\(\mathfrak G\)-uniform scalar integrability and strong laws of large numbers for Pettis integrable functions with values in a separable locally convex space (Q1975245)

The generalized Vitali convergence theorem and a strong law of large numbers (SLLN) for pairwise independent Pettis integrable random elements with values in a separable locally convex space \(E\) are given. SLLN is proved under the assumptions that the sequence \(((1/n) \sum^n_{i=1} P_n)_n\) is tight and \({\mathfrak G}\)-uniformly scalarly integrable and moreover for almost all \(\omega\in\Omega\) the sequence \(((1/n) \sum^n_{i=1} \delta_{X_i(\omega)})_n\) is \({\mathfrak G}\)-uniformly scalarly integrable. This theorem contains a known result for a separable Banach space. It is also proved a generalization of this SLLN, namely sufficient conditions for SLLN are given in terms of compactness conditions in a suitable topology on the set of Pettis integrable probabilities on \(E\).