## On some subsets of $$L_ 1(\mu,E)$$.(English)Zbl 0799.46023

Let $$(E,\| \cdot \|)$$ be a Banach space and let $$(\Omega, \Sigma, \mu)$$ be a $$\sigma$$-finite measure space. If $$1 \leq p < \infty$$, then $$\| f \|_ p$$ denotes $$(\int_ \Omega \| f \|^ pd \mu)^{1/p}$$, and $$L_ p (\mu,E)$$ denotes $$\{f : \Omega \to E : \| f \|_ p < \infty\}$$. If $$F$$ is a set, then a class of subsets of $$F$$ with prescribed properties is denoted by $${\mathcal H} (F)$$, and a subset $$K$$ of $$L_ 1 (\mu,E)$$ is said to be a $$\mu - {\mathcal H}$$ set if
(i) $$K$$ is bounded,
(ii) $$\lim .\sup .\{\int_ A \| f \| d \mu : f \in K$$, $$\mu (A) \to 0\} = 0$$;
(iii) if $$A \in \Sigma$$, then $$K(A) = \{\int_ A fd \mu : f \in K\} \subseteq {\mathcal H} (E)$$.
The Banach space $$E$$ is said to have the $$P(\mu, {\mathcal H})$$ property if each $$\mu - {\mathcal H}$$ set is an element of $${\mathcal H} (L_ 1 (\mu,E))$$.
In the results of this paper, the author indicates equivalent characterizations of the property $$P(\mu, {\mathcal H})$$, where $${\mathcal H}$$ is one of a variety of properties including
(a) class of bounded sets: $${\mathcal B}$$;
(b) Class of weakly relatively compact sets: $${\mathcal W}$$;
(c) Class of weakly conditionally compact sets: $${\mathcal W}^* {\mathcal C}$$;
(d) class of weakly compact sets: $${\mathcal K}$$;
(e) class of Dunford-Pettis subsets of $$E:{\mathcal V}$$.
In particular, the general results include the statement that: $$E$$ contains no complemented copy of $$\ell_ 1$$ if and only if $$E$$ has property $$P(\mu, {\mathcal V}^*)$$, and $$E$$ contains no copy of $$\ell_ 1$$ if and only if $$E$$ has property $$P(\mu, {\mathcal W}^* {\mathcal C})$$.

### MSC:

 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 4.6e+41 Spaces of vector- and operator-valued functions
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### References:

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