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Countably additive restrictions of vector-valued quasi measures with respect to range preservation
No full textThis paper furtherly develops the question of the existence of countably additive restrictions. The problem has already been solved for scalar charges; here we face the two dimensional case.
Let m:P(X) --> R2 be a continuous quasimeasure with nonnegative components, i.e. a pair m=(m1,m2) of nonnegative strongly non atomic finitely additive set functions on the power set of X.
We prove that if the range of the pair is closed (which is not always true in the finitely additive setting) then there exists an algebra A such that m is countably additive on it, strongly non atomic, and the ranges m(A) and R(m) (on the whole power set) are the same. In order to obtain this result we prove a geometric properties of zonoids in R2 labelled as Hereditarily Overlapping Boundary Property. Also we give a limit theorem, showing that the R(m) can be approximated by means of subranges m(Ak) where each Ak is a σ-algebra on which m is non atomic and countably additiv
On integration with respect to LCTVS-valued finitely additive measures
No full textIn this paper we introduce and develop an integration theory of a real-valued function with respect to a strongly bounded and finitely additive set function m with values in a locally convex topological vector space (LCTVS) X, using both a weak and a strong approach. Then we investigate the relationships among these two integrals; in particular, assuming the sequential completeness of X, the separability of the range of m and a form of Lebesgue integrability on [0,∞) by seminorms of the distribution functions we prove that all the integrals coincide. Finally we show the above integrals can be considered as Burkill-Cesari integrals
L'integrale di Fubini-Tonelli alla Weierstrass-Burkill lungo coppie di curve continue
No full textWe prove some existence theorems for the Fubini-Tonelli integral in the Weierstrass sense (with respect to Vinti's "intermediate" setting) along CBV Curves, and for a class of separately convex functions satisfying assumptions weaker than those taken into account by other authors: for this larger class we prove that the associated function verifies the hypotheses of a new integrability theorem
Topological properties of the range of a group-valued finitely additive measure
No full textThe aim of this paper is the investigation of the topological properties of the range of a finitely additive measure with values in a topological group. Via an example the standard strong continuity (or s-bundedness) versus the assumption of semiconvexity are investigated, and since they prove to be not comparable in the framework of topological groupsc(contrary to the case of Banach valued, and more in general LCTVS-valued finitely additive measures) results concerning arcwise connectedness and relative compactness of the range are given under both assumptions, by means of ad hoc extension techniques. The result in the semiconvex case is proven along an ad hoc constructive technique
A FURTHER GENERALIZATION OF MIDPOINT CONVEXITY OF MULTIMAPS TOWARDS COMMON FIXED POINT THEOREMS AND APPLICATIONS
No full textWe furtherly generalize midpoint convexity for multivalued maps and derive Fixed Point Theorems and Common Fixed Point Theorems without requiring strong compactness. As an application we obtain some Best Approximation results, minimax and variational inequalities
On minimax theorems for sets closed in measure
No full textThe paper is devoted to Ky Fan minimax equality for convex subsets of L^1 that are closed in measure; in general such sets do not carry any frmal compactness properties for any reasonable topology. The minimax theorem is proven under mild convexity assumptions, such as finite midpoint convex-likeness plus quasi convexity. In the last section we sketch some possible directions for application
Radon-Nikodym Theorems for vector-valued finitely additive measures
No full textThis paper studies the existence of a density function in the Dunford sense for a finitely additive measure with values in a locally convex topological vector space (LCTVS) X . A finitely additive measure m from a σ-algebra to a locally convex is said to be dominated if there exists a finitely additive extended real-valued measure λ such that m is measurable and λ-integrable for every y in X′, the function f is bounded on each member of an increasing sequence of measurable sets that fill Ω in measure, and ∫Edy= for every measurable set E. Here the duals of X have been endowed with the strong topology.
We find several conditions involving the separability of X′, subsets of {m(A)/λ(A):A measurable}, and the weak derivatives d/dλ that ensure the existence of Dunford-type derivatives. In the final section we give conditions for the existence of Dunford-type derivatives for Banach-space-valued finitely additive measures
Sul rango di una massa vettoriale
No full textThis paper focuses on the structure of the range of a strongly non atomic (i.e. strongly bounded) vector-valued finitely additive measures, in both the finite dimensional and the infinite dimensional case. In the finite-dimensional case we prove that the Lyapounoff Theorem has to be weakened in the finitely additive case; more precisely the range is always convex, but, as an example shows, it is not closed (and therefore not compact) in general. On the contrary, in the infinite dimensional case, where even in the countably additive case one just describes the weak or strong closure of the range, the use of the Stone extension and a density argument lead to conclusion completely similar to those of Bartle-Dunford-Schwartz, Kluvanek, Uhl
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