Rybakov's theorem for vector measures in Fréchet spaces

Fernández, Antonio; Naranjo, Francisco

doi:10.1016/s0019-3577(97)83348-4

Cited by 13 publications

(15 citation statements)

References 8 publications

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“…During the last 25 years, several papers have analysed the properties of the space of integrable functions L 1 (m) (see Cubera [4,5,6], Fernández et al [9], Fernández-Naranjo [10], Okada [18], Okada-Ricker [21], Okada-RickerRodríguez-Piazza [23], and Stefansson [30]). In particular, the papers by Curbera [4][5][6] began a systematic study of the structure of L 1 (m) spaces.…”

Section: Introductionmentioning

confidence: 99%

Spaces of p-integrable Functions with Respect to a Vector Measure

Fernández¹,

Mayoral²,

Naranjo³

et al. 2006

Positivity

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Section: Introductionmentioning

confidence: 99%

Spaces of p-integrable Functions with Respect to a Vector Measure

Fernández¹,

Mayoral²,

Naranjo³

et al. 2006

Positivity

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“…Compare our result with [6 where to say that v is X -continuous with respect to a positive measure X on E means that every A.-null set is av-null set. Measuresof the form |jc'v| with*' e D v (whenthey exist) are called Rybakov control measures for v. Conditions on the space X for which (4.1) holds have been studied in [13], where it is shown that if X admits a continuous norm, then every X-valued vector measure has a Rybakov control measure. In this section we study sufficient conditions on v and X in order that the space L'(v, X) has the Dunford-Pettis property.…”

Section: Proof By the Representation Theorem [12 Proposition 24 (Vmentioning

confidence: 99%

Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure

Fernández¹,

Naranjo²

1998

J Aust Math Soc A

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We consider the space L' (v, X) of all real functions that are integrable with respect to a measure v with values in a real Frechet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of t°° in the space of all linear continuous operators from a complete DF-space Y to a Frechet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure v and the space X that imply that L' (v, X) has the Dunford-Pettis property. Applications of these results to Frechet AL-spaces and Kothe sequence spaces are also given.1991 Mathematics subject classification (Amer. Math. Soc): primary 46A04, 46A40,46G10,47B07.

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“…If necessary redefine / and each f n{k) , k e N, to be zero on this w-null set so that f n{k) ->• / pointwise everywhere on Q. (4). Since E e E and y' e X' are arbitrary we see that the requirements of Definition 1 are satisfied, that is / e L\m).…”

Section: ((Mx'))mentioning

confidence: 99%

“…This is possible due to a recent result of Fernandez and Naranjo [4] characterizing such (real) Frechet spaces as precisely those in which Rybakov's theorem holds. This class of spaces contains all Banach spaces and so, in particular, we provide a simple and transparent proof of the completeness of the L' -space for Banach space-valued measures.…”

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confidence: 99%