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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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