On the Extension of Set-Valued Set Functions

2009

In this paper, we study different types of non-additive set multifunctions (such as: uniformly autocontinuous, null-additive, null-null-additive), presenting relationships among them and some of their properties regarding atoms and pseudo-atoms. We also study non-atomicity and non-pseudo-atomicity of regular null-additive set multifunctions defined on the Baire (Borel respectively) δ-ring of a Hausdorff locally compact space and taking values in the family of non-empty closed subsets of a real normed space.

2004

Introduction. Let Σ be an algebra of subsets of a given set Ω. Assume λ1, · · · , λn, μ are (finitely additive) real-valued measures over Σ. By “linearity theorems”we mean theorems which, under suitable conditions, ensure that μ is a linear combination of the measures λi. In [M-M, Theorem 20], the authors, among many other interesting results, proved such a linearity theorem for σ-additive measures on a σ-algebra. The linearity theorem is then applied [M-M, Theorem 21] to characterize those measure games for which the core is made of measures which can be written as μ = ∑ i αiλi. We recall that measure games, which play an important role in economic theory (see [A-S], [H-N]), are cooperative games ν of the special form ν = g(λ1, · · · , λn). Let us observe that [M-M, Theorem 20], in its turn, generalizes the uniqueness theorem of [M] to a multivariate setting. In [A-B] we proved that the uniqueness theorem above cited holds true more generally for measures defined on a very general ...

Journal of the Operations Research Society of Japan, 2004

The concepts of M-convex functions and Mn-convex functions play central roles in the theory of discrete convex analysis which has been applied to mathematical economics. On the oLher hand, sub-stit･utability, which is a key property guaranteeing the existence of a stable matching in generalized stable marriage models, is knazun as a nice property in mathematical economics. In this paper, we introduce new properties, which are extensions ofsubstitutability, and present new charactepizations of Mta-convex set functions by these properties.

Transactions of the American Mathematical Society, 1983

A family B \mathcal {B} of Borel subsets of a space X X is (boundedly) Borel additive if, for some countable ordinal α \alpha , the union of every subfamily of B \mathcal {B} is a Borel set of class α \alpha in X X . A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of B \mathcal {B} (of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when X X is a complete metric space or under additional set-theoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric sp...

2015

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that c0 (or any E^-space) is a Xspace, i.e. cannot be realized as the quotient of a nonlocally convex f-space by a one-dimensional subspace.

Set-Valued and Variational Analysis, 2020

The notion of set-valued means is introduced. Set-valued counterparts of the arithmetic, quasi-arithmetic and Lagrangian means are investigated and various properties of them are presented.

Mathematische Zeitschrift, 1979

We show that there is a probability space X and a bounded scalarly measurable function from X to #~ which has no Pettis integral (Theorem 2B). Our method relies on a new decomposition theorem for additive functionals defined on power sets (Theorem 1H). As another corollary we prove the existence of an indefinite Pettis integral with non-totally-bounded range (Example 2D).

Economic Theory Bulletin, 2013

We present a result on convexity and weak compactness of the range of a vector measure with values in a Banach space, based on the Maharam classification of measure spaces. Our result extends a recent result of Khan and Sagara (Illinois J. Math. 2013). We apply our result to integration of Banach space valued correspondences and to the core-Walras equivalence problem in coalitional exchange economies with an infinite-dimensional commodity space. Keywords Liapounoff's theorem • Vector measures • Correspondences • Blocking power of small coalitions • Core-Walras equivalence • Coalitional economies JEL Classification C02 • C60 • C71 • D51

Advances in Pure Mathematics, 2013

Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.

Computational Intelligence, Theory …, 2006