes their
result by showing what happens without the nonatomicity of the underlying Pi s. In
this case,
νF1 = νF2 ⇐⇒ co (DF1 ) = co (DF2 ) .
The lack of nonatomicity is thus reected by the use of convex hulls.
They consider X = Rl , but their proof can be easily adapted to Polish spaces.
5
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
8
As to Artstein [1], given a probability measure π on K, he says that a probability
measure P on X is selectionable with respect to π if there exist a probability space
(S, , P), a random set F : S ⇒ X with PF = π, and f ∈ Sel F such that Pf = P .
The set of all probability measures selectionable with respect to π is denoted by Sπ .
He shows that
(3.5) Sπ = {P ∈ M : P (C ) ≤ π ({K ∈ K : K ∩ C 6= ∅}) ∀closed C ⊆ X} ,
and that Sπ = DF provided P is nonatomic and F is a random set such that PF = π.
Our result shows that to determine the set Sπ it is enough to consider any random
set F such that PF = π, and not necessarily one dened on a nonatomic probability
space. In fact Eq. (3.5) implies Sπ = Pν , and so Eq. (3.3) of Corollary 3.4
yields Sπ = co (DF ). Notice that it is possible to proceed the other way around
when P is nonatomic. That is, one can start from the results of [13] and [1] to
obtain the second part of Corollary 3.4 and derive from it (3.1) of Theorem 3.2 for
a bounded and continuous u : X → R. Moving to general bounded Borel functions
is a nontrivial step, and it is one of our contributions.
Finally, let 0 be the sub-σ-algebra of generated by a random set F : S ⇒ X
on (S, , P).6 Along with DF , Hess [14] considers the set DF0 of the probability
distributions of all 0 -measurable selections, and he shows that co (DF ) = co (DF0 ).7
Eq. (3.3) of Corollary 3.4 suggests an alternative proof of this result since νF does
not depend on the choice of or 0 .
F
4.
Additional Results
4.1. Change of Variable. The Aumann integral of a multifunction
with respect to P is dened by
Z
Z
HdP ≡
hdP : h ∈ Sela.s. H
H
:
S ⇒ R
and h integrable
.
Next result complements Theorem 5 of Hildenbrand [15], which considers the composition of a function with a multifunction; in contrast, here we consider the composition u ◦ F of a multifunction with a function, given by (u ◦ F ) (s) ≡ u (F (s))
for each s ∈ S .
Theorem 4.1. Let (S, , P) be a probability space, X a Polish space, and F : S ⇒ X
a random set. Then, for each u ∈ B (X ),
Z
(u ◦ F ) dP =
Z
udPf : f ∈ Sel F
.
That is the smallest σ -algebra relative to which F is measurable.
He considers the more general case of a weakly measurable (see [16]) closed-valued
multifunction.
6
7
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
9
In particular,
inf
Z
(u ◦ F ) dP =
Z
udν F
sup
and
Z
(u ◦ F ) dP =
Z
udνF ,
where the inf (resp., sup) is attained if u is lower (resp., upper) Weierstrass.
If P is nonatomic, the integral (u ◦ F ) dP is convex (see [3]), and so Theorem
4.1 implies that
R
Z
(u ◦ F ) dP =
Z
Z
udν F ,
udνF ,
when u is both lower and upper Weierstrass.
4.2. Conditional Distributions. If A ∈ B and P (F −1 (A)) 6= 0, the upper conditional distribution of B given A is
νF (B ∩ A)
,
νF (B|A) ≡
νF (A)
while the lower conditional distribution of B given A is its conjugate
ν F (B ∪ Ac ) − ν F (Ac )
.
ν F (B|A) ≡
1 − ν F (Ac )
If A is closed, and the probability space F −1 (A) , ∩ F −1 (A) , PF 1 (A) is considered, νF (·|A) and ν F (·|A) are, respectively, the upper and lower distributions
induced by F ∩ A (see Dempster [8]). Next we show that these updating rules are
maximum-likelihoods (see also Gilboa and Schmeidler [11, p. 42]).
−
Corollary 4.2. Let (S, , P) be a probability space, X a Polish space, and F : S ⇒
X a random set. For all A, B ∈ B such that νF (A) 6= ∅,
νF (B|A)
=
=
max Pf (B|A) : f ∈ Sel F, Pf (A) = h∈max
P h ( A)
Sel F
max P (B|A) : P ∈ PνF , P (A) = max Q (A) .
Q∈PνF
Appendix A. Some Useful Tools
Cb (X ) (resp., Ub (X )) denotes the space of all real-valued, bounded, and continuous (resp., upper semicontinuous) functions on X , with the supnorm topology. We
set
Ub+ (X )
Cb+ (X ) ≡ {u ∈ Cb (X ) : u (x) ≥ 0 ∀x ∈ X} ;
and B + (X ) are dened in the same way.
The topological dual of Cb (X ) is (isometrically isomorphic to) rba (X ) the space
of regular, bounded, additive set functions dened on the algebra generated by
closed sets (see [10, p. 262]). We can identify M with the subset of countably
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
additive elements of
weak* topology.
rba+ (X )
that take value 1 on
X,
thus endowing
M
10
with a
A.1. Choquet Capacitability Theorem: Functional Form. The following functional form of Choquet Capacitability Theorem can be derived from Dellacherie and
Meyer [6, p. 160].
Lemma A.1 (Dellacherie-Meyer-Mokobodzki). Let X be a Polish space and γ :
B + (X ) → R a functional such that:
(f.1) γ (u) ≤ γ (w) for all u, w ∈ B + (X ) such that u ≤ w,
(f.2) γ (wn ) ↑ γ (w) for all wn , w ∈ B + (X ) such that wn ↑ w,
(f.3) γ (un ) ↓ γ (u) for all un , u ∈ Ub+ (X ) such that un ↓ u.
Then
γ (w) = sup γ (u) : u ∈ Ub+ (X ) , u ≤ w
∀w ∈ B + (X ) .
A.2. Variations on Capacities. The following lemma can be obtained from Lemma
A.1 and the results ofRHuber and Strassen [17]. For each u ∈ Cb (X ), ν~ (u) denotes
the Choquet integral udν .
Lemma A.2.n Let ν be a capacity on a Polish space X . o
(i) Pν = P ∈ rba (X ) : P~ (u) ≤ ν~ (u) ∀u ∈ Cb (X ) ; in particular, it is weak*
compact and convex.
(ii) Pν = {P ∈ M : P (C ) ≤ ν (C ) ∀closed C ⊆ X}.
(iii) If ν is alternating of order 2, then Pν is nonempty and
Z
udν
= Pmax
∈P
Z
udP
ν
∀u ∈ Ub (X ) ∪ {1B }B∈B .
(iv) If ν is alternating of order 2 and P is a subset of Pν such that
Z
udν
= sup
Z
= sup
Z
udP
∀u ∈ Ub (X ) ,
wdP
∀w ∈ B (X ) .
P ∈P
then
Z
wdν
P ∈P
Proof: (i) By [17, Lemma 2.1],
Set
n
Pν = P ∈ rba+ (X ) : P~ (1) = 1, P~ (u) ≤ ν~ (u)
∀u ∈ Cb (X ) .
n
Qν ≡ P ∈ rba (X ) : P~ (u) ≤ ν~ (u)
o
+
∀u ∈ Cb (X ) .
o
If P ∈ Pν , for all u ∈ Cb+ (X ) and all c ∈ R we have P~ (u + c) = P~ (u) + c ≤
ν~ (u) + c = ν~ (u + c), hence Pν ⊆ Qν . Conversely, if P ∈ Qν , then P~ (u) ≤ ν~ (u)
for all u ∈ Cb (X ) ⊇ Cb+ (X ); if u ≥ 0, then −u ≤ 0, P~ (−u) ≤ ν~ (−u) ≤ 0, hence
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
11
P~
is positive, that is P ∈ rba+ (X ); P~ (−1) ≤ ν~ (−1) = −1, hence P~ (1) ≥ 1,
since P~ (1) ≤ 1, we have P~ (1) = 1 and Qν ⊆ Pν . Convexity is immediate, weak*
compactness follows from [17, Lemma 2.2].
(ii) Set
Qν ≡ {P ∈ M : P (C ) ≤ ν (C ) ∀closed C ⊆ X} .
By denition, Pν ⊆ Qν . Conversely, if u ∈ Cb (X ) and P ∈ Qν , then
P~ (u)
=
Z
+∞
0
Z
+∞
≤
0
P ({u ≥ t}) dt +
Z
ν ({u ≥ t}) dt +
Z
0
−∞
0
−∞
[P ({u ≥ t}) − 1] dt
[ν ({u ≥ t}) − 1] dt = ν~ (u) ,
since {u ≥ t} is closed for all t ∈ R. By (i), P ∈ Pν .
(iii) [17, Lemma 2.4] shows that: If ν is 2-alternating, and u ∈ Ub (X ), then there
exist Q ∈ Pν such that ν {u ≥ t} = Q {u ≥ t} and ν {u > t} = Q {u > t} for all
t ∈ R. In particular, Pν is nonempty and
Z
udQ =
Z
udν ≥
sup
Z
udP
P ∈Pν
yields udν = maxP ∈P udP. The case u ∈ {1B }B∈B is [17, Lemma 2.5].
(iv) Since wn ↑ w (resp. un ↓ u) implies {wn > t} ↑ {w > t} (resp. {un ≥ t} ↓
{u ≥ t}) for all t, using the Levi Monotone Convergence Theorem, it is easy to show
R
that the functional ν^ (·) = ·dν satises (f.1)-(f.3)
of Lemma A.1 on B + (X ). Next
R
we show that the functional γ (·) = supP ∈P ·dP satises (f.1)-(f.3) on B + (X ) too.
Then, since ν^ and γ coincide on Ub+ (X ), by Lemma A.1 they coincide on B + (X ),
and so on B (X ). Obviously, γ satises (f.1).
R
If wn , w ∈ BR+ (X ) and wn ↑ w, then clearly γ (wn ) = supP ∈P wn dP ↑ ` and
` ≤ supP ∈P wdP = γ (w). For all n ≥ 1, there exists Pn ∈ P such that
R
R
R
R
R
supPR∈P wn dP − n1 ≤ wn dPn ≤ supP ∈P wn dP , then wn dPn → ` and wn dPn +
1
≥ wn dP for all P ∈ P . Passing to the limits (use Levi's Theorem on the right)
n
R
R
as n → ∞ we obtain ` ≥ wdP for all P ∈ P ; therefore ` ≥ supP ∈P wdP = γ (w).
And γ satises (f.2).
Finally, γ satises (f.3) since, by assumption, γ|U + (X ) = ν^|U + (X ) and ν^ satises (f.3).
R
R
ν
Q.E.D.
b
b
Next lemma shows that the denition of innitely monotone cocapacity we gave
at p. 4 coincides with that of Philippe et al. [25, p. 771].
Lemma A.3. Let X be a Polish space, and υ : B → R a set function satisfying
(c.1), (c.2), (c.3'), (c.5'). Then υ satises (c.4') if and only if
(A.1)
υ (G) = sup {υ (K ) : K compact, K ⊆ G}
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
12
for all open subsets G of X .
Proof: Assume (c.4'). Then, υ is the conjugate of an innitely alternating capacity
ν.
By [17, Lemma 2.5], υ (·) = minP ∈P P (·). Moreover, by [17, Lemma 2.2], Pν is
compact, and [25, Proposition 1] guarantees that υ satises (A.1). Conversely, let
Gn be a sequence of open sets such that Gn ↑ G. Clearly, υ (Gn ) ↑ ` ≤ υ (G); if
S
` < υ (G), by (A.1), there exists a compact set K such that K ⊆ G = ∞
n=1 Gn and
υ (K ) > `. But, then there exists m ≥ 1 such that K ⊆ Gm , and υ (Gm ) ≥ υ (K ) >
`, a contradiction.
Q.E.D.
ν
A.3. An Implicit Function Result. Next lemma is a variation on an implicit
function theorem of Himmelberg [16].
Lemma A.4. Let (S, , P) be a probability space, X a Polish space, F : S ⇒ X
a random set, and u : X → R a Borel function. For every g ∈ Sela.s. u ◦ F , there
exists f ∈ Sel F such that P-a.s. g = u ◦ f .
Proof: Let g ∈ Sela.s. u◦F , and let E ∈ be such that P(E ) = 1 and g (s) ∈ u (F (s))
for all s ∈ E . Take f 0 ∈ Sel F (which is nonempty by the Selection Theorem
of Kuratowski and Ryll-Nardzewski) and g0 = u ◦ f 0 , then g0 ∈ Sel u ◦ F . Set
g 00 (s) = g (s) for all s ∈ E and g 00 (s) = g 0 (s) for all s ∈ E c . Then g 00 ∈ Sel u ◦ F and
P-a.s. g 00 = g . By [16, Theorem 7.4], there exist a measurable function f 00 : S → X
and L ∈ with P(L) = 1 such that: f 00 (s) ∈ F (s) and g00 (s) = u (f 00 (s)) for all
s ∈ L. Set f (s) = f 00 (s) for all s ∈ L and f (s) = f 0 (s) for all s ∈ Lc . Then f ∈ Sel F
and P-a.s. g = g00 = u ◦ f .
Q.E.D.
Appendix B. Proofs
B.1. Lemma 2.2. (i) For all B ∈ B
νF (B )
=
=
=
P ({s ∈ S : F (s) ∩ B 6= ∅})
P ({s ∈ S : F (s) ∈ {K ∈ K : K ∩ B 6= ∅}})
PF ({K ∈ K : K ∩ B 6= ∅}) = νPF (B ) .
(ii) For each innitely alternating capacity ν on X there exists a probability
measure π on K such that ν = νπ . But, the nonatomicity of the Lebesgue measure
and the fact that K is Polish guarantee that there exists a random variable F :
[0, 1] → K such that π = PF where P is the Lebesgue measure.
Q.E.D.
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
B.2. Lemma 3.3. Set
t ∈ R,
m (s )
{m ≥ t}
= maxx∈F (s) u (x) for all
=
=
=
=
{s ∈ S : ∃x ∈ F (s)
s∈S
: max
x∈F (s)
s ∈ S.
13
Notice that for all
u (x) ≥ t
with u (x) ≥ t}
{s ∈ S : F (s) ∩ {u ≥ t} =
6 ∅}
F −1 ({u ≥ t}) ∈ P ,
and analogously {m > t} = F −1 ({u > t}) ∈ P . Therefore, m : S → R is P measurable and m (s) ∈ u (F (s)) for all s ∈ S . The P -measurability of m implies
that there exists m0 : S → R which is measurable and m0 (s) = m (s) for P-almost
all s ∈ S . The fact that m (s) ∈ u (F (s)) for all s ∈ S implies that m0 ∈ Sela.s. u ◦ F .
By Lemma A.4, there exists f ∈ Sel F such that u◦f = m0 = m P-a.s.. In particular,
f ∈ Sel F and for all t ∈ R
Pf ({u ≥ t})
=
=
P ({u ◦ f ≥ t}) = P ({m ≥ t})
P F −1 ({u ≥ t}) = νF ({u ≥ t}) .
Analogously, Pf ({u > t}) = νF ({u > t}) for all t ∈ R.
Q.E.D.
B.3. Theorem 3.2. We rst prove that for each upper Weierstrass u it holds:
Z
(B.1)
udνF
= fmax
∈Sel F
Z
udPf .
Let f ∈ Sel F and B ∈ B. If f (s) ∈ B , since f (s) ∈ F (s), then F (s) ∩ B 6= ∅, that
is, s ∈ F −1 (B ). Then, from {f ∈ B} ⊆ F −1 (B ), we have
Pf (B ) = P ({f ∈ B}) ≤ P F −1 (B ) = νF (B )
∀B ∈ B.
Hence {Pf : f ∈ Sel F } ⊆ Pν . Moreover, by Lemma 3.3, there exists
such that νF ({u ≥ t}) = Ph ({u ≥ t}) for all t ∈ R. Therefore
F
Z
udPh
=
Z
udνF ≥
sup
Z
udP ≥
P ∈PνF
sup
h ∈
Sel F
Z
udPf ,
f ∈Sel F
as wanted. To obtain Eq. (3.1) it is now sucient to use Eq. (B.1) and point (iv) of
Lemma A.2. To obtain Eq. (3.2), set U (K ) = supx∈K u (x) for all K ∈ K. Clearly
U : K → R is bounded. Notice that for all t ∈ R,
{U > t}
=
K ∈ K : sup u (x) > t
=
=
{K ∈ K : ∃x ∈ K
x∈K
with u (x) > t}
{K ∈ K : K ∩ {u > t} =
6 ∅} ,
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
14
and the latter set is universally measurable in K. Moreover
PF ({U > t})
=
=
=
=
P ({s ∈ S : F (s) ∈ {U > t}})
P ({s ∈ S : F (s) ∈ {K ∈ K : K ∩ {u > t} =
6 ∅}})
P ({s ∈ S : F (s) ∩ {u > t} =
6 ∅})
P F −1 ({u > t}) = νF ({u > t}) .
It is now enough to apply the Choquet integral formula.
Q.E.D.
B.4. Corollary 3.4. By Theorem 3.2 and Lemma A.2, for all u ∈ Cb (X ), we have
max
Z
f ∈Sel F
udPf
= Pmax
∈P
Z
udP,
νF
but Pν is weak* compact, convex and contains {Pf : f ∈ Sel F }. The Separating
Hyperplane Theorem { for the dual pair (Cb (X ) , rba (X )) { guarantees that Pν is
the weak* closure of the convex hull of {Pf : f ∈ Sel F }. This proves Eq. (3.3).
Lemma A.2.ii and Lemma 2.2.i together imply
F
F
PνF
=
=
=
=
{P ∈ M : P (C ) ≤ νF (C )
{P ∈ M : P (C ) ≤ νPF (C )
∀closed C ⊆ X}
∀closed C ⊆ X}
{P ∈ M : P (C ) ≤ PF ({K ∈ K : K ∩ C 6= ∅})
∀closed C ⊆ X}
SPF .
On the other hand, [1, Theorem 2.2] guarantees that there exists a suitable nonatomic
probability space (T, , Q) and a random set G : T ⇒ X such that QG = PF and
SP = DG (in particular DG is weak* closed). By Lemma 2.2.i, QG = PF implies
νG = νF . Finally, if P is nonatomic, [13, Theorem 1] implies
F
DF
= DG = DG = SP = Pν
F
F
.
That is, Eq. (3.4).
Q.E.D.
B.5. Theorem 4.1. By denition
Z
(u ◦ F ) dP =
Z
gdP : g ∈ Sela.s. u ◦ F
and g integrable
.
Let g ∈ Sela.s. u ◦ F and g integrable. By Lemma A.4, there
exists Rf ∈ Sel F such
R
that P-a.s. g = u ◦ f ; in particular u ◦ f is integrable and gdP = u ◦ f dP. This
shows that
Z
(u ◦ F ) dP ⊆
Z
u ◦ f dP : f ∈ Sel F
and u ◦ f integrable
.
RANDOM CORRESPONDENCES AS BUNDLES OF RANDOM VARIABLES
15
The converse inclusion is trivial. Moreover, since u is bounded, u ◦ f is integrable
for all f ∈ Sel F . Finally,
Z
(u ◦ F ) dP =
Z
=
Z
u ◦ f dP : f ∈ Sel F
u ◦ f dP : f ∈ Sel F
and u ◦ f integrable
=
Z
udPf : f ∈ Sel F
.
This concludes the proof of the rst part of the theorem. The second one immediately follows from Theorem 3.2.
Q.E.D.
B.6. Corollary 4.2. Since B∩A ⊆ A, there exists g ∈ Sel F such that νF (B ∩ A) =
Pg (B ∩ A) and νF (A) = Pg (A). Hence,
νF (B|A)
=
Pg (B|A)
≤ sup Pf (B|A) : f ∈ Sel F, Pf (A) = max Ph (A)
h∈Sel F
≤ sup P (B|A) : P ∈ PνF , P (A) = max Q (A) .
Q∈PνF
On the other hand, if P
and P (A) = maxQ∈P Q (A) = νF (A), then
P (B ∩ A)
P (B ∩ A)
=
≤ νF (B|A) .
P (B|A) =
P (A)
νF (A)
That is sup P (B|A) : P ∈ Pν , P (A) = maxQ∈P Q (A) ≤ νF (B|A). Q.E.D.
∈ PνF
νF
νF
F
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Department of Economics, University of Sussex, Falmer, Brighton, United Kingdom
E-mail address : [email protected]
Bocconi, Milano, Italy
Istituto di Metodi Quantitativi and IGIER, Universita
E-mail address : [email protected]
di Torino,
Dipartimento di Statistica e Matematica Applicata and ICER, Universita
Torino, Italy
E-mail address : [email protected]