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Conical measures and properties of a vector measure determined by its range

L. Rodríguez-Piazza, M. Romero-Moreno (1997)

Studia Mathematica

We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability...

Conical measures and vector measures

Igor Kluvánek (1977)

Annales de l'institut Fourier

Every conical measure on a weak complete space E is represented as integration with respect to a σ -additive measure on the cylindrical σ -algebra in E . The connection between conical measures on E and E -valued measures gives then some sufficient conditions for the representing measure to be finite.

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm | | · | | ℱ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space ( ℳ , | | · | | ℱ ) * is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals on...

Continuous selection theorems

Michał Kisielewicz (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Continuous approximation selection theorems are given. Hence, in some special cases continuous versions of Fillipov's selection theorem follow.

Controlled convergence theorems for Henstock-Kurzweil-Pettis integral on m -dimensional compact intervals

Sokol B. Kaliaj, Agron D. Tato, Fatmir D. Gumeni (2012)

Czechoslovak Mathematical Journal

In this paper we use a generalized version of absolute continuity defined by J. Kurzweil, J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exch. 17 (1992), 110–139. By applying uniformly this generalized version of absolute continuity to the primitives of the Henstock-Kurzweil-Pettis integrable functions, we obtain controlled convergence theorems for the Henstock-Kurzweil-Pettis integral. First, we present a controlled convergence theorem for...

Convergence of conditional expectations for unbounded closed convex random sets

Charles Castaing, Fatima Ezzaki, Christian Hess (1997)

Studia Mathematica

We discuss here several types of convergence of conditional expectations for unbounded closed convex random sets of the form E ℬ n X n where ( ℬ n ) is a decreasing sequence of sub-σ-algebras and ( X n ) is a sequence of closed convex random sets in a separable Banach space.

Convergence theorems for measures with values in Riesz spaces

Domenico Candeloro (2002)

Kybernetika

In some recent papers, results of uniform additivity have been obtained for convergent sequences of measures with values in l -groups. Here a survey of these results and some of their applications are presented, together with a convergence theorem involving Lebesgue decompositions.

Convergence theorems for set-valued conditional expectations

Nikolaos S. Papageorgiou (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper we prove two convergence theorems for set-valued conditional expectations. The first is a set-valued generalization of Levy’s martingale convergence theorem, while the second involves a nonmonotone sequence of sub σ -fields.

Convergence theorems for the Birkhoff integral

Marek Balcerzak, Monika Potyrała (2008)

Czechoslovak Mathematical Journal

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence ( f n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f n ’s is involved and we assume f n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of ( f n ) to f is assumed.

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