Research Article
Some Condition for Scalar and Vector
Measure Games to Be Lipschitz
F. Centrone
1and A. Martellotti
21Dipartimento di Studi per l’Economia e l’Impresa, Universit`a del Piemonte Orientale “A. Avogadro”, Via Perrone 18, 28100 Novara, Italy
2Dipartimento di Matematica e Informatica, Universit`a di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
Correspondence should be addressed to F. Centrone; francesca.centrone@eco.unipmn.it Received 12 April 2014; Accepted 10 September 2014; Published 1 October 2014
Academic Editor: Biren N. Mandal
Copyright © 2014 F. Centrone and A. Martellotti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide a characterization for vector measure games] = 𝑓 ∘ 𝜇 in 𝑝𝑁𝐴∞, with𝜇 vector of nonatomic probability measures,
analogous to the one of Tauman for games in𝑝𝑁𝐴, and also provide a necessary and sufficient condition for a particular class of
vector measure games to belong to𝐴𝐶∞.
1. Introduction
Measure games, that is, transferable utility (TU) games of the form] = 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 on𝑓 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 between scalar measure
games where𝜇 is a nonnegative scalar measure and vector measure games where𝜇 is an 𝑛-dimensional measure with
nonnegative components. The extension to signed measure is also customary.
The most classical space related to measure games is 𝑝𝑁𝐴, 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 in𝑝𝑁𝐴 [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 subspace𝐴𝐶∞ ⊂ 𝐵𝑉 of the so-called Lipschitz games. Then clearly one may define also the space𝑝𝑁𝐴∞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 function𝑓 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 form𝑓∘𝜇 do not belong to𝑝𝑁𝐴∞; on the other side the linear span of games with𝑓 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 that𝑓 ∘ 𝜇 ∈ 𝐴𝐶∞ if and only if𝑓 is Lipschitz. Here we introduce a generalization of the Lipschitz condition, namely, lipschitzianity in link
directions, which proves to characterize vector measure
Volume 2014, Article ID 849685, 10 pages http://dx.doi.org/10.1155/2014/849685
games𝑓 ∘ 𝜇 in 𝐴𝐶∞, when the range of𝜇 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 form𝑓 ∘ 𝜇 where 𝜇 is a signed measure.
Another interesting subspace is the class𝐵𝐶 of Burkill-Cesari (𝐵𝐶) integrable games, introduced in [6]. The investi-gation 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 not𝐵𝑉-continuous) and that it differs from the Aumann and Shapley value. Actually, the𝐵𝐶 integral and the space 𝐵𝐶 turned out to be fruitful to provide a proper subspace of 𝐴𝐶 strictly larger than𝑝𝑁𝐴 on which a value can be defined; remember that existence and uniqueness of a value on𝑝𝑁𝐴 are well known, while the question is still open on𝐴𝐶. 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 Section 3 we characterize vector measure games in𝑝𝑁𝐴∞; 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. InSection 4we first characterize a particular class of vector measure games in 𝐴𝐶∞ and, as a corollary, also measure games where𝜇 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 a𝐵𝐶 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]).
A transferable utility (TU) game] is a real valued function onΣ such that ](0) = 0.
The set of all nonatomic measures on(Ω, Σ) is denoted by𝑁𝐴, the cone of nonnegative measures of 𝑁𝐴 by 𝑁𝐴+, while the set of probability measures in𝑁𝐴 is indicated by 𝑁𝐴1. Given𝜇 ∈ 𝑁𝐴, the variation measure is denoted by |𝜇|.
For a vector measure𝜇 = (𝜇1, . . . , 𝜇𝑛) ∈ (𝑁𝐴)𝑛, the variation measure is defined by|𝜇| = ∑𝑛𝑖=1|𝜇𝑖|.
A game] is said to be Lipschitz if there exists 𝜇 ∈ 𝑁𝐴+ such that for every link𝑆 ⊂ 𝑇 ⊂ Ω in Σ it holds
|] (𝑇) − ] (𝑆)| ≤ 𝜇 (𝑇) − 𝜇 (𝑆) . (1) The space of Lipschitz games is denoted by𝐴𝐶∞for it is a Banach space under the norm‖]‖∞defined in the following way; for every𝜇 ∈ 𝑁𝐴+such that (1) holds, write−𝜇 ⪯ ] ⪯ 𝜇. Then set
‖]‖∞= inf {𝜇 (Ω) , 𝜇 ∈ 𝑁𝐴+, −𝜇 ⪯ ] ⪯ 𝜇} . (2)
To simplify notation, given a game ] ∈ 𝐴𝐶∞, and for fixed𝐻 ∈ Σ we will denote as usual by Π(𝐻) the class of finite partitions of𝐻 consisting of elements in Σ; for every 𝐷 ∈ Π(𝐻), say, 𝐷 = {𝐷1, . . . , 𝐷𝑘}, let T𝐷 = {𝑇𝐷 = {𝑇1, . . . , 𝑇𝑘} with 𝑇𝑖⊂ 𝐷𝑐 𝑖, 𝑇𝑖∈ Σ}. Then denote 𝑆 (], 𝐷, 𝑇𝐷) =∑𝑘 𝑖=1 [] (𝐷𝑖∪ 𝑇𝑖) − ] (𝑇𝑖)] . (3) For a game] ∈ 𝐴𝐶∞define, according to [7], the measures
]∗(𝐻) = sup {𝑆 (], 𝐷, 𝑇𝐷) , 𝐷 ∈ Π (𝐻) , 𝑇𝐷∈ T𝐷} ,
]∗(𝐻) = inf {𝑆 (], 𝐷, 𝑇𝐷) , 𝐷 ∈ Π (𝐻) , 𝑇𝐷∈ T𝐷} .
(4) Then
‖]‖∞= sup {]∗(𝐻) + ]∗(𝐻𝑐) , 𝐻 ∈ Σ} . (5) It is clear that if ] is a monotone game, then ]∗, ]∗ are nonnegative and in this case‖]‖∞= ]∗(Ω).
Note that for every𝐻 ∈ Σ, 𝐷 ∈ Π(𝐻), 𝑇𝐷∈ T𝐷one has 𝑆(],𝐷,𝑇𝐷) ≤ ‖]‖∞. (6)
On the other side, since]∗, ]∗are measures, there are Hahn decompositions(𝑃1, 𝑁1), (𝑃2, 𝑁2) which allow us to rewrite (5) as
‖]‖∞= sup {]∗(𝐸 ∩ 𝑃1) − ]∗(𝐸 ∩ 𝑁1) + ]∗(𝐸𝑐∩ 𝑃2)
−]∗(𝐸𝑐∩ 𝑁2) , 𝐸 ∈ Σ} . (7)
Hence, if for a game] one has |𝑆(], 𝐷, 𝑇𝐷)| ≤ 𝜀 for each 𝐷 ∈ Π(𝐻), 𝑇𝐷∈ T𝐷and every𝐻 ∈ Σ, then clearly ‖]‖∞≤ 4𝜀.
Given a vector measure𝜇 = (𝜇1, . . . , 𝜇𝑛) ∈ (𝑁𝐴+)𝑛 and denoting its range by𝑅(𝜇), a vector measure game is a game ] of the form] = 𝑓∘𝜇, where 𝑓 is a real valued function defined on the range of𝜇 and 𝑓(0) = 0. In the case 𝑛 = 1, this type of games is called scalar measure games.
Analogously to what is usually done in the space 𝐵𝑉, the symbol𝑝𝑁𝐴∞ denotes the‖ ⋅ ‖∞-closure of the space generated by all powers of nonatomic probability measures.
Given a convex subset 𝑋 of R𝑛, a vector𝑧 is called 𝑋
admissible if 𝑧 = 𝑥 − 𝑦 for some 𝑥, 𝑦 ∈ 𝑋. A real
valued function 𝑓 defined on 𝑋 is said to be continuously
differentiable on𝑋 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 of𝑋. The space of continuously differentiable functions on a set𝑋 will be denoted byC1(𝑋).
Given a convex compact subset𝐾 of R𝑛, a point𝑥 ∈ 𝐾 is said to be exposed if{𝑥} is the intersection of 𝐾 with some supporting hyperplane of𝐾.
As in [8], given a monotone nonatomic game 𝜆, one defines the mesh of a partition𝐷 as
𝛿𝜆(𝐷) = max {𝜆 (𝐼) , 𝐼 ∈ 𝐷} . (8) A game] is Burkill-Cesari (BC) integrable with respect to 𝛿𝜆if the following limit exists, for each𝐸 ∈ Σ:
𝐸 → ∫
𝐸] = lim𝛿𝜆(𝐷) → 0
𝐷∈Π(𝐸)
∑
We denote by𝐵𝐶 the space of games ] such that there exists 𝜆 ∈ 𝑁𝐴+so that] is 𝐵𝐶 integrable with respect to the mesh
𝛿𝜆.
The 𝐵𝐶 integral does not depend upon the integration mesh (see Proposition 5.2 in [6]); in other words, for every 𝜆 ∈ 𝑁𝐴+ such that] is 𝛿𝜆-𝐵𝐶 integrable, the 𝐵𝐶 integral remains the same. Moreover, the𝐵𝐶 integral of a game ] is a finitely additive measure.
3. Measure Games in
𝑝𝑁𝐴
∞In [5] we have obtained a characterization (Proposition 12) of scalar measure games (made through a nonnegative measure) for the whole space𝐴𝐶∞. 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 to𝑝𝑁𝐴∞. To our knowledge, this is also the unique characterization of vector measure games in𝑝𝑁𝐴∞so far. It is well known that𝑝𝑁𝐴∞is strictly contained in𝑝𝑁𝐴 and that many games of interest belong to this space [4].
Our first goal is to obtain a characterization for vector measure games in 𝑝𝑁𝐴∞, 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. Let 𝜇 ∈ (𝑁𝐴1)𝑛,𝑓 ∈ C1([𝑅(𝜇)]). Then for every
𝜀 > 0 there exists a polynomial in 𝑛 variables 𝑝 such that (𝑓 ∘ 𝜇) − (𝑝 ∘ 𝜇)∞< 𝜀. (10)
Proof. According to [1] for every 𝜀 > 0 there exists a polynomial𝑝 on 𝑅(𝜇) such that the norm ‖𝑓−𝑝‖𝐼< 𝜀, where one defines 𝑓 − 𝑝𝐼= 𝑓 − 𝑝𝑢+ 𝑛 ∑ 𝑖=1 𝜕𝑥𝜕𝑓𝑖− 𝜕𝑝 𝜕𝑥𝑖𝑢 (11) and‖ ⋅ ‖𝑢is 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:
𝑆[(𝑝 ∘ 𝜇)− (𝑓 ∘ 𝜇),𝐷,𝑇𝐷] ≤ ∑ 𝑖 [(𝑝 ∘ 𝜇)(𝐷𝑖 ∪ 𝑇𝑖) − (𝑓 ∘ 𝜇) (𝐷𝑖∪ 𝑇𝑖)] − [(𝑝 ∘ 𝜇)(𝑇𝑖) − (𝑓 ∘ 𝜇) (𝑇𝑖)] = ∑ 𝑖 ∇(𝑓 − 𝑝)[𝑐𝑖𝜇 (𝐷𝑖∪ 𝑇𝑖) + (1 − 𝑐𝑖) 𝜇 (𝑇𝑖) ] ⋅ 𝜇 (𝐷𝑖) ≤ ∑ 𝑖 𝑛 ∑ 𝑗=1 ( 𝜕𝑓 𝜕𝑥𝑗 − 𝜕𝑝 𝜕𝑥𝑗) [𝑐𝑖𝜇 (𝐷𝑖∪ 𝑇𝑖) + (1 − 𝑐𝑖) 𝜇 (𝑇𝑖)] × 𝜇𝑗(𝐷𝑖) . (12)
Now𝑐𝑖𝜇(𝐷𝑖∪ 𝑇𝑖) + (1 − 𝑐𝑖)𝜇(𝑇𝑖) ∈ 𝑅(𝜇) for each 𝑖 = 1, . . . , 𝑛 and hence ( 𝜕𝑓 𝜕𝑥𝑗 − 𝜕𝑝 𝜕𝑥𝑗) [𝑐𝑖𝜇 (𝐷𝑖∪ 𝑇𝑖) + (1 − 𝑐𝑖) 𝜇 (𝑇𝑖)] ≤ 𝜕𝑓 𝜕𝑥𝑗 − 𝜕𝑝 𝜕𝑥𝑗𝑢< 𝜀. (13) Thus 𝑆[(𝑝 ∘ 𝜇)− (𝑓 ∘ 𝜇),𝐷,𝑇𝐷] ≤ 𝜀∑ 𝑖 ∑ 𝑗 𝜇𝑗(𝐷𝑖) = 𝜀∑ 𝑗 𝜇𝑗(𝐻) = 𝜀 𝜇(𝐻). (14)
Tauman [2] gave the definition of‖ ⋅ ‖𝐵𝑉-continuity at𝜇 for a function𝑓 defined on the range of a vector measure 𝜇 ∈ (𝑁𝐴1)𝑛, which we recall below.
Given𝜇 ∈ (𝑁𝐴)𝑛, denote with𝐵(𝜇) the set of measures 𝜂 ∈ (𝑁𝐴)𝑛having the same range of𝜇 and, for 𝜀 > 0, define
𝐵 (𝜇, 𝜀) = {𝜂 ∈ 𝐵 (𝜇) such that 𝜂 − 𝜇(Ω) < 𝜀}. (15) Fixed 𝜇 ∈ (𝑁𝐴1)𝑛, a function 𝑓 on 𝑅(𝜇) (with 𝑓(0) = 0), is ‖ ⋅ ‖𝐵𝑉-continuous at 𝜇, if for every 𝜀 > 0 there exists𝛿 > 0 such that for each 𝜂 ∈ 𝐵(𝜇, 𝛿) there follows ‖(𝑓 ∘ 𝜇) − (𝑓 ∘ 𝜂)‖𝐵𝑉 < 𝜀. Tauman proved that a vector measure game𝑓 ∘ 𝜇, 𝜇 ∈ (𝑁𝐴1)𝑛, belongs to𝑝𝑁𝐴 if and only if𝑓 is ‖ ⋅ ‖𝐵𝑉-continuous at𝜇.
We translate the same definition using the‖ ⋅ ‖∞-norm; that is, we say that a function𝑓 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 in𝑝𝑁𝐴∞.
Theorem 2. 𝑓 is ‖ ⋅ ‖∞-continuous at𝜇 ∈ (𝑁𝐴1)𝑛if and only
if(𝑓 ∘ 𝜇) ∈ 𝑝𝑁𝐴∞.
Proof. We first prove the sufficient condition; namely, we
assume that𝑓 is ‖ ⋅ ‖∞-continuous at𝜇 ∈ (𝑁𝐴1)𝑛.
Note first that without loss of generality we can assume that𝑅(𝜇) has nonempty interior.
As in [2], fix𝑎 > 0 such that the cube
𝐶 = 𝜇 (Ω)2 + [−𝑎2,𝑎2]𝑛⊂ 𝑅𝑜(𝜇), (16) and for𝛿 ∈ ]0, 1[ define
𝑓𝛿(𝑥) = 1
𝑎𝑛∫𝐶𝑓 [(1 − 𝛿) 𝑥 + 𝛿𝑦] 𝑑𝑦 −
1
𝑎𝑛∫𝐶𝑓 (𝛿𝑦) 𝑑𝑦.
(17) 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. Let 𝜂 ∈ (𝑁𝐴1)𝑛 and letΩ
1 ⊂ Ω2 ⊂ Ω be a link
with bothΩ1, Ω\Ω2uncountable. Then, for every𝛿 ∈]0, 1[ and for every𝑦 ∈ 𝑅(𝜂) there exists 𝜂𝑦 ∈ 𝐵(𝜂, 2𝑛𝛿) with 𝜂𝑦(𝐻) =
(1 − 𝛿)𝜂(𝐻) + 𝛿𝑦 for each 𝐻 ∈ Σ with Ω1⊂ 𝐻 ⊂ Ω2.
Let𝛿 be momentarily fixed, and let {𝑇, 𝑆 ∪ 𝑇} be any link inΣ, with 𝑇, Ω \ (𝑆 ∪ 𝑇) uncountable. Consider the difference 𝛾 = ]𝛿− ] where ]𝛿= 𝑓𝛿∘ 𝜇, ] = 𝑓 ∘ 𝜇. We have 𝛾 (𝑆 ∪ 𝑇) − 𝛾 (𝑇) = 𝑓𝛿[𝜇 (𝑆 ∪ 𝑇)] − 𝑓 [𝜇 (𝑆 ∪ 𝑇)] − 𝑓𝛿[𝜇 (𝑇)] + 𝑓 [𝜇 (𝑇)] = 𝑎1𝑛 ∫ 𝐶𝑓 [(1 − 𝛿) 𝜇 (𝑆 ∪ 𝑇) + 𝛿𝑦] 𝑑𝑦 − 𝑓 [𝜇 (𝑆 ∪ 𝑇)] −𝑎1𝑛 ∫ 𝐶𝑓 [(1 − 𝛿) 𝜇 (𝑇) + 𝛿𝑦] 𝑑𝑦 + 𝑓 [𝜇 (𝑇)] = 1 𝑎𝑛 ∫𝐶{𝑓 [(1 − 𝛿) 𝜇 (𝑆 ∪ 𝑇) + 𝛿𝑦] − 𝑓 [𝜇 (𝑆 ∪ 𝑇)] − 𝑓 [(1 − 𝛿) 𝜇 (𝑇) + 𝛿𝑦] + 𝑓 [𝜇 (𝑇)]} 𝑑𝑦. (18) Now by Lemma 3 applied with𝑇 = 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
𝛾 (𝑆 ∪ 𝑇) − 𝛾 (𝑇) = 𝑎1𝑛 ∫
𝐶{[(𝑓 ∘ 𝜂𝑦) (𝑆 ∪ 𝑇) − (𝑓 ∘ 𝜇) (𝑆 ∪ 𝑇)]
− [(𝑓 ∘ 𝜂𝑦) (𝑇) − (𝑓 ∘ 𝜇) (𝑇)]} 𝑑𝑦. (19)
By the assumption corresponding to𝜀 > 0 there exists 𝛿𝑜> 0 such that for𝑚 ∈ 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 that𝑇𝑖, Ω \ (𝐷𝑖 ∪ 𝑇𝑖) are uncountable. Then we have 𝑆 (𝛾, 𝐷, 𝑇𝐷) = ∑ 𝑖 𝛾 (𝐷𝑖∪ 𝑇𝑖) − 𝛾 (𝑇𝑖) = 𝑎1𝑛∑ 𝑖 ∫ 𝐶{[(𝑓 ∘ 𝜂𝑦) (𝐷𝑖∪ 𝑇𝑖) − (𝑓 ∘ 𝜇) (𝐷𝑖∪ 𝑇𝑖)] − [(𝑓 ∘ 𝜂𝑦) (𝑇𝑖) − (𝑓 ∘ 𝜇) (𝑇𝑖)]} 𝑑𝑦 = 1 𝑎𝑛∫𝐶𝑆 {[(𝑓 ∘ 𝜂𝑦) − (𝑓 ∘ 𝜇)] , 𝐷, 𝑇𝐷} 𝑑𝑦. (20)
Since𝜂𝑦 ∈ 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 ∈ T𝐷that satisfies the cardinality conditions
at each link that proves that𝑓𝛿∘ 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 choice𝐻♯ ∈ Σ, 𝐷♯∈ Π(𝐻♯), 𝑇♯
𝐷∈ T𝐷so that each link(𝑇𝑖♯, 𝐷𝑖♯∪𝑇𝑖♯) fulfills
the cardinality requirements for each link, and𝑆[𝛾, 𝐷, 𝑇𝐷] = 𝑆[𝛾, 𝐷♯, 𝑇𝐷♯♯].
To this aim, let𝜀, 𝛿, 𝐻 ∈ Σ, 𝐷 ∈ Π(𝐻), 𝑇𝐷 ∈ T𝐷 be fixed, with𝐷 = {𝐷1, . . . , 𝐷𝑘}. Without loss of generality we can assume that each𝐷𝑖is not𝜇-null.
Let 𝐼 ⊂ {1, . . . , 𝑘} be the set of indexes for which the corresponding link(𝑇𝑖, 𝐷𝑖∪𝑇𝑖) does not fulfill the cardinality requirements; then𝐼 can be split into two disjoint subsets 𝐼1 and𝐼2, where𝐼1is the set of indexes for whichΩ \ (𝐷𝑖∪ 𝑇𝑖) is at most countable.
For each𝑖 ∈ 𝐼2one can choose an uncountable𝜇-null set 𝑁𝑖inΩ \ (𝐷𝑖∪ 𝑇𝑖) and then replace 𝑇𝑖with ̃𝑇𝑖= 𝑇𝑖∪ 𝑁𝑖; since 𝑖 ∈ 𝐼2, we have𝑇𝑖countable, and, therefore,𝜇(𝑇𝑖) = 0; thus the replacement does not affect the corresponding summand in𝑆(𝛾, 𝐷, 𝑇𝐷).
For each index𝑖 ∈ 𝐼1choose an uncountable𝜇-null set 𝑁𝑖⊂ 𝐷𝑖and replace𝐷𝑖with ̃𝐷𝑖= 𝐷𝑖\𝑁𝑖, 𝑇𝑖with ̃𝑇𝑖= 𝑇𝑖∪𝑁𝑖. Then we can replace𝐷 with 𝐷♯= {𝐷♯𝑖, 𝑖 = 1, . . . , 𝑛} setting
𝐷𝑖♯= {𝐷̃𝑖 if𝑖 ∈ 𝐼1,
𝐷𝑖 otherwise (21)
and analogously𝑇𝐷with𝑇𝐷♯ = {𝑇𝑖♯, 𝑖 = 1, . . . , 𝑛} where 𝑇𝑖♯= {𝑇̃𝑖 if𝑖 ∈ 𝐼,
𝑇𝑖 otherwise (22)
and finally set 𝐻♯ = ⋃𝑘𝑖=1𝐷♯𝑖. By construction then 𝑆(𝛾, 𝐷, 𝑇𝐷) = 𝑆(𝛾, 𝐷♯, 𝑇♯
𝐷♯).
To prove the converse implication, according to [1, Lemma 7.2 page 41], it is enough to prove that monomials of the form𝑐𝜇𝛼, with𝑐 ∈ R, 𝜇 ∈ 𝑁𝐴1,𝛼 ∈ N+, are‖ ⋅ ‖∞ -continuous at each𝜇 ∈ 𝑁𝐴1.
We first prove that if𝜇 ∈ 𝐵(𝑚, 𝛿), 𝐻 ∈ Σ, 𝐷 = {𝐷𝑖}𝑖=1,...,𝑘∈ Π(𝐻), 𝑝 ∈ N, then 𝑘 ∑ 𝑖=1𝑚 𝑝(𝐷 𝑖) − 𝜇𝑝(𝐷𝑖) < 𝑝𝛿. (23)
In fact, for each summand we have 𝑚𝑝(𝐷
𝑖) − 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,
(24) where the last inequality follows from Lipschitz condition on [0, 1] and the fact that 𝑚 ∈ 𝑁𝐴1. Then in𝐵(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 in𝑆(𝑚𝛼− 𝜇𝛼, 𝐷, 𝑇𝐷)
𝑚𝛼(𝑇𝑖∪ 𝐷𝑖) − 𝜇𝛼(𝑇𝑖∪ 𝐷𝑖) − 𝑚𝛼(𝑇𝑖) + 𝜇𝛼(𝑇𝑖) = 𝑚𝛼(𝐷𝑖) − 𝜇𝛼(𝐷𝑖) +𝛼−1∑ 𝑗=1(𝛼𝑗)[𝑚 𝛼−𝑗(𝐷 𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) 𝜇𝑗(𝑇𝑖) ] . (25)
In the sum𝑆(𝑚𝛼− 𝜇𝛼, 𝐷, 𝑇𝐷) there will then appear a sum of the form in (23) which we can estimate with𝛼𝛿.
Let us now treat the summands of the form [𝑚𝛼−𝑗(𝐷 𝑖)𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖)𝜇𝑗(𝑇𝑖)]. Clearly 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)] (26)
and so in the whole|𝑆(𝑚𝛼 − 𝜇𝛼, 𝐷, 𝑇𝐷)| we can apply the following bound from above:
∑ 𝑖 𝑚 𝛼−𝑗(𝐷 𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) 𝜇𝑗(𝑇𝑖) ≤ ∑ 𝑖 𝑚𝛼−𝑗(𝐷𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝑗(𝑇𝑖) +𝜇𝑗(𝑇 𝑖) 𝑚𝛼−𝑗(𝐷𝑖) − 𝜇𝛼−𝑗(𝐷𝑖). (27)
Now each term|𝑚𝑗(𝑇𝑖) − 𝜇𝑗(𝑇𝑖)| ≤ 𝑗𝛿 again by Lipschitz condition and the choice of𝑚 ∈ 𝐵(𝜇, 𝛿). Also 𝑚𝛼−𝑗(𝐷𝑖) ≤ 𝑚(𝐷𝑖) since 𝑚 ∈ 𝑁𝐴1and 𝜇(𝑇
𝑖) ≤ 1 for the same reason.
Hence, because of (23), ∑ 𝑖 𝑚 𝛼−𝑗(𝐷 𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) 𝜇𝑗(𝑇𝑖) ≤ 𝑗𝛿∑ 𝑖 𝑚 (𝐷𝑖) + ∑ 𝑖 𝑚 𝛼−𝑗(𝐷 𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) < 𝑗𝛿 + (𝛼 − 𝑗) 𝛿 = 𝛼𝛿. (28) Finally ∑ 𝑖 𝛼−1 ∑ 𝑗=1(𝛼𝑗)𝑚 𝛼−𝑗(𝐷 𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) 𝜇𝑗(𝑇𝑖) =𝛼−1∑ 𝑗=1(𝛼𝑗)∑𝑖 𝑚 𝛼−𝑗(𝐷 𝑖) 𝑚𝑗(𝑇𝑖) − 𝜇𝛼−𝑗(𝐷𝑖) 𝜇𝑗(𝑇𝑖) ≤ (2𝛼− 1) 𝛼𝛿 (29)
from which we conclude according to (25) and (23) 𝑆(𝑚𝛼− 𝜇𝛼, 𝐷, 𝑇
𝐷) ≤ 𝛼2𝛼𝛿 (30)
which only depends upon𝛼. 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 each𝜀 > 0 there exists 𝛿 > 0 such that ‖(𝑓 ∘ 𝜇) − (𝑓𝛿∘ 𝜇)‖∞< 𝜀.
On the other side, as 𝑓𝛿 ∈ C1[𝑅(𝜇)], we can apply Lemma 1 and find a polynomial 𝑝𝜀 such that ‖(𝑓𝛿∘ 𝜇) − (𝑝
𝜀∘ 𝜇)‖∞< 𝜀.
In other words we have proven the following statement.
Theorem 4. Let 𝜇 ∈ (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
(𝑓 ∘ 𝜇) − (𝑝 ∘ 𝜇)∞< 𝜀. (31)
4. Measure Games in
𝐴𝐶
∞Throughout this section we will deal with maps𝑓 : R𝑛+ → R with𝑓(0) = 0, namely, admissible maps.
For two vectorsx, y ∈ R𝑛+ 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 be Lipschitz in increasing directions when𝐿 > 0 exists such that for each pair x ≪ y in
R𝑛+one has
𝑓(x) − 𝑓(y) ≤ 𝐿 ⋅ x − y. (32)
Definition 6. Let𝜇 : Σ → R𝑛+be a measure; a pairx, y ∈ 𝑅(𝜇) is said to be a𝜇-link direction when there exists a link (𝑆, 𝑆∪𝑇) inΣ such that 𝜇(𝑆) = x, 𝜇(𝑆 ∪ 𝑇) = y.
A map𝑓 : R𝑛+ → R is then said to be Lipschitz in
𝜇-link directions if there exists𝐿 > 0 such that for every 𝜇-link
directionx, y there holds
𝑓(x) − 𝑓(y) ≤ 𝐿 ⋅ x − y. (33) Note that if 𝑓 is Lipschitz in increasing directions, then it is Lipschitz in 𝜇-link directions for each 𝑛-dimensional nonnegative measure𝜇.
Let now𝑛 ∈ N+and𝜇 ∈ (𝑁𝐴+)𝑛be fixed, and consider the spaces
(i)A = {𝑓 : R𝑛+ → R admissible and Lipschitz in the increasing directions},
(ii)L(𝜇) = {𝑓 : R𝑛+ → R admissible and Lipschitz in 𝜇-link directions}.
Note that if𝑓 ∈ A, then 𝑓 ∘ 𝜇 ∈ 𝐴𝐶∞for every𝜇 ∈ (𝑁𝐴+)𝑛, while we can assert the same for𝑓 ∈ L(𝜇) only for precisely 𝜇.
Now, given𝑓, 𝜇, let ] = 𝑓 ∘ 𝜇 and define the subsets of 𝑁𝐴+. Consider
𝐴 (]) = {𝜆 ∈ 𝑁𝐴+𝜆 ≪ 𝜇,−𝜆 ⪯ ] ⪯ 𝜆},
Note that if] is a vector measure game in 𝐴𝐶∞, then𝐴(]) ̸= 0. Indeed, let𝜆 ∈ 𝑁𝐴+be such that−𝜆 ⪯ ] ⪯ 𝜆 and consider the Jordan decomposition𝜆 = 𝜆1+ 𝜆2with𝜆1 ≪ |𝜇|, 𝜆2 ⊥ |𝜇|. Then there are two disjoint setsΩ1, Ω2which support𝜆1, 𝜆2, respectively. This means that𝜇 vanishes on Ω2; hence,𝜆1 ∈ 𝐴(]).
In fact, for each link𝑆 ⊂ 𝑇, setting 𝑆1 = 𝑆 ∩ Ω1,𝑇1 = 𝑇 ∩ Ω1since𝜇(𝑆1) = 𝜇(𝑆), 𝜇(𝑇1) = 𝜇(𝑇) one has
|] (𝑇) − ] (𝑆)| = ] (𝑇1) − ] (𝑆1) ≤ 𝜆 (𝑇1) − 𝜆 (𝑆1)
= 𝜆1(𝑇1) − 𝜆1(𝑆1) = 𝜆1(𝑇) − 𝜆1(𝑆) . (35) We can now state the following.
Proposition 7. Let 𝜇 ∈ (𝑁𝐴+)𝑛and] = 𝑓 ∘ 𝜇. The following
are equivalent:
(i)𝑓 ∈ L(𝜇);
(ii) there exists𝑐 > 0 such that 𝑐|𝜇| ∈ 𝐴(]); (iii)𝐴(]) ∩ 𝐶(𝜇) ̸= 0.
Proof. (i) implies (ii), since, if𝑓 ∈ L(𝜇), then immediately
𝐿|𝜇| ∈ 𝐴(]) ∩ 𝐶(𝜇).
The implication (ii)⇒ (iii) is immediate.
Finally, let 𝜆 ∈ 𝐴(]) ∩ 𝐶(𝜇). This immediately implies that there exists𝐿 > 0 such that 𝑑𝜆/𝑑|𝜇| ≤ 𝐿|𝜇|-a.e., and this in turn implies that𝜆 ≤ 𝐿|𝜇| on Σ. On the other side, as mentioned inSection 2, we have that|](𝑆 ∪ 𝑇) − ](𝑇)| ≤ 𝜆(𝑆) for every pair of disjoint sets𝑆, 𝑇 ∈ Σ. Then immediately, on each𝜇-link direction x ≪ y and every link 𝑆, 𝑆 ∪ 𝑇 such that 𝜇(𝑆) = x, 𝜇(𝑆 ∪ 𝑇) = y one finds
𝑓(x) − 𝑓(y) ≤ 𝜆(𝑆) ≤ 𝐿𝜇(𝑆) = 𝐿 ⋅ x − y (36) that says that𝑓 ∈ L(𝜇).
Each of the conditions ofProposition 7implies that] ∈ 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 for𝑛 = 1.
In the more general case, we have the following result.
Theorem 8. Let 𝜇 ∈ (𝑁𝐴+)𝑛 be such that its range𝑅(𝜇) has
only finitely many exposed points. Then a vector measure game
] = 𝑓 ∘ 𝜇 ∈ 𝐴𝐶∞if and only if𝑓 ∈ L(𝜇).
Proof. We only need to prove that if] ∈ 𝐴𝐶∞, then𝑓 ∈ L(𝜇).
Let 𝑃1, . . . , 𝑃𝑘 be the exposed points of 𝑅(𝜇), and let 𝐴1, . . . , 𝐴𝑘be sets inΣ such that 𝜇(𝐴𝑖) = 𝑃𝑖,𝑖 = 1, . . . , 𝑘.
Consider the partition generated by 𝐴1, . . . , 𝐴𝑘, say, 𝐷 = {Ω1, . . . , Ω𝑛}, and assume that each Ω𝑖has nonnull 𝜇-measure.
If Σ𝑜denotes 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 each𝑖, x𝑖,y𝑖∈ {𝑡𝜇(Ω𝑖), 𝑡 ∈ [0, 1]}.
Moreover, easilyx𝑖≪ y𝑖.
Set nowx = v𝑜, v𝑖 = ∑𝑖𝑗=1y𝑗+ ∑𝑛𝑗=𝑖+1x𝑗, 𝑖 = 1, . . . , 𝑛 − 1, v𝑛= y.
Fix𝜆 ∈ 𝐴(]) and 𝑖 = 1, . . . , 𝑛.
We know that there are𝑡𝑖 ≤ 𝑡𝑖 ∈ [0, 1] such that x𝑖 = 𝑡𝑖𝜇(Ω𝑖), y𝑖= 𝑡𝑖𝜇(Ω𝑖). Choose then 𝑆𝑖 ⊆ 𝑆𝑖 ⊂ Ω𝑖such 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 𝜇 (𝑇𝑖) =𝑖−1∑ 𝑗=1 y𝑗+ ∑𝑛 𝑗=𝑖+1 x𝑗. (37)
This is possible forx𝑗, y𝑗 ∈ 𝑅(𝜇|Ω𝑗) and for 𝑖 ̸= 𝑗, Ω𝑖∩Ω𝑗= 0.
Then𝑆𝑖∪ 𝑇𝑖,𝑆𝑖 ∪ 𝑇𝑖is a link and hence ] (𝑆𝑖∪ 𝑇𝑖) − ] (𝑆𝑖 ∪ 𝑇𝑖) ≤ 𝜆(𝑆𝑖\ 𝑆𝑖) = (𝑡𝑖 − 𝑡𝑖) 𝜆 (Ω𝑖) . (38) But𝜇(𝑆𝑖 ∪ 𝑇𝑖) = 𝜇(𝑆𝑖) + 𝜇(𝑇𝑖) = v𝑖−1,𝜇(𝑆𝑖 ∪ 𝑇𝑖) = v𝑖. Thus (38) becomes 𝑓(k𝑖) − 𝑓 (k𝑖−1) ≤ (𝑡𝑖 − 𝑡𝑖) 𝜆 (Ω𝑖) . (39) Now k𝑖− k𝑖−1 = y𝑖− x𝑖 = (𝑡𝑖 − 𝑡𝑖) 𝜇 (Ω𝑖) = (𝑡𝑖 − 𝑡𝑖) 𝜇(Ω𝑖) . (40) Thus 𝑓(k𝑖) − 𝑓 (k𝑖−1) ≤𝜇(Ω𝜆 (Ω𝑖) 𝑖)k𝑖− k𝑖−1. (41)
Sincev𝑖− v𝑖−1= y𝑖− x𝑖, we have that‖v𝑖− v𝑖−1‖ ≤ ‖y − x‖. In conclusion we have that
𝑓(x) − 𝑓(y) ≤ 𝐿y − x (42) with𝐿 = 𝑛 ⋅ ∑𝑛𝑖=1(𝜆(Ω𝑖)/|𝜇|(Ω𝑖)).
Observe that the above result includes two interesting cases: the case of 𝜇𝑖 ⊥ 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, fromTheorem 8one derives the following corollary that generalizes Proposition 12 in [5] to the case of signed scalar measure games, namely, games of the form] = 𝑓 ∘ 𝜇 with𝜇 signed measure.
Corollary 9. Let 𝜇 : Σ → R be a nonatomic signed measure,
and let𝑓 : [−𝜇−(Ω), 𝜇+(Ω)] → R. Then 𝑓 ∘ 𝜇 ∈ 𝐴𝐶∞if and only if𝑓 is Lipschitz on 𝑅(𝜇).
Proof. Consider the nonatomic two-dimensional measure
𝑚 = (𝜇+, 𝜇−) and the function 𝑔 : R2 → R defined by
𝑔(𝑥, 𝑦) = 𝑓(𝑥 − 𝑦). Thus ] = 𝑓 ∘ 𝜇 = 𝑔 ∘ 𝑚. Since 𝑅(𝑚) is a rectangle, if] ∈ 𝐴𝐶∞, fromTheorem 8𝑔 ∈ L(𝑚), namely, a
constant𝐿 > 0 exists, such that for each 𝑚-link direction x, y 𝑔(x) − 𝑔(y) ≤ 𝐿y − x. (43) Then𝑓 is also Lipschitz with constant 𝐿.
So far, we have not been able to answer the question whether the condition inTheorem 8 actually characterizes vector measure games in𝐴𝐶∞.
We now turn our attention to a smaller subspace of𝐴𝐶∞. In [5, 6] we have introduced and studied the space𝐵𝐶 of Burkill-Cesari (𝐵𝐶) integrable games. In particular in [5] we considered the space𝑄 = 𝐵𝐶 ∩ 𝐴𝐶∞of Lipschitz games that are indeed𝐵𝐶 integrable.
Here we will consider the subspaceV of vector measure games in𝑄.
First of all, observe that𝐴𝐶∞ \ 𝐵𝐶 and 𝐵𝐶 \ 𝐴𝐶∞are nonempty. Indeed a scalar measure game𝑓 ∘ 𝜇 ∈ 𝐴𝐶∞ if and only if𝑓 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 ] = |𝜇|; then easily ] ∈ 𝐴𝐶∞.
In fact, if𝑆, 𝑇 is a link, then
|] (𝑇) − ] (𝑆)| − 𝜇(𝑇)|−|𝜇(𝑆)
≤ 𝜇 (𝑇) − 𝜇 (𝑆) ≤ 𝜇(𝑇 \ 𝑆) (44) and so|𝜇| ∈ 𝐴(]). On the other side, ] ∉ 𝐵𝐶; indeed, the following extension of Proposition 14 in [5] can be derived fromCorollary 9.
Proposition 10. Take 𝜇 ∈ 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, since] ∈ 𝐴𝐶∞ we know fromCorollary 9
that 𝑓 is Lipschitz; hence the ratios 𝑓(𝑥)/𝑥, 𝑥 ̸= 0 are bounded. Assume that] ∈ 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 ofΩ.
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 𝑓 (𝑥𝑛) 𝑥𝑛 − ℓ1 < 𝜀 3𝜇+(𝐹), 𝑓 (−𝑥𝑛) −𝑥𝑛 − ℓ2 < 𝜀 3𝜇−(𝐹). (45)
By means of the continuity of𝑓 at 0, choose next ̃𝑛 > 𝑛 such that|𝑓(𝑥)| < 𝜀/3 whenever 𝑥 ≤ 𝑥̃𝑛; alsõ𝑛 can be chosen so that|ℓ𝑖||𝑥̃𝑛| < 𝜀/3 and such that 𝑥̃𝑛(𝜆(𝐹)/𝜇+(𝐹)) < 𝛿(𝜀/3) where𝛿 is the parameter of 𝛿𝜆𝐵𝐶 integrability.
Choose now the following 𝐷 ∈ Π(𝐹 ∩ 𝑃); by means of Lyapunov Theorem, divide𝐹 into finitely many sets, say 𝐼1, . . . , 𝐼𝑘, each with(𝜇, 𝜆)(𝐼𝑗) = (𝑥̃𝑛, (𝜆(𝐹)/𝜇+(𝐹))𝑥̃𝑛), until 𝜇((𝐹 ∩ 𝑃) \ ⋃𝑘𝑗=1𝐼𝑗) ≤ 𝑥̃𝑛and then choose𝐼𝑘+1 = (𝐹 ∩ 𝑃) \
⋃𝑘𝑗=1𝐼𝑗; thus easily𝜆(𝐼𝑘+1) = (𝜆(𝐹)/𝜇+(𝐹))𝜇(𝐼𝑘+1) for 𝜆 (𝐼𝑘+1) = 𝜆 (𝐹) −∑𝑘 𝑗=1 𝜆 (𝐼𝑗) = 𝜆 (𝐹) − 𝑘𝜇𝜆 (𝐹)+(𝐹)𝑥̃𝑛 = 𝜆 (𝐹) 𝜇+(𝐹)[𝜇+(𝐹) − 𝑘𝑥̃𝑛] = 𝜇𝜆 (𝐹)+(𝐹)𝜇 (𝐼𝑘+1) . (46)
Then for𝐷 = {𝐼1, . . . , 𝐼𝑘, 𝐼𝑘+1} one has 𝛿𝜆(𝐷) < 𝛿(𝜀/3). We have then ∑ 𝐼∈𝐷𝑓[𝜇(𝐼)] − ℓ1𝜇 (𝐼) =∑𝑘 𝑛=1𝑓(𝑥̃𝑛 ) − ℓ1𝑥̃𝑛 + 𝑓[𝜇(𝐼𝑘+1) ] − ℓ1𝜇 (𝐼𝑘+1) ≤∑𝑘 𝑛=1𝑓(𝑥̃𝑛 ) − ℓ1𝑥̃𝑛 + 𝑓[𝜇(𝐼𝑘+1)] +ℓ1𝑥̃𝑘 =∑𝑘 𝑛=1 𝑓 (𝑥̃𝑛) − ℓ1𝑥̃𝑛 𝑥̃𝑛 𝑥̃𝑛+ 𝜀 3+ 𝜀 3. (47)
As for the first sum we have the following estimate:
𝑘 ∑ 𝑛=1 𝑓 (𝑥̃𝑛) − ℓ1𝑥̃𝑛 𝑥̃𝑛 𝑥̃𝑛< 𝜀 3𝜇+(𝐹) 𝑘 ∑ 𝑛=1 𝑥̃𝑛 = 𝜀 3⋅ 𝜇 ((𝐹 ∩ 𝑃) \ 𝐼𝑘+1) 𝜇+(𝐹) < 𝜀 3. (48) In conclusion ∑ 𝐼∈𝐷𝑓[𝜇(𝐼)] − ℓ1𝜇 (𝐼) < 𝜀. (49)
Clearly we can repeat this construction with −𝑥̃𝑛 to find a partition𝐷∗ ∈ Π(𝐹 ∩ 𝑁) with 𝛿𝜆(𝐷∗) < 𝛿(𝜀/3) as above; again
∑
Hence𝐷 ∪ 𝐷∗∈ Π(𝐹) and 𝐼∈𝐷∪𝐷∑ ∗] (𝐼) − [ℓ1𝜇 (𝐹 ∩ 𝑃) + ℓ2𝜇 (𝐹 ∩ 𝑁)] = 𝐼∈𝐷∪𝐷∑ ∗] (𝐼) − 𝜇 +(𝐹) (ℓ 1+ ℓ2) ≤ 2𝜀. (51)
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 symmetrically𝐽(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 whole𝐹 with 𝛿𝜆(𝐷𝑜) < 𝜀 but ∑𝐷𝑜](𝐼) = 0. Since ℓ1 ̸= ℓ2, the game] 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 of𝑓(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 with𝐵𝐶 integrability we are taking into account the𝜇-admissible directions.
For example, consider on R2+ the classical example of nondifferentiable map 𝑓 (𝑥, 𝑦) = { { { { { { { { { 𝑥2𝑦 𝑥4+ 𝑦2 whenever(𝑥, 𝑦) ̸= O, 0 if(𝑥, 𝑦)= O (52)
which is not differentiable on the whole orthant, being not continuous atO. However, if 𝜇 ∈ 𝑁𝐴+and𝜂 = (𝜇, 𝜇), then 𝑅(𝜂) reduces to a line segment and 𝑓 ∘ 𝜂 = 𝜂/(1 + 𝜂2) ∈ 𝑄
since it can be represented as𝑔∘𝜇 with 𝑔(𝑡) = 𝑡/(1+𝑡2) which is differentiable.
Indeed the following necessary condition derives from the𝐵𝐶 integrability.
Proposition 11. Let 𝜇 = (𝜇1, . . . , 𝜇𝑛) be in (𝑁𝐴+)𝑛, and let𝑓 :
R𝑛
+ → R be in L(𝜇). If 𝑓 ∘ 𝜇 ∈ V, then 𝑓 admits directional
derivative𝑑𝑓u(O) along every admissible direction u (i.e., such that𝜇(𝐹) = 𝑡u for some 𝐹 ∈ Σ, 𝜇(𝐹) ̸= 0 and some 𝑡 > 0); moreover, the convergence
lim
ℎ → 0
𝑓 (ℎu)
ℎ = 𝑑u𝑓 (O) (53)
is uniform with respect tou.
The proof of the existence of each directional derivative is substantially the same as that ofProposition 10, while the uniformity of the limit is deduced from the assumption of𝐵𝐶 integrability, where the defining limit
∫
𝐸] = lim𝛿𝜆(𝐷) → 0
𝐷∈Π(𝐸)
∑
𝐼∈𝐷] (𝐼) (54)
is uniform with respect to𝐸 ∈ Σ.
Observe that the requirement𝑓 ∈ L(𝜇) in the previous statement could be weakened by requiring that for each radial directionu 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
differ-entiable atO, since it is not continuous, despite the fact that it admits directional derivative at every directionu on the positive orthant, given by
𝑑u𝑓 (O) = { { { { { { { cos2𝜗 sin𝜗 if sin𝜗 ̸= 0, 0 if sin𝜗 = 0, (55)
whereu = (cos 𝜗, sin 𝜗).
A simple computation in fact provides, for sin𝜗 ̸= 0,
𝑓 (𝑡 cos 𝜗, 𝑡 sin 𝜗)𝑡 − 𝑑u𝑓 (O) =
𝑡4cos6𝜗
|sin 𝜗| (𝑡4cos4𝜗 + sin2𝜗).
(56) For the sake of simplicity set𝛼 = sin 𝜗, 𝛽 = cos2𝜗. Then we want to show that the ratios𝛽3𝑡4/|𝛼|(𝑡4𝛽2 + 𝛼2) can be made arbitrarily small, somehow independently on the values of𝛼, 𝛽. Easily 𝛽3𝑡4 |𝛼| (𝑡4𝛽2+ 𝛼2) ≤ 𝛽3𝑡4 |𝛼|3 (57)
which in turn is smaller than 𝜀 for 𝑡4 < 𝜀 ⋅ (|𝛼|3/𝛽3) = 𝜀(| sin 𝜗|3/cos6𝜗); one immediately checks that the map 𝜗 → | sin 𝜗|3/cos6𝜗 is increasing for 0 < 𝜗 < 𝜋/2.
Consider now𝛾 : [0, 1] → R defined as 𝛾 (𝑥) = {𝑥2 for 0 ≤ 𝑥 ≤ 𝑥𝑜,
2𝑥𝑜𝑥 − 𝑥2
𝑜 for 𝑥 > 𝑥𝑜 (58)
with0 < 𝑥𝑜< 1.
Then consider the “reverse” function𝐺 : [0, 1] → R defined as𝐺(𝑥) = 1 − 𝛾(1 − 𝑥); then the subset of R2+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 such𝜇 have slopes not exceeding 2𝑥𝑜; this is enough to achieve the required uniformity for the
vector measure game𝑓 ∘ 𝜇.
We will now prove that the weakened assumption is satisfied.
Fix an admissible directionu = (𝑢1, 𝑢2), that is, such that
spanu 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, 𝑥2𝑜); whichever 𝛿 > 0, 𝛿 <
𝑥𝑜/3 we choose, we can always find a subset 𝐼 of 𝐹 such that𝜇(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 that𝜇(𝐽𝑘) = 𝑚(𝐹 \ 𝐼)/𝑛 < 𝛿, 𝑘 = 1, . . . , 𝑛.
Thus, as𝜇(𝐽𝑘) = (1/𝑛)(𝑥𝑜− 𝛿, 𝑥2𝑜− 𝛿2), one finds (𝑓 ∘ 𝜇) (𝐽𝑘) = 𝑓 (𝑥𝑜− 𝛿, 𝑥2𝑜− 𝛿2 𝑛 ) = (𝑥𝑜− 𝛿) 3(𝑥 𝑜+ 𝛿) 𝑛3 × ((𝑥𝑜− 𝛿) 4 𝑛4 + (𝑥𝑜− 𝛿)2(𝑥𝑜+ 𝛿)2 𝑛2 ) −1 . (59) Consider then𝐷 = {𝐼, 𝐽1, . . . , 𝐽𝑛} ∈ Π(𝐹); we have
∑ 𝐹𝑖∈𝐷 (𝑓 ∘ 𝜇) (𝐹𝑖) = (𝑓 ∘ 𝜇) (𝐼) + 𝑛 (𝑓 ∘ 𝜇)(𝐽1) = 12+(𝑥𝑜− 𝛿) 3(𝑥 𝑜+ 𝛿) 𝑛2 × ((𝑥𝑜− 𝛿) 4 𝑛4 + (𝑥𝑜− 𝛿)2(𝑥𝑜+ 𝛿)2 𝑛2 ) −1 = 12+ 𝑥2𝑜− 𝛿2 (𝑥𝑜+ 𝛿)2+ ((𝑥𝑜− 𝛿)2/𝑛2) . (60) Hence, letting𝑛 → ∞ 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 sets𝐹1, . . . , 𝐹𝑛each having𝜇(𝐹𝑗) = 𝜇(𝐹)/𝑛 and 𝑛 large enough to have𝜇(𝐹𝑗) < 𝛿. On the decompositions of this type then
∑ 𝐷 (𝑓 ∘ 𝜇) (𝐹𝑗) = 𝑛 (𝑓 ∘ 𝜇) (𝐹1) = 𝑛𝑓 (𝑥𝑛𝑜,𝑥𝑛𝑜2) = 𝑛𝑥4 𝑥4𝑜/𝑛3 𝑜/𝑛4+ 𝑥4𝑜/𝑛2 = 1 + 1/𝑛1 2 → 1−. (61)
Therefore, the𝐵𝐶 integral does not exist.
One may be interested also in the naturally arising space V∞of games obtained as‖ ⋅ ‖∞-limit of sequences of games inV.
Several questions arise in this space; the first open one is
Are 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 ofV, because we could not reach so far any satisfac-tory result relative to‖ ⋅ ‖∞-convergence; to be more precise
given an admissible function 𝑓 on R𝑛+ and a sequence of vector measures(𝜇𝑘)𝑘⊂ (𝑁𝐴+)𝑛, which convergence of(𝜇𝑘)𝑘 ensures that the sequence𝑓 ∘ 𝜇𝑘→ 𝑓 ∘ 𝜇?‖⋅‖∞
And, dually, one can also ask: given a sequence of admissible functions𝑓𝑘 : R𝑛+ → R and a vector measure 𝜇, what kind of convergence of the sequence(𝑓𝑘)𝑘ensures that the sequence of games𝑓𝑘∘ 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
𝑑lip(𝑓, 𝑔)
= inf {𝐿 > 0, such that 𝐿 is a Lipschitz constant for 𝑓 − 𝑔 in the increasing directions} .
(62)
Then 𝑑lip defines a metric on A. The analogous 𝑑lip is a
semimetric on eachL(𝜇). Then we have the following.
Proposition 13. Let 𝜇 ∈ (𝑁𝐴+)𝑛and let𝑓 be admissible. If 𝑓
is a𝑑lip cluster point forL(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
inL(𝜇)) with 𝑑lip(𝑓, 𝑔) < 𝛿.
This means that𝑓 − 𝑔 is Lipschitz in the 𝜇-link directions with constant𝐿 < 𝛿.
Fix now𝐹 ∈ Σ, 𝐷 ∈ Π(𝐹), 𝑇𝐷 ∈ T𝐷; for the game(𝑓 ∘ 𝜇) − (𝑔 ∘ 𝜇) one computes 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 (63)
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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