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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
##### Keywords:
normed space; Pettis integrable functions
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##### References:
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