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

Bombal, Fernando

Czech. Math. J. 41, No. 1, 170-179 (1991).

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 \(P(\mu, {\mathcal W}^* {\mathcal C})\).

Reviewer: G.O.Okikiolu (London)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40	Spaces of vector- and operator-valued functions

Keywords:

class of bounded sets; weakly relatively compact sets; weakly conditionally compact sets; weakly compact sets; Dunford-Pettis subsets

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