On Some Aspects of Vector Measures

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.

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.

Lithuanian Mathematical Journal, 1980

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].

Publicacions Matemàtiques, 1991

Transactions of the American Mathematical Society, 1980

Let G be a locally compact group and A an arbitrary Banach space. L p ( G , A ) {L^p}(G,A) will denote the space of p-integrable A-valued functions on G. M ( G , A ) M(G,A) will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of L p ( G , A ) {L^p}(G,A) . Using this result, we prove that if μ ∈ M ( G , A ) \mu \, \in \, M(G,A) , such that either x → μ x x\, \to \, {\mu _x} or x → x μ x{ \to _x}\mu is continuous, then μ ∈ L 1 ( G , A ) \mu \, \in \, {L^1}(G,A) .

Czechoslovak Mathematical Journal, 2001

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Proceedings of the American Mathematical Society, 1995

We consider the space L'(i>) of real functions which are integrable with respect to a measure v with values in a Banach space X. We study type and cotype for Lx{v). We study conditions on the measure v and the Banach space X that imply that Ll(v) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in Lx(v).

Topology and its Applications, 2008

Publications de l'Institut Math?matique (Belgrade), 2006

The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.

Given a non atomic, finite and complete measure space (Ω, Σ, µ) and a Banach space X, we define the modulus of continuity for a vector measure F as the function ω F (t) = sup µ(E)≤t |F |(E) and in-

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1976

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.

Journal of Mathematical Analysis and Applications, 1999

In this paper we resume the most important results that we obtained in our papers [1,2,5,6,7] concerning a broad class of measures that we defined in dealing with a bangbang control problem. Let M be the σ−algebra of the Lebesgue measurable subsets of [0, 1] and µ : M → R n be a non-atomic vector measure. A well known Theorem of Lyapunov (see [11]) states that the range of µ, defined by R(µ) = {µ(E) : E ∈ M}, is closed and convex or, equivalently, that given a measurable function ρ with values in [0, 1] there exists a set E in M such that (*) X ρ dµ = µ(E). Lyapunov's Theorem has been widely applied in bang-bang control theory [10] and, more recently, in some non-convex problems of the Calculus of Variations [3]. As an example we mention the following bang-bang existence result:

2016

1 Measure Spaces 1 1.1 Algebras and σ–algebras of sets................. 1 1.1.1 Notation and preliminaries................ 1 1.1.2 Algebras and σ–algebras................. 2