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On vector measures which have everywhere infinite variation or noncompact range

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Drewnowski L., Lipecki Z.

Instytut Matematyczny Polskiej Akademii Nauk

Abstract Let X be an infinite-dimensional Banach space, let Σ be a σ-algebra of subsets of a set S, and denote by ca(Σ,X) the Banach space of X-valued measures on Σ equipped with the uniform norm. We say that a nonzero μ ∈ ca(Σ,X) is everywhere of infinite variation [has everywhere noncompact range] if |μ|(A) = ∞ or 0 [{μ(E): E ∈ Σ, E ⊂ A} is not relatively compact or equals {0}] for every A ∈ Σ. Let λ be a nonatomic probability measure on Σ, and denote by ca(Σ,λ,X) the closed subspace of ca(Σ,X) consisting of λ-continuous measures. Analogously as above, we define measures in ca(Σ,λ,X) that are λ-everywhere of infinite variation or have λ-everywhere noncompact range. Using the Dvoretzky-Rogers theorem, we give two constructions of an absolutely convergent series of λ-simple measures in ca(Σ,λ,X) such that the sum of each of its subseries is λ-everywhere of infinite variation. In particular, the normed space P(λ,X) of Pettis λ-integrable functions with values in X lacks property (K), and so is incomplete. These results refine and improve some earlier results of E. Thomas, and L. Janicka and N. J. Kalton. One of the constructions also yields the existence of an infinite-dimensional closed subspace in ca(Σ,λ,X) all of whose nonzero members are λ-everywhere of infinite variation. Moreover, modifying some ideas of R. Anantharaman and K. M. Garg, we prove that the measures that are λ-everywhere of infinite variation form a dense $G_δ$-set in ca(Σ,λ,X). From this we derive an analogous result on measures that are everywhere of infinite variation and the closed subspace of ca(Σ,X) consisting of nonatomic measures. Similar results concerning measures that have [λ-] everywhere noncompact range are also established. In this case, the existence of X-valued measures with noncompact range must, however, be postulated. We also prove that the measures of σ-finite variation form an $F_{σδ}$-, but not $F_σ$-, subset of ca(Σ,λ,X), and the same is true for P(λ,X) provided that X is separable. Finally, we consider the special case when X is a Banach lattice and, for X nonisomorphic to an AL-space, we note analogues of some of the results above for positive X-valued measures on Σ.

Ograniczanie wyników

1 Drewnowski L.

1 Lipecki Z.

1 1995

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On vector measures which have everywhere infinite variation or noncompact range