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Embedding \(c_0\) in the space of Pettis integrable functions. (English) Zbl 0963.46025
Quaest. Math. 21, No. 3-4, 261-267 (1998); correction ibid. 29, No. 1,. 133-134 (2006).
Let \(P_1(\mu,X)\) (resp. \(P(\mu,X))\) be the normed space, generally noncomplete, of \(\mu\)-measurable (resp. weakly \(\mu\)-measurable) Pettis integrable functions defined on a probability space \((S,\Sigma, \mu)\) with values in the Banach space \(X\). The author shows that if \(\mu\) has infinite range and \(X\) contains \(c_0\), then so does \(P_1(\mu, X)\) which furthermore is complemented in \(P(\mu, X)\). Conversely, if \(P_1(\mu, X)\) contains a copy of \(c_0\), so does \(X\). This generalizes a result of S. Diaz, A. Fernandez, M. Florencio and P. F. Paul [Quest. Math. 16, No. 1, 61-66 (1993; Zbl 0830.46014)].

MSC:
46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
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