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On sums of Pettis integrable random elements. (English) Zbl 1036.46010
Let \(X\) be a Banach space and \((\Omega, \Sigma, \mu)\) be a complete probability space. The author investigates the relationship between different types of convergence in \(P_1(\mu,X)\), the space of all \(X\)-valued \(\mu\)-measurable Pettis integrable functions, of sums of random elements of the form \(\sum_{i=1}^{n}\varepsilon_if_i\) (where \(\varepsilon_i\) takes the values \(\pm1 \) with the same probability and \(f_i \in P_1(\mu,X)\) for each \(i \in \mathbb N\)) and the existence of weakly unconditionally Cauchy subseries of \(\sum_{n=1}^{\infty}f_n(\omega)\) for \(\mu\)-almost all \(\omega \in \Omega\).

MSC:
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B09 Probabilistic methods in Banach space theory
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