# Some Condition for Scalar and Vector Measure Games to Be Lipschitz

Authors: F. Centrone; A. Martellotti
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

We provide a characterization for vector measure gamesν=f∘μ in pNA∞, with μ vector of nonatomic probability measures, analogous to the one of Tauman for games in pNA, and also provide a necessary and sufficient condition for a particular class of vector measure games to belong to AC∞.


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## Body

## 1. Introduction

Measure games, that is, transferable utility (TU) games of the formν=f∘μ, where μ is a nonatomic measure on σ-algebra Σ of a space Ω and f is a function defined on the range of μ, with f(0)=0, arise in several contexts including game theory, mathematical economics, and even finance (where, under suitable hypotheses on f and μ, they are termed as distorted probabilities). One of the reasons of their popularity lies in the fact that they generate fundamental spaces of games and that many games of interest, such as, for instance, market games of finite type, fall into this category. Classically in the literature, one distinguishes betweenscalar measure games where μ is a nonnegative scalar measure andvector measure games where μ is an n-dimensional measure with nonnegative components. The extension to signed measure is also customary.The most classical space related to measure games ispNA, that is, the closed linear subspace of BV generated by all powers (with respect to pointwise multiplication) of nonatomic probability measures. Several results exist, concerning scalar measure games in pNA [1, 2], while the only characterization of vector measure games in this space is the one of Tauman [2].Besides theBV-norm, one encounters in the literature the ∥·∥∞-norm that defines the subspace AC∞⊂BV of the so-calledLipschitz games. Then clearly one may define also the space pNA∞ as the ∥·∥∞-closure of the space generated by all powers of nonatomic probability measures. In this space, the only characterization of vector measure games we are aware of up to now is due to Milchtaich [3], and it requires the function f to be continuously differentiable.All these results put in evidence how measure games are difficult to characterize once the differentiability assumption is dropped, though economically significant measure games that do not fall into this category exist in the literature. For example, Milchtaich's characterization shows that iff is piecewise linear, then measure games of the form f∘μ do not belong to pNA∞; on the other side the linear span of games with f piecewise linear and μ vector of mutually singular nonatomic probability measures plays a role for example in value theory [4].In this paper, we face the problem of characterizing measure games both inpNA∞ and in AC∞. Our starting point is the characterization for scalar measure games in AC∞ given in [5]; there we proved that f∘μ∈AC∞ if and only if f is Lipschitz. Here we introduce a generalization of the Lipschitz condition, namely,lipschitzianity in link directions, which proves to characterize vector measure games f∘μ in AC∞, when the range of μ has finitely many exposed points and which thus covers interesting cases in literature. As a consequence, we extend the characterization in [5] to measure games of the form f∘μ where μ is a signed measure.Another interesting subspace is the classBC of Burkill-Cesari (BC) integrable games, introduced in [6]. The investigation on this space has been further developed in [5]; there it has been proved that the Burkill-Cesari integral is a ∥·∥∞-continuous (semi)value (but in general not BV-continuous) and that it differs from the Aumann and Shapley value. Actually, the BC integral and the space BC turned out to be fruitful to provide a proper subspace of AC strictly larger than pNA on which a value can be defined; remember that existence and uniqueness of a value on pNA are well known, while the question is still open on AC. In force of these results, a better understanding of the structure of the space BC∩AC∞, starting from its simplest elements, that is, measure games, seems to be an interesting task.The outline of the paper is as follows. In Section3 we characterize vector measure games in pNA∞; although the statement of the result is formally analogous to the one of Tauman, the proof differs from his, and this is essentially due to the difficulties arising in handling the ∥·∥∞-norm on this space. In Section 4 we first characterize a particular class of vector measure games in AC∞ and, as a corollary, also measure games where μ is a signed measure are characterized through lipschitzianity. The final part of the section is devoted to the investigation of BC integrable Lipschitz measure games; we completely characterize the one-dimensional ones and provide a necessary condition in larger dimensions; also a topological condition is given, to ensure that a Lipschitz measure game is ∥·∥∞-close to a BC integrable one.


## 2. Preliminaries

In the following we will deal with the following elements, as in [1].(
Ω
,
Σ
) denotes a measurable space isomorphic to ([0,1],B) (where B denotes the Borel σ-algebra on [0,1]).Atransferable utility (TU)gameν is a real valued function on Σ such that ν(∅)=0.The set of all nonatomic measures on(Ω,Σ) is denoted by NA, the cone of nonnegative measures of NA by NA+, while the set of probability measures in NA is indicated by NA1. Given μ∈NA, the variation measure is denoted by |μ|. For a vector measure μ=(μ1,…,μn)∈(NA)n, the variation measure is defined by |μ|=∑i=1n‍|μi|.A gameν is said to be Lipschitz if there exists μ∈NA+ such that for every link S⊂T⊂Ω in Σ it holds
(1)|ν(T)-ν(S)|≤μ(T)-μ(S).
The space of Lipschitz games is denoted by AC∞ for it is a Banach space under the norm ∥ν∥∞ defined in the following way; for every μ∈NA+ such that (1) holds, write -μ⪯ν⪯μ. Then set
(2)∥ν∥∞=inf⁡{μ(Ω),μ∈NA+,-μ⪯ν⪯μ}.
To simplify notation, given a game ν∈AC∞, and for fixed H∈Σ we will denote as usual by Π(H) the class of finite partitions of H consisting of elements in Σ; for every D∈Π(H), say, D={D1,…,Dk}, let TD={TD={T1,…,Tk}withTi⊂Dic,Ti∈Σ}. Then denote
(3)S(ν,D,TD)=∑i=1k‍[ν(Di∪Ti)-ν(Ti)].
For a game ν∈AC∞ define, according to [7], the measures
(4)ν*(H)=sup⁡{S(ν,D,TD),D∈Π(H),TD∈TD},ν*(H)=inf⁡{S(ν,D,TD),D∈Π(H),TD∈TD}.
Then
(5)∥ν∥∞=sup⁡{|ν*|(H)+|ν*|(Hc),H∈Σ}.
It is clear that if ν is a monotone game, then ν*,ν* are nonnegative and in this case ∥ν∥∞=ν*(Ω).Note that for everyH∈Σ, D∈Π(H), TD∈TD one has
(6)|S(ν,D,TD)|≤∥ν∥∞.
On the other side, since ν*,ν* are measures, there are Hahn decompositions (P1,N1), (P2,N2) which allow us to rewrite (5) as
(7)∥ν∥∞=sup⁡{ν*(E∩P1)-ν*(E∩N1)+ν*(Ec∩P2)ffffffk-ν*(Ec∩N2),E∈Σ}.
Hence, if for a game ν one has |S(ν,D,TD)|≤ε for each D∈Π(H),TD∈TD and every H∈Σ, then clearly ∥ν∥∞≤4ε.Given a vector measureμ=(μ1,…,μn)∈(NA+)n and denoting its range by R(μ), avector measure game is a game ν of the form ν=f∘μ, where f is a real valued function defined on the range of μ and f(0)=0. In the case n=1, this type of games is calledscalar measure games.Analogously to what is usually done in the spaceBV, the symbol pNA∞ denotes the ∥·∥∞-closure of the space generated by all powers of nonatomic probability measures.Given a convex subsetX of Rn, a vector z is called Xadmissible if z=x-y for some x,y∈X. A real valued function f defined on X is said to becontinuously differentiable on X if for each X admissible z the derivative df(x+hz)/dh exists at each point in the relative interior of X and it can be continuously extended at each point of X. The space of continuously differentiable functions on a set X will be denoted by C1(X).Given a convex compact subsetK of Rn, a point x∈K is said to beexposed if {x} is the intersection of K with some supporting hyperplane of K.As in [8], given a monotone nonatomic game λ, one defines themesh of a partition D as
(8)δλ(D)=max⁡{λ(I),I∈D}.A gameν isBurkill-Cesari (BC)integrable with respect to δλ if the following limit exists, for each E∈Σ:
(9)E⟼∫E‍ν=lim⁡δλ(D)→0D∈Π(E)⁡∑I∈D‍ν(I).
We denote by BC the space of games ν such that there exists λ∈NA+ so that ν is BC integrable with respect to the mesh δλ.TheBC integral does not depend upon the integration mesh (see Proposition  5.2 in [6]); in other words, for every λ∈NA+ such that ν is δλ-BC integrable, the BC integral remains the same. Moreover, the BC integral of a game ν is a finitely additive measure.


## 3. Measure Games inpNA∞

In [5] we have obtained a characterization (Proposition  12) of scalar measure games (made through a nonnegative measure) for the whole space AC∞. Anyway the Lipschitz condition on f is not sufficient to ensure that a measure game belongs to pNA∞, that is, to the ∥·∥∞-closure of the space generated by all powers of nonatomic probability measures. Indeed, in [3] Milchtaich has shown that continuous differentiability is required, and it guarantees that a vector measure game belongs to pNA∞. To our knowledge, this is also the unique characterization of vector measure games in pNA∞ so far. It is well known that pNA∞ is strictly contained in pNA and that many games of interest belong to this space [4].Our first goal is to obtain a characterization for vector measure games inpNA∞, similar to the one given by Tauman for pNA [2]. We point out that the technique of the proof is different, due to the difficulties in dealing with approximations in the ∥·∥∞-norm instead of in the BV-norm.We begin with a Lemma.Lemma 1.
Letμ∈(NA1)n, f∈C1([R(μ)]). Then for every ε>0 there exists a polynomial in n variables p such that
(10)∥(f∘μ)-(p∘μ)∥∞<ε.Proof.
According to [1] for every ε>0 there exists a polynomial p on R(μ) such that the norm ∥f-p∥I<ε, where one defines
(11)∥f-p∥I=∥f-p∥u+∑i=1n‍∥∂f∂xi-∂p∂xi∥u
and ∥·∥u is the usual uniform norm on continuous functions.
Without loss of generality we assumep(0)=0.
FixH∈Σ, D∈Π(H), TD∈TD. From the Mean Value Theorem we have the following:
(12)|S[(p∘μ)-(f∘μ),D,TD]|≤∑i‍|[(p∘μ)(Di∪Ti)-(f∘μ)(Di∪Ti)]hhhhg-[(p∘μ)(Ti)-(f∘μ)(Ti)]|=∑i‍|∇(f-p)[ciμ(Di∪Ti)+(1-ci)μ(Ti)]·μ(Di)|≤∑i‍∑j=1n‍|(∂f∂xj-∂p∂xj)[ciμ(Di∪Ti)+(1-ci)μ(Ti)]|×μj(Di).
Now ciμ(Di∪Ti)+(1-ci)μ(Ti)∈R(μ) for each i=1,…,n and hence
(13)|(∂f∂xj-∂p∂xj)[ciμ(Di∪Ti)+(1-ci)μ(Ti)]|≤∥∂f∂xj-∂p∂xj∥u<ε.
Thus
(14)|S[(p∘μ)-(f∘μ),D,TD]|≤ε∑i‍∑j‍μj(Di)=ε∑j‍μj(H)=ε|μ|(H).Tauman [2] gave the definition of ∥·∥BV-continuity at μ for a function f defined on the range of a vector measure μ∈(NA1)n, which we recall below.Givenμ∈(NA)n, denote with B(μ) the set of measures η∈(NA)n having the same range of μ and, for ε>0, define
(15)B(μ,ε)={η∈B(μ)suchthat|η-μ|(Ω)<ε}.
Fixed μ∈(NA1)n, a function f on R(μ) (with f(0)=0), is ∥·∥BV-continuous atμ, if for every ε>0 there exists δ>0 such that for each η∈B(μ,δ) there follows ∥(f∘μ)-(f∘η)∥BV<ε. Tauman proved that a vector measure game f∘μ, μ∈(NA1)n, belongs to pNA if and only if f is ∥·∥BV-continuous at μ.We translate the same definition using the∥·∥∞-norm; that is, we say that a function f on R(μ) is ∥·∥∞-continuous atμ if for every ε>0 there exists δ>0 such that for each η∈B(μ,δ) there follows ∥(f∘μ)-(f∘η)∥∞<ε. Then we are able to prove that the analogous characterization of Tauman holds for vector measure games in pNA∞.Theorem 2.
f is ∥·∥∞-continuous at μ∈(NA1)n if and only if (f∘μ)∈pNA∞.Proof.
We first prove the sufficient condition; namely, we assume thatf is ∥·∥∞-continuous at μ∈(NA1)n.
Note first that without loss of generality we can assume thatR(μ) has nonempty interior.
As in [2], fix a>0 such that the cube
(16)C=μ(Ω)2+[-a2,a2]n⊂Ro(μ),
and for δ∈]0,1[ define
(17)fδ(x)=1an∫C‍f[(1-δ)x+δy]dy-1an∫C‍f(δy)dy.
Thus fδ(0)=0 and according to [2, Lemma  7] fδ∈C1[R(μ)].
We also need the following structure Lemma [2, Lemma  9].
Lemma  3.Letη∈(NA1)nand letΩ1⊂Ω2⊂Ωbe a link with bothΩ1,Ω∖Ω2uncountable. Then, for everyδ∈]0,1[and for everyy∈R(η)there existsηy∈B(η,2nδ)withηy(H)=(1-δ)η(H)+δyfor eachH∈ΣwithΩ1⊂H⊂Ω2.
Letδ be momentarily fixed, and let {T,S∪T} be any link in Σ, with T, Ω∖(S∪T) uncountable. Consider the difference γ=νδ-ν where νδ=fδ∘μ,ν=f∘μ. We have
(18)γ(S∪T)-γ(T)=fδ[μ(S∪T)]-f[μ(S∪T)]-fδ[μ(T)]+f[μ(T)]=1an∫C‍f[(1-δ)μ(S∪T)+δy]dy-f[μ(S∪T)]-1an∫C‍f[(1-δ)μ(T)+δy]dy+f[μ(T)]=1an∫C‍{f[(1-δ)μ(S∪T)+δy]-f[μ(S∪T)]jjjjjjjjjjj-f[(1-δ)μ(T)+δy]+f[μ(T)]}dy.
Now by Lemma 3 applied with T=Ω1, S∪T=Ω2, and η=μ for every y∈C there is ηy∈B(μ,2nδ) such that ηy(S∪T)=(1-δ)μ(S∪T)+δy and ηy(T)=(1-δ)μ(T)+δy. Hence, continuing the computation (18) we reach
(19)γ(S∪T)-γ(T)=1an∫C‍{[(f∘ηy)(S∪T)-(f∘μ)(S∪T)]jjjjjjjjjjjjjjjjjjk-[(f∘ηy)(T)-(f∘μ)(T)]}dy.
By the assumption corresponding to ε>0 there exists δo>0 such that for m∈B(μ,δo) one has ∥(f∘m)-(f∘μ)∥∞<ε.
Let us now fixδ<δo/2n, and let H∈Σ, D∈Π(H), TD∈TD be fixed also, in such a way that Ti, Ω∖(Di∪Ti) are uncountable.
Then we have(20)S(γ,D,TD)=∑i‍γ(Di∪Ti)-γ(Ti)=1an∑i‍∫C‍{[(f∘ηy)(Di∪Ti)-(f∘μ)(Di∪Ti)]hhhhhhhhhh-[(f∘ηy)(Ti)-(f∘μ)(Ti)]}dy=1an∫C‍S{[(f∘ηy)-(f∘μ)],D,TD}dy.
Since ηy∈B(μ,2nδ)⊂B(μ,δo) we know that ∥(f∘ηy)-(f∘μ)∥∞<ε, and from (6), |S{[(f∘ηy)-(f∘μ)],D,TD}|<ε for each y∈[0,1]. Hence |S(γ,D,TD)|<ε for each pair D∈Π(H),TD∈TD that satisfies the cardinality conditions at each link that proves that fδ∘μ approximates f∘μ in the ∥·∥∞-norm.
To conclude our proof, we will prove that for everyH∈Σ, D∈Π(H), TD∈TD can be replaced by a choice H♯∈Σ, D♯∈Π(H♯), TD♯∈TD so that each link (Ti♯,Di♯∪Ti♯) fulfills the cardinality requirements for each link, and S[γ,D,TD]=S[γ,D♯,TD♯♯].
To this aim, letε,δ,H∈Σ,D∈Π(H),TD∈TD be fixed, with D={D1,…,Dk}. Without loss of generality we can assume that each Di is not μ-null.
LetI⊂{1,…,k} be the set of indexes for which the corresponding link (Ti,Di∪Ti) does not fulfill the cardinality requirements; then I can be split into two disjoint subsets I1 and I2, where I1 is the set of indexes for which Ω∖(Di∪Ti) is at most countable.
For eachi∈I2 one can choose an uncountable μ-null set Ni in Ω∖(Di∪Ti) and then replace Ti with Ti~=Ti∪Ni; since i∈I2, we have Ti countable, and, therefore, μ(Ti)=0; thus the replacement does not affect the corresponding summand in S(γ,D,TD).
For each indexi∈I1 choose an uncountable μ-null set Ni⊂Di and replace Di with Di~=Di∖Ni, Ti with Ti~=Ti∪Ni.
Then we can replaceD with D♯={Di♯,i=1,…,n} setting
(21)Di♯={Di~ifi∈I1,Diotherwise
and analogously TD with TD♯={Ti♯,i=1,…,n} where
(22)Ti♯={Ti~ifi∈I,Tiotherwise
and finally set H♯=⋃i=1k‍Di♯. By construction then S(γ,D,TD)=S(γ,D♯,TD♯♯).
To prove the converse implication, according to [1, Lemma  7.2 page 41], it is enough to prove that monomials of the form cμα, with c∈R, μ∈NA1, α∈N+, are ∥·∥∞-continuous at each μ∈NA1.
We first prove that ifμ∈B(m,δ), H∈Σ, D={Di}i=1,…,k∈Π(H), p∈N, then
(23)∑i=1k‍|mp(Di)-μp(Di)|<pδ.
In fact, for each summand we have
(24)|mp(Di)-μp(Di)|≤mp-1(Di)|m(Di)-μ(Di)|+μ(Di)|mp-1(Di)-μp-1(Di)|≤m(Di)|m-μ|(Di)+μ(Di)(p-1)δ,
where the last inequality follows from Lipschitz condition on [0,1] and the fact that m∈NA1. Then in B(m,δ) we have further |mp(Di)-μp(Di)|<δm(Di)+(p-1)μ(Di)δ, whence for the whole sum we obtain (23).
Let now as usualH∈Σ, D∈Π(H), D={Di}, TD∈TD be fixed. Consider the single summand in S(mα-μα,D,TD)(25)mα(Ti∪Di)-μα(Ti∪Di)-mα(Ti)+μα(Ti)=mα(Di)-μα(Di)+∑j=1α-1‍(αj)[mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)].
In the sum S(mα-μα,D,TD) there will then appear a sum of the form in (23) which we can estimate with αδ.
Let us now treat the summands of the form[mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)]. Clearly
(26)mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)=mα-j(Di)[mj(Ti)-μj(Ti)]+μj(Ti)[mα-j(Di)-μα-j(Di)]
and so in the whole |S(mα-μα,D,TD)| we can apply the following bound from above:
(27)∑i‍|mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)|≤∑i‍mα-j(Di)|mj(Ti)-μj(Ti)|+μj(Ti)|mα-j(Di)-μα-j(Di)|.
Now each term |mj(Ti)-μj(Ti)|≤jδ again by Lipschitz condition and the choice of m∈B(μ,δ). Also mα-j(Di)≤m(Di) since m∈NA1 and μ(Ti)≤1 for the same reason. Hence, because of (23),
(28)∑i‍|mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)|≤jδ∑i‍m(Di)+∑i‍|mα-j(Di)-μα-j(Di)|<jδ+(α-j)δ=αδ.
Finally
(29)∑i‍∑j=1α-1‍(αj)|mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)|=∑j=1α-1‍(αj)∑i‍|mα-j(Di)mj(Ti)-μα-j(Di)μj(Ti)|≤(2α-1)αδ
from which we conclude according to (25) and (23)
(30)|S(mα-μα,D,TD)|≤α2αδ
which only depends upon α. Hence the proof is complete.Note that, in proving the “if” part of the above theorem, we have incidentally proven that, for a measure gamef∘μ∈pNA∞, for each ε>0 there exists δ>0 such that ∥(f∘μ)-(fδ∘μ)∥∞<ε.On the other side, asfδ∈C1[R(μ)], we can apply Lemma 1 and find a polynomial pε such that ∥(fδ∘μ)-(pε∘μ)∥∞<ε.In other words we have proven the following statement.Theorem 4.
Letμ∈(NA1)n,f:R(μ)→R. The following are equivalent. (i)
(f∘μ) is in pNA∞;(ii)
for eachε>0 there exists a polynomial p:Rn→R vanishing at 0 such that
(31)∥(f∘μ)-(p∘μ)∥∞<ε.


## 4. Measure Games inAC∞

Throughout this section we will deal with mapsf:R+n→R with f(0)=0, namely,admissible maps.For two vectorsx,y∈R+n we will adopt the notation x≪_y for the usual componentwise order. We introduce the following definitions, which extend Lipschitz condition when n>1.Definition 5.
The mapf is said to beLipschitz in increasing directions when L>0 exists such that for each pair x≪_y in R+n one has
(32)|f(x)-f(y)|≤L·∥x-y∥.Definition 6.
Letμ:Σ→R+n be a measure; a pair x,y∈R(μ) is said to be a μ-link direction when there exists a link (S,S∪T) in Σ such that μ(S)=x, μ(S∪T)=y.
A mapf:R+n→R is then said to beLipschitz inμ-link directions if there exists L>0 such that for every μ-link direction x,y there holds
(33)|f(x)-f(y)|≤L·∥x-y∥.
Note that if f is Lipschitz in increasing directions, then it is Lipschitz in μ-link directions for each n-dimensional nonnegative measure μ.
Let nown∈N+ and μ∈(NA+)n be fixed, and consider the spaces (i)
A={f:R+n→R admissible and Lipschitz in the increasing directions},(ii)
L(μ)={f:R+n→R admissible and Lipschitz in μ-link directions}.Note that if f∈A, then f∘μ∈AC∞ for every μ∈(NA+)n, while we can assert the same for f∈L(μ) only for precisely μ.
Now, givenf,μ, let ν=f∘μ and define the subsets of NA+. Consider
(34)A(ν)={λ∈NA+λ≪|μ|,-λ⪯ν⪯λ},C(μ)={λ≪|μ|:dλd|μ|∈L∞(|μ|)}.
Note that if ν is a vector measure game in AC∞, then A(ν)≠∅. Indeed, let λ∈NA+ be such that -λ⪯ν⪯λ and consider the Jordan decomposition λ=λ1+λ2 with λ1≪|μ|, λ2⊥|μ|. Then there are two disjoint sets Ω1,Ω2 which support λ1,λ2, respectively. This means that μ vanishes on Ω2; hence, λ1∈A(ν).
In fact, for each linkS⊂T, setting S1=S∩Ω1, T1=T∩Ω1 since μ(S1)=μ(S), μ(T1)=μ(T) one has
(35)|ν(T)-ν(S)|=|ν(T1)-ν(S1)|≤λ(T1)-λ(S1)=λ1(T1)-λ1(S1)=λ1(T)-λ1(S).
We can now state the following.Proposition 7.
Letμ∈(NA+)n and ν=f∘μ. The following are equivalent: (i)
f∈L(μ);(ii)
there existsc>0 such that c|μ|∈A(ν);(iii)
A(ν)∩C(μ)≠∅.Proof.
(i) implies (ii), since, iff∈L(μ), then immediately L|μ|∈A(ν)∩C(μ).
The implication (ii)⇒ (iii) is immediate.
Finally, letλ∈A(ν)∩C(μ). This immediately implies that there exists L>0 such that dλ/d|μ|≤L|μ|-a.e., and this in turn implies that λ≤L|μ| on Σ. On the other side, as mentioned in Section 2, we have that |ν(S∪T)-ν(T)|≤λ(S) for every pair of disjoint sets S,T∈Σ. Then immediately, on each μ-link direction x≪_y and every link S,S∪T such that μ(S)=x, μ(S∪T)=y one finds
(36)|f(x)-f(y)|≤λ(S)≤L|μ|(S)=L·∥x-y∥
that says that f∈L(μ).Each of the conditions of Proposition7 implies that ν∈AC∞. It is, therefore, rather natural to ask whether condition f∈L(μ) characterizes vector measure games f∘μ in AC∞. We already know from Proposition  12 in [5] that this is the case for n=1.In the more general case, we have the following result.Theorem 8.
Letμ∈(NA+)n be such that its range R(μ) has only finitely many exposed points. Then a vector measure game ν=f∘μ∈AC∞ if and only if f∈L(μ).Proof.
We only need to prove that ifν∈AC∞, then f∈L(μ).
LetP1,…,Pk be the exposed points of R(μ), and let A1,…,Ak be sets in Σ such that μ(Ai)=Pi, i=1,…,k.
Consider the partition generated byA1,…,Ak, say, D={Ω1,…,Ωn}, and assume that each Ωi has nonnull μ-measure.
IfΣo denotes the algebra generated by D, we have that μ(Σo)=R(μ); in fact by additivity each Pi∈μ(Σo) and from Lyapunov Theorem μ(Σo) is convex; hence μ(Σo)⊃co⁡{P1,…,Pn}=R(μ) where the last equality is deduced from Straszewicz Theorem.
Letx≪_y be a fixed μ-link direction. Then x=x1+⋯+xn, y=y1+⋯+yn, where, for each i, xi, yi∈{tμ(Ωi),t∈[0,1]}. Moreover, easily xi≪_yi.
Set nowx=vo,vi=∑j=1i‍yj+∑j=i+1n‍xj,i=1,…,n-1,vn=y.
Fixλ∈A(ν) and i=1,…,n.
We know that there areti′≤ti′′∈[0,1] such that xi=ti′μ(Ωi), yi=ti′′μ(Ωi). Choose then Si′⊆Si′′⊂Ωi such that for the (n+1)-dimensional measure (λ,μ) one has (λ,μ)(Si′)=ti′(λ,μ)(Ωi),(λ,μ)(Si′′)=ti′′(λ,μ)(Ωi), and let Ti⊂Ω∖Ωi with
(37)μ(Ti)=∑j=1i-1‍yj+∑j=i+1n‍xj.
This is possible forxj,yj∈R(μ|Ωj) and for i≠j,Ωi∩Ωj=∅.
ThenSi′∪Ti, Si′′∪Ti is a link and hence
(38)|ν(Si′∪Ti)-ν(Si′′∪Ti)|≤λ(Si′′∖Si′)=(ti′′-ti′)λ(Ωi).
But μ(Si′∪Ti)=μ(Si′)+μ(Ti)=vi-1, μ(Si′′∪Ti)=vi. Thus (38) becomes
(39)|f(vi)-f(vi-1)|≤(ti′′-ti′)λ(Ωi).
Now
(40)∥vi-vi-1∥=∥yi-xi∥=(ti′′-ti′)∥μ(Ωi)∥=(ti′′-ti′)|μ|(Ωi).
Thus
(41)|f(vi)-f(vi-1)|≤λ(Ωi)|μ|(Ωi)∥vi-vi-1∥.
Since vi-vi-1=yi-xi, we have that ∥vi-vi-1∥≤∥y-x∥.
In conclusion we have that(42)|f(x)-f(y)|≤L∥y-x∥
with L=n·∑i=1n‍(λ(Ωi)/|μ|(Ωi)).Observe that the above result includes two interesting cases: the case ofμi⊥μj for each pair of components μi,μj of μ and the case of vector measures μ which can be represented as integral measures (μ1,μ1(φ)) where φ is a (n-1)-dimensional simple function.Also, from Theorem8 one derives the following corollary that generalizes Proposition 12 in [5] to the case of signed scalar measure games, namely, games of the form ν=f∘μ with μ signed measure.Corollary 9.
Letμ:Σ→R be a nonatomic signed measure, and let f:[-μ-(Ω),μ+(Ω)]→R. Then f∘μ∈AC∞ if and only if f is Lipschitz on R(μ).Proof.
Consider the nonatomic two-dimensional measurem=(μ+,μ-) and the function g:R2→R defined by g(x,y)=f(x-y). Thus ν=f∘μ=g∘m. Since R(m) is a rectangle, if ν∈AC∞, from Theorem 8g∈L(m), namely, a constant L>0 exists, such that for each m-link direction x, y(43)|g(x)-g(y)|≤L∥y-x∥.
Then f is also Lipschitz with constant L.So far, we have not been able to answer the question whether the condition in Theorem8 actually characterizes vector measure games in AC∞.We now turn our attention to a smaller subspace ofAC∞. In [5, 6] we have introduced and studied the space BC of Burkill-Cesari (BC) integrable games. In particular in [5] we considered the space Q=BC∩AC∞ of Lipschitz games that are indeed BC integrable.Here we will consider the subspaceV of vector measure games in Q.First of all, observe thatAC∞∖BC and BC∖AC∞ are nonempty. Indeed a scalar measure game f∘μ∈AC∞ if and only if f is Lipschitz on [0,μ(Ω)], while, according to Proposition  14 in [5], f∘μ∈BC if and only if f admits right hand side derivative at 0.Therefore, for instance, iff(x)=1-1-x and P∈NA1, (f∘P)∈BC∖AC∞.On the other side, letμ be a signed measure with μ(Ω)=0, and let ν=|μ|; then easily ν∈AC∞.In fact, ifS,T is a link, then
(44)|ν(T)-ν(S)|-||μ(T)|-|μ(S)||≤|μ(T)-μ(S)|≤|μ|(T∖S)
and so |μ|∈A(ν). On the other side, ν∉BC; indeed, the following extension of Proposition 14 in [5] can be derived from Corollary 9.Proposition 10.
Takeμ∈NA and let f:[-μ-(Ω),μ+(Ω)]→R. Then f∘μ∈V if and only if f is Lipschitz on R(μ) and f′(0) exists.Proof.
The proof of the sufficiency goes along the same lines of the proof of Theorem  6.1 in [6].
Conversely, sinceν∈AC∞ we know from Corollary 9 that f is Lipschitz; hence the ratios f(x)/x, x≠0 are bounded. Assume that ν∈BC but f′(0) does not exist. We have then the following cases: (i)
at least one betweenf+′(0) and f-′(0) does not exist;(ii)
f-′(0)≠f+′(0).The first case can be treated analogously to proof of Proposition  14 in [5], simply working with a set F⊂P, F⊂N, respectively, where (P,N) is a Hahn decomposition of Ω.
Assume then thatf-′(0)=l1≠l2=f+′(0).
Let thenxn↓0 and let us fix F∈Σ with μ+(F)=μ-(F)>0, so that μ(F)=0. Choose ε∈]0,μ+(F)]. Then there exists n¯∈N such that for each n>n¯(45)|f(xn)xn-l1|<ε3μ+(F),|f(-xn)-xn-l2|<ε3μ-(F).
By means of the continuity of f at 0, choose next n~>n¯ such that |f(x)|<ε/3 whenever x≤xn~; also n~ can be chosen so that |li||xn~|<ε/3 and such that xn~(λ(F)/μ+(F))<δ(ε/3) where δ is the parameter of δλBC integrability.
Choose now the followingD∈Π(F∩P); by means of Lyapunov Theorem, divide F into finitely many sets, say I1,…,Ik, each with (μ,λ)(Ij)=(xn~,(λ(F)/μ+(F))xn~), until μ((F∩P)∖⋃j=1k‍Ij)≤xn~ and then choose Ik+1=(F∩P)∖⋃j=1k‍Ij; thus easily λ(Ik+1)=(λ(F)/μ+(F))μ(Ik+1) for
(46)λ(Ik+1)=λ(F)-∑j=1k‍λ(Ij)=λ(F)-kλ(F)μ+(F)xn~=λ(F)μ+(F)[μ+(F)-kxn~]=λ(F)μ+(F)μ(Ik+1).
Then for D={I1,…,Ik,Ik+1} one has δλ(D)<δ(ε/3).
We have then(47)∑I∈D‍|f[μ(I)]-l1μ(I)|=∑n=1k‍|f(xn~)-l1xn~|+|f[μ(Ik+1)]-l1μ(Ik+1)|≤∑n=1k‍|f(xn~)-l1xn~|+|f[μ(Ik+1)]|+|l1|xk~=∑n=1k‍|f(xn~)-l1xn~xn~|xn~+ε3+ε3.
As for the first sum we have the following estimate:
(48)∑n=1k‍|f(xn~)-l1xn~xn~|xn~<ε3μ+(F)∑n=1k‍xn~=ε3·μ((F∩P)∖Ik+1)μ+(F)<ε3.
In conclusion
(49)∑I∈D‍|f[μ(I)]-l1μ(I)|<ε.
Clearly we can repeat this construction with -xn~ to find a partition D*∈Π(F∩N) with δλ(D*)<δ(ε/3) as above; again
(50)∑I∈D*‍|f[μ(I)]-l2μ(I)|<ε.
Hence D∪D*∈Π(F) and
(51)|∑I∈D∪D*‍ν(I)-[l1μ(F∩P)+l2μ(F∩N)]|=|∑I∈D∪D*‍ν(I)-μ+(F)(l1+l2)|≤2ε.
On the other side, we can choose a decomposition D∈Π(F∩P) with δλ(D)<ε/2 and then for each I∈D choose symmetrically J(I)⊂F∩N with μ[J(I)]=-μ(I) so that D*={J(I)}∈Π(F∩N).
Then we can defineDo={I∪J(I),I∈D} to produce a decomposition of the whole F with δλ(Do)<ε but ∑Do‍ν(I)=0. Since l1≠l2, the game ν is not δλ-BC integrable.Up to this point we have been able to characterize scalar measure games and signed measure games inV by means of the existence of f′(0). What can be said for more general vector measure games?We will see that differentiability at zero is not necessary, because we are considering admissible functions defined on the whole positive orthant ofRn, while with BC integrability we are taking into account the μ-admissible directions.For example, consider onR+2 the classical example of nondifferentiable map
(52)f(x,y)={x2yx4+y2whenever(x,y)≠O,0if(x,y)=O
which is not differentiable on the whole orthant, being not continuous at O. However, if μ∈NA+ and η=(μ,μ), then R(η) reduces to a line segment and f∘η=η/(1+η2)∈Q since it can be represented as g∘μ with g(t)=t/(1+t2) which is differentiable.Indeed the following necessary condition derives from theBC integrability.Proposition 11.
Letμ=(μ1,…,μn) be in (NA+)n, and let f:R+n→R be in L(μ). If f∘μ∈V, then f admits directional derivative dfu(O) along every admissible direction u (i.e., such that μ(F)=tu for some F∈Σ,μ(F)≠0 and some t>0); moreover, the convergence
(53)lim⁡h→0f(hu)h=duf(O)
is uniform with respect to u.The proof of the existence of each directional derivative is substantially the same as that of Proposition10, while the uniformity of the limit is deduced from the assumption of BC integrability, where the defining limit
(54)∫E‍ν=lim⁡δλ(D)→0D∈Π(E)⁡∑I∈D‍ν(I)
is uniform with respect to E∈Σ.Observe that the requirementf∈L(μ) in the previous statement could be weakened by requiring that for each radial direction u that crosses ∂R(μ) at a point P≠O, the ratios f(tu)/t are bounded in ]0,1].Unfortunately, even with this weakening, the condition expressed in the above Proposition does not characterize vector measure games inBC. To get convinced, we present the following example.Example 12.
Considerf(x,y) as above which is not differentiable at O, since it is not continuous, despite the fact that it admits directional derivative at every direction u on the positive orthant, given by
(55)duf(O)={cos⁡2ϑsinϑifsin⁡ϑ≠0,0ifsin⁡ϑ=0,
where u=(cos⁡ϑ,sinϑ).
A simple computation in fact provides, forsinϑ≠0,
(56)|f(tcos⁡ϑ,tsinϑ)t-duf(O)|=t4cos⁡6ϑ|sinϑ|(t4cos⁡4ϑ+sin2ϑ).
For the sake of simplicity set α=sinϑ,β=cos⁡2ϑ. Then we want to show that the ratios β3t4/|α|(t4β2+α2) can be made arbitrarily small, somehow independently on the values of α,β.
Easily(57)β3t4|α|(t4β2+α2)≤β3t4|α|3
which in turn is smaller than ε for t4<ε·(|α|3/β3)=ε(|sinϑ|3/cos⁡6ϑ); one immediately checks that the map ϑ→|sinϑ|3/cos⁡6ϑ is increasing for 0<ϑ<π/2.
Consider nowγ:[0,1]→R defined as
(58)γ(x)={x2for0≤x≤xo,2xox-xo2forx>xo
with 0<xo<1.
Then consider the “reverse” functionG:[0,1]→R defined as G(x)=1-γ(1-x); then the subset of R+2 given by {(x,y),x∈[0,1],γ(x)≤y≤G(x)} is a zonoid, that is, the range of a nonatomic measure μ (see [9]), and the admissible directions for such μ have slopes not exceeding 2xo; this is enough to achieve the required uniformity for the vector measure game f∘μ.
We will now prove that the weakened assumption is satisfied.
Fix an admissible directionu=(u1,u2), that is, such that span u intersects ∂R(μ) at a point P≠O.
Observe then that all the ratios along theu direction are bounded.
However the game is notBC integrable; in fact, let F be a set for which μ(F)=(xo,xo2); whichever δ>0, δ<xo/3 we choose, we can always find a subset I of F such that μ(I)=(δ,δ2) (thanks to the Hereditarily overlapping boundary property [10]).
Now one can always decomposeF∖I into n sets J1,…,Jn in such a way that μ(Jk)=m(F∖I)/n<δ, k=1,…,n.
Thus, asμ(Jk)=(1/n)(xo-δ,xo2-δ2), one finds
(59)(f∘μ)(Jk)=f(xo-δ,xo2-δ2n)=(xo-δ)3(xo+δ)n3×((xo-δ)4n4+(xo-δ)2(xo+δ)2n2)-1.
Consider then D={I,J1,…,Jn}∈Π(F); we have
(60)∑Fi∈D‍(f∘μ)(Fi)=(f∘μ)(I)+n(f∘μ)(J1)=12+(xo-δ)3(xo+δ)n2×((xo-δ)4n4+(xo-δ)2(xo+δ)2n2)-1=12+xo2-δ2(xo+δ)2+((xo-δ)2/n2).
Hence, letting n→∞ we have that ∑Fi∈D‍(f∘μ)(Fi) can be made strictly greater than 1.
On the other side, we can decomposeF into finitely many sets F1,…,Fn each having μ(Fj)=μ(F)/n and n large enough to have μ(Fj)<δ. On the decompositions of this type then
(61)∑D‍(f∘μ)(Fj)=n(f∘μ)(F1)=nf(xon,xo2n)=nxo4/n3xo4/n4+xo4/n2=11+1/n2⟶1-.
Therefore, the BC integral does not exist.
One may be interested also in the naturally arising spaceV∞ of games obtained as ∥·∥∞-limit of sequences of games in V.
Several questions arise in this space; the first open one isAre games inV∞still vector measure games?
And even in the negative, can one at least characterize those vector measure games in terms of properties of the functionf that defines them?
It is indeed rather difficult to characterize the∥·∥∞-closure of V, because we could not reach so far any satisfactory result relative to ∥·∥∞-convergence; to be more precise given an admissible function f on R+n and a sequence of vector measures (μk)k⊂(NA+)n, which convergence of (μk)k ensures that the sequence f∘μk→∥·∥∞f∘μ?
And, dually, one can also ask: given a sequence of admissible functionsfk:R+n→R and a vector measure μ, what kind of convergence of the sequence (fk)k ensures that the sequence of games fk∘μ→∥·∥∞f∘μ?
We can give so far only a sufficient condition; to this extent we first set the following structure on the already defined spaceA; set
(62)dlip(f,g)=inf⁡{L>0,suchthatLisaLipschitzconstantforjjjjjjjf-gintheincreasingdirections}.
Then dlip defines a metric on A. The analogous dlip is a semimetric on each L(μ).
Then we have the following.Proposition 13.
Letμ∈(NA+)n and let f be admissible. If f is a dlip cluster point for L(μ), then f∘μ∈V∞.Proof.
Fixε>0, and choose σ=ε/4; set δ=σ/(μ1(Ω)+⋯+μn(Ω)). Next, let g∈L′(μ) (the set of differentiable functions in L(μ)) with dlip(f,g)<δ.
This means thatf-g is Lipschitz in the μ-link directions with constant L<δ.
Fix nowF∈Σ, D∈Π(F), TD∈TD; for the game (f∘μ)-(g∘μ) one computes
(63)|S[(f∘μ)-(g∘μ),D,TD]|≤∑i‍|[(f-g)∘μ](Di∪Ti)-[(f-g)∘μ](Ti)|<δ∑i‍∥μ(Di∪Ti)-μ(Ti)∥=δ∑i‍∥μ(Di)∥=δ[μ1(F)+⋯+μn(F)]≤σ
from which we deduce ∥(f∘μ)-(g∘μ)∥∞≤4σ=ε. Since g∈L′(μ) the game g∘μ∈V thanks to Proposition  3 in [5], and the proof is complete.



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*Source: 849685-2014-10-01.xml*