On Some Aspects of Vector Measures (original) (raw)
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On some open problems involving range of vector measures
Filomat, 2012
The close connection between the geometry of a Banach space and the properties of vector measures acting into it is now fairly well-understood. The present paper is devoted to a discussion of some of these developments and certain problems arising naturally in this circle of ideas which are either open or have been partially resolved. Emphasis shall be laid mainly on those aspects of this theory which involve properties of the range of these vector measures.
VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE
Function Spaces - Proceedings of the Sixth Conference, 2003
Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Köthe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions f X : Ω → [0, ∞) are in the lattice E. The notion of E-variation of a measure -which allows to recover the pvariation (for E = L p ), Φ-variation (for E = L Φ ) and the general notion introduced by Gresky and Uhl-is introduced. The space of measures of bounded E-variation V E (X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X)) * = V E (X * ), that V E (X) can be identified with space of cone absolutely summing operators from E into X and that E(X) = V E (X) if and only if X has the RNP property.
Ranges of vector measures in Fréchet spaces
Indagationes Mathematicae, 2000
Characterizations are given of those Frechet spaces E such that every compact subset of E lies in the range of an E-valued measure of bounded variation, respectively in the range of a measure of bounded variation with values in a superspace of E. Extending results for Banach spaces due to Pifieiro and Rodriguez-Piazza, we prove that this property characterizes nuclear spaces, respectively hilbertizable spaces, in the framework of Frechet spaces.
Remarks on the semivariation of vector measures with respect to Banach spaces
Bulletin of the Australian Mathematical Society, 2007
Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.
Studia Mathematica, 2014
It is proved that if X is infinite-dimensional, then there exists an infinitedimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that ca(B, λ, X) \ Mσ, the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear Algebra Appl. 428 (2008)].
The de la Vallée Poussin Theorem for Vector Valued Measure Spaces
Rocky Mountain Journal of Mathematics, 2000
The purpose of this paper is to extend the de la Vallée Poussin theorem to cabv(µ, X), the space of measures defined in Σ with values in the Banach space X which are countably additive, of bounded variation and µ-continuous, endowed with the variation norm.
Vector measures and the strong operator topology
Proceedings of The American Mathematical Society, 2009
A fundamental result of Nigel Kalton is used to establish a result for operator valued measures which has improved versions of the Vitali-Hahn-Saks Theorem, Phillips's Lemma, the Orlicz-Pettis Theorem and other classical results as straightforward corollaries.
Some Properties of Variational Measures
Real Analysis Exchange
Recently several authors have established a remarkable property of the variational measures associated with a function. Expressed in classical language, this property asserts that if a function is ACG * on all sets of Lebesgue measure zero then the function must be globally ACG *. This article is an exposition of some ideas related to this property with the intention of bringing it to the attention of a wider audience than these original papers might attract. If f : [a, b] → R then a necessary and sufficient condition for the identity f (x)−f (a) = x a f (t) dt in the sense of the Denjoy-Perron integral is that µ f is σ-finite and absolutely continuous with respect to Lebesgue measure on [a, b].