Dentability and the Radon-Nikodým property

Huff, Robert

doi:10.1215/s0012-7094-74-04111-8

Cited by 38 publications

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“…E is an arbitrary positive number, then there exists an element x~X and a subset F of E such that V-~m(F ) > 0 and F has the property (T -a (D~ (Tx))~ ~) . 5) If 2 is a ~-algebra of subsets of a set ~ and m: ~ + X is a measure with values in the space X, then the triple (~, E, m) has the property D(T).…”

Section: ) If I Is a O-algebra Of Subsets Of A Set ~ Mmentioning

confidence: 99%

“…The proof of the implication I) ~ 2) is based on a construction of Maynard [4], modified by Huff [5] for establishing the following result:…”

Section: Consequently V~(f)< I/(kn--l)mentioning

confidence: 99%

“…Let us recall that a bounded subset B of X is called dentable (s-dentable) (see [4,5]) if it is T-dentable (Ts-dentable) for the identity operator T: X § X. It is easily seen that if U ~ L (X, Y) and UB is a dentable (an s-dentable) set, where B is a bounded subset of the space X, then B is a U-dentable (Us-dentable) set.…”

Section: Consequently V~(f)< I/(kn--l)mentioning

confidence: 99%

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Geometric characterization of RN-operators

Reinov¹

1977

Mathematical Notes of the Academy of Sciences of the USSR

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Let X and Y be Banach spaces and T E L(X. Y). An operator T: X § Y is called an RN-operator if it transforms every X-valued measure ~ of bounded variation into a Yvalued measure having a derivative with respect to the variation of the measure m. The notions of T-dentability and Ts-dentability of bounded sets in Banach spaces are introduced and in their terms are given conditions equivalent to the condition that T is an RN-operator (Theorem i).It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace X0 of X the set (TIX0)*(Y*) is separable (Theorem 2). RN-Operators, introduced in [i], are studied.Characteristics of RN-operators are deduced in certain geometrical terms (Theorem i), and the class of operators whose adjoint operators are RN-operators is described.In particular, we obtain a proof, which is not based on the Dunford--Phillips--Pettice theorem, of the fact -that every weakly compact operar tor is an RN-operator.Let X and Y be Banach spaces and L(X, Y) be the space of continuous linear operators from X into Y.If A is a bounded subset of X, then we will denote the closed convex hull of the set A by co(A) and the s-convex hull of the set A by s-co(A):

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Section: ) If I Is a O-algebra Of Subsets Of A Set ~ Mmentioning

confidence: 99%

“…The proof of the implication I) ~ 2) is based on a construction of Maynard [4], modified by Huff [5] for establishing the following result:…”

Section: Consequently V~(f)< I/(kn--l)mentioning

confidence: 99%

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Geometric characterization of RN-operators

Reinov¹

1977

Mathematical Notes of the Academy of Sciences of the USSR

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“…(X The first equivalence is the theorem of Davis-Phelps-Huff-Maynard [3], [6], [10] and the second is simple.…”

Section: First If a E σ + And / E S Thenmentioning

confidence: 99%

“…Other conditions which are equivalent to the RNP in a Banach space A" are: Every closed bounded (convex) set in Xis dentable [3], [6]. Other conditions which are equivalent to the RNP in a Banach space A" are: Every closed bounded (convex) set in Xis dentable [3], [6].…”

Section: Remarkmentioning

confidence: 99%

On integration and the Radon-Nikodym theorem in quasicomplete locally convex topological vector spaces.

1977

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we shall extend the Dunford second integral or equivalently the strongly measurable Pettis integral to quasicomplete locally convex spaces. We shall then give a Radon-Nikodym theorem in this setting which extends a theorem for Banach spaces due to Rieffei. It is shown that the other conditions on the average r nge of a vector measure which are equivalent to the existence of a strongly measurable density in Banach spaces fail to be equivalent in general.Examples are given of locally convex spaces with the Radon-Nikodym property. We briefly discuss the relationship between the Radon-Nikodym property in Banach spaces and more general locally convex spaces.The Radon-Nikodym property and Radon-Nikodym theorems have been considered for Frechet spaces [1], [2], [15] and more generally for Strict Mackey Convergence spaces [5]. In these spaces the Banach space results generally carry over directly or with only slight modifications. In fact the proofs usually involve an embedding in an appropriate Banach space and an application of the Banach space result. Notation and Definitions.We shall denote by E a quasicomplete (closed bounded sets are complete) locally convex topological vector space, by E* the algebraic dual and by E' the continuous dual of E. By {ρ} we shall mean a fundamental collection of continuous seminorms which determine the topology of E. Throughout, we shall let (T, Z, μ) represent a positive finite, complete measure space, m: Σ -> E a countably additive vector measure, and Σ + = {S e Σ: μ (S) > 0}. Definition 1. A s (m) = \ -^-^-: S' c= S, S' E Σ + >, is called the average r nge of m on S.[ Definition 2. A set property, P, on Σ is said to be local if for every S in Σ + , there is an S' c:S,S'mZ + such that S f has P. Definition 3. m has locally small average r nge if for S in Σ + , ρ in {ρ} and ε>0, there is an S" c 5, S" in Σ + such that A s ,(m) has ρ-diameter less than ε. Journal f r Mathematik. Band 292 17 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/5/15 12:05 AM

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