scholarly journals The bounded vector measure associated to a conical measure and pettis differentiability

2001 ◽  
Vol 70 (1) ◽  
pp. 10-36
Author(s):  
L. Rodriguez-Piazza ◽  
M. C. Romero-Moreno
Keyword(s):  
Vector Measure ◽  
Locally Convex Space ◽  
Finite Intersection ◽  
Vector Measures ◽  
Weak Space ◽  
Countably Additive Vector Measure ◽  
Differentiable Vector ◽  
Locally Convex ◽  
Additive Vector ◽  
Additive Vector Measure

AbstractLet X be a locally convex space. Kluvánek associated to each X-valued countably additive vector measure a conical measure on X; this can also be done for finitely additive bounded vector measures. We prove that every conical measure u on X, whose associated zonoform Ku is contained in X, is associated to a bounded additive vector measure σ(u) defined on X, and satisfying σ(u)(H) ∈ H, for every finite intersection H of closed half-spaces. When X is a complete weak space, we prove that σ(u) is countably additive. This allows us to recover two results of Kluvánek: for any X, every conical measure u on it with Ku ⊆ X is associated to a countably additive X-valued vector measure; and every conical measure on a complete weak space is localizable. When X is a Banach space, we prove that σ(u) is countably additive if and only if u is the conical measure associated to a Pettis differentiable vector measure.

Completeness of the L1-space of closed vector measures

1990 ◽  
Vol 33 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Uniform Convergence ◽  
Convex Space ◽  
Vector Measure ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

The notion of a closed vector measure m, due to I. Kluv´;nek, is by now well established. Its importance stems from the fact that if the locally convex space X in which m assumes its values is sequentially complete, then m is closed if and only if its L1-space is complete for the topology of uniform convergence of indefinite integrals. However, there are important examples of X-valued measures where X is not sequentially complete. Sufficient conditions guaranteeing the completeness of L1(m) for closed X-valued measures m are presented without the requirement that X be sequentially complete.

scholarly journals Remarks on the range of a vector measure

1994 ◽  
Vol 36 (2) ◽  
pp. 157-161 ◽  
Author(s):  
Jesús M. F. Castilo ◽  
Fernando Sánchez
Keyword(s):  
Vector Measure ◽  
Null Sequence ◽  
Property A ◽  
Weakly Compact ◽  
Countably Additive Vector Measure ◽  
Additive Vector ◽  
Standing Problem ◽  
Additive Vector Measure

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

An ordered suprabarrelled space

1991 ◽  
Vol 51 (1) ◽  
pp. 106-117
Author(s):  
J. C. Ferrando ◽  
M. López-Pellicer
Keyword(s):  
Convex Space ◽  
Affirmative Answer ◽  
Locally Convex Space ◽  
Vector Measures ◽  
Locally Convex

AbstractA locally convex space E is said to be ordered suprabarrelled if given any increasing sequence of subspaces of E covering E there is one of them which is suprabarrelled. In this paper we show that the space m0(X, Σ), where X is any set and Σ is a σ-algebra on X, is ordered suprabarrelled, given an affirmative answer to a previously raised question. We also include two applications of this result to the theory of vector measures.

scholarly journals On copies of the null sequence Banach space in some vector measure spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 443-447
Author(s):  
J.C. Ferrando ◽  
J.M. Amigó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Measure ◽  
Null Sequence ◽  
Vector Measures ◽  
Measure Spaces ◽  
Additive Vector

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

Generalized Köthe function spaces. I

1969 ◽  
Vol 65 (3) ◽  
pp. 601-611 ◽  
Author(s):  
Nguyen Phuong-Các
Keyword(s):  
Vector Space ◽  
Function Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Special Cases ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Differentiable Vector ◽  
Locally Convex ◽  
Kernel Theorem

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E-valued functions φ(t) such that for all elements e′ of the dual E′ of E. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E. Therefore one usually starts from some function space which has interesting properties.

scholarly journals Some properties of vector measures taking values in a topological vector space

1987 ◽  
Vol 43 (2) ◽  
pp. 224-230
Author(s):  
Efstathios Giannakoulias
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Necessary Condition ◽  
Vector Measures ◽  
Topological Vector Spaces ◽  
Locally Convex ◽  
Pettis Integrability

AbstractIn this paper we study some properties of vector measures with values in various topological vector spaces. As a matter of fact, we give a necessary condition implying the Pettis integrability of a function f: S → E, where S is a set and E a locally convex space. Furthermore, we prove an iff condition under which (Q, E) has the Pettis property, for an algebra Q and a sequentially complete topological vector space E. An approximating theorem concerning vector measures taking values in a Fréchet space is also given.

scholarly journals Lattice Copies ofℓ2inL1of a Vector Measure and Strongly Orthogonal Sequences

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
E. Jiménez Fernández ◽  
E. A. Sánchez Pérez
Keyword(s):  
Vector Measure ◽  
Complete Characterization ◽  
Positive Vector ◽  
Domain Space ◽  
Integrable Functions ◽  
Countably Additive Vector Measure ◽  
Space Of Integrable Functions ◽  
Square Integrable Functions ◽  
Additive Vector

Letmbe anℓ2-valued (countably additive) vector measure and consider the spaceL2(m) of square integrable functions with respect tom. The integral with respect tomallows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of stronglym-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spacesL1(m) andL2(m). Under certain requirements, our main result establishes that a normalized sequence inL2(m) with a weakly null sequence of integrals has a subsequence that is stronglym-orthonormal inL2(m∗), wherem∗is anotherℓ2-valued vector measure that satisfiesL2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to anℓ2-valued positive vector measure contains a lattice copy ofℓ2.

scholarly journals Bounded vector measures on effect algebras

2005 ◽  
Vol 72 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Hong Taek Hwang ◽  
Longlu Li ◽  
Hunnam Kim
Keyword(s):  
Convex Space ◽  
Effect Algebra ◽  
Locally Convex Space ◽  
Effect Algebras ◽  
Vector Measures ◽  
Orthogonal Sequence ◽  
Locally Convex

Let (L, ⊥, ⊕,0,1) be an effect algebra and X a locally convex space with dual X′. A function μ: L → X is called a measure if μ(a ⊕ b) = μ(a) + μ(b) whenever a⊥b in L and it is bounded if is bounded for each orthogonal sequence {an} in L. We establish five useful conditions that are equivalent to boundedness for vector measures on effect algebras.

Measures on effect algebras

2019 ◽  
Vol 69 (1) ◽  
pp. 159-170
Author(s):  
Giuseppina Barbieri ◽  
Francisco J. García-Pacheco ◽  
Soledad Moreno-Pulido
Keyword(s):  
Effect Algebras ◽  
Vector Measures ◽  
Finite Dimensional ◽  
Riesz Decomposition ◽  
Nikodym Property ◽  
Additive Measures ◽  
Finite Dimensional Banach Space ◽  
The Riesz Decomposition Property ◽  
Additive Vector ◽  
Additive Vector Measure

Abstract We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which σ-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.

scholarly journals The range of an o-weakly compact mapping

1985 ◽  
Vol 39 (3) ◽  
pp. 391-399 ◽  
Author(s):  
P. G. Dodds
Keyword(s):  
Convex Set ◽  
Vector Measure ◽  
Compact Convex ◽  
Locally Convex Space ◽  
Weakly Compact ◽  
Compact Mapping ◽  
Compact Convex Set ◽  
Locally Convex ◽  
Order Continuous

AbstractIt is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

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