Conditional expectation of Pettis integrable random sets: existence and convergence theorems
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Tusi Mathematical Research Group
ORIGINAL PAPER
Conditional expectation of Pettis integrable random sets: existence and convergence theorems M. El Allali1 • F. Ezzaki1 Received: 25 February 2020 / Accepted: 15 September 2020 Tusi Mathematical Research Group (TMRG) 2020
Abstract Existence of the conditional expectation of Pettis integrable random sets is proved. A Representation theorem of Pettis integrable set-valued regular martingales by regular martingales selectors is provided. As applications, Levy’s theorem convergence results are studied for Pettis integrable random sets. At the end of this paper, monotone convergence theorem and Fatou’s lemma are established for a sequence of conditional expectations of Pettis integrable random sets. Keywords Pettis integration Random sets Conditional expectation Fatou’s lemma Monotone convergence theorem Linear topology
Mathematics Subject Classification 46G12 60G42 60G46
1 Introduction Theory of conditional expectation of Bochner integrable random sets is the basic foundation in the study of set-valued martingales, mils, pramart and other set-valued random processes. This line of theoretical research has attracted many authors; it can be traced in the works of Hiai-Umegagi [21], Hiai [20] , Castaing -Valadier [6], Hess [18], Valadier [29], Ezzaki [14], Castaing et al. [7], Castaing Ezzaki and Tahri [8], Ezzaki and Tahri [15] and others. Communicated by Timur Oikhberg. & F. Ezzaki [email protected] M. El Allali [email protected] 1
Faculty of Sciences and Technologies, Laboratory of Modeling and Mathematical Structures, University Sidi Mohamed Ben Abdellah, BP 2202, Fez, Morocco
M. E. Allali and F. Ezzaki
Contrary to Bochner integration, the conditional expectation of a Pettis integrable random variable does not generally exist without an additional condition; see Rybakov [24], Talagrand [25] and Egghe [10]. The existence of this operator for Pettis integrable random sets has been studied by Faik [16] under the assumption that the Banach space has strongly separable dual and possesses the Radon Nikodym property (RNP). This problem is solved later by Akhiat, Castaing and Ezzaki [2] for Pettis integrable random sets with convex weakly compact values, and by Ziat [31] for Pettis integrable random sets with closed bounded (resp. convex weakly compact) values in a Banach space with RNP (Resp. in a Banach space whose dual has the RNP). Recently, the same problem has been studied by Akhiat et al. [3] in a Banach space without RNP, and with the assumption that every countable subset of selectors of F is uniformly Pettis continuous. In this paper, we derive several existence theorems of the conditional expectation of Pettis integrable random sets. The first one is established for Pettis integrable random sets with convex and closed values in a Banach space without RNP and without the aforementioned condition of the selectors of F. The result presented here is general enough to include the same result
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