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Bocce criterion and convergence in Pettis norme in \(P_E^1(\mu)\). (Critère de Bocce et convergence en norm de Pettis dans \(P_E^1(\mu)\).) (French) Zbl 1041.46021
In the first part of the paper, scalar uniform integrability with respect to the Pettis integral is characterized with the so-called sequential Pettis-Bocce criterion. The authors refer to the paper of E. Balder, M. Girardi, and V. Jalby [Stud. Math. 111, No. 3, 241–262 (1994; Zbl 0809.28006)] and assert that their proofs are adaptations of Balder-Girardi-Jalby’s ones. However, they are verbatim the same. They work in the space \(P_E^1(\mu)\) of the Pettis integrable functions, when [BGJ] worked in the subspace \(L^1_E(\mu)\) (with the Pettis norm), but the results and the proofs are exactly the same. Their Lemme 2.1 is the Lemma 5.10 of [BGJ]: the only difference is that it seems that they assume the sequence scalarly uniformly Pettis integrable, but they forgot this word in the statement of the lemma, and in the second part of the proof, though it appears in the first line of the proof; however, the proof of [BGJ] only used scalar uniform Pettis integrability. Their Lemme 2.2, Lemme 2.3, Théorème 3.1 are respectively Lemma 2.5, Lemma 5.9 and Theorem 5.11 of [BGJ], with the same proofs (in Théorème 3.1, scalar uniform Pettis integrability is assumed instead of uniform Pettis integrability, but the proof of [BGJ] only used the first one).
In the second part, the authors introduce the notions of \({\mathcal R}_k(E)\)-tight and \({\mathcal R}_w(E)\)-tight subsets of \(P_E^1(\mu)\); these notions are connected to the existence of some measurable multifunction with values in the closed convex subsets of \(E\), and they study the relationship with uniform Pettis integrability and the Pettis-Bocce criterion. The main result is Théorème 4.1, where it is proved that a sequence \((f_n)_n\) of Pettis integrable selections of a measurable multifunction \(\Gamma\) with values in the compact convex sets of the separable Banach space \(E\) satisfies the sequential Pettis-Bocce criterion when four conditions hold: (i) \((f_n)_n\) is uniformly Pettis integrable; (ii) \(\{\omega\mapsto \sup_{x\in \Gamma(\omega)} x^\ast(x)\); \(x^\ast\in B_{E^\ast}\}\) is uniformly integrable; (iii) \(\int_B f_n\,d\mu\to \int_B f\,d\mu\) weakly for every measurable subset \(B\) (the measure space is assumed to be finite), with \(f\) Pettis integrable; (iv) \(f(\omega)\) is an extreme point of \(\Gamma(\omega)\) for almost all \(\omega\).
There are several misprints in the paper.
Reviewer: Daniel Li (Lens)
MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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