We provide some new results on the weak convergence of sequences or nets lying in L p (( T , ∑, μ ), X ) ≡ L p ( μ , X ), 1 ⩽ p < ∞, i.e., the space of equivalence classes of X -valued ( X is a Banach space) Bochner integrable functions on the finite measure space ( T , ∑, μ ). Our theorems generalize in several directions recent resuls on weak sequential convergence in L 1 ( μ , X ) obtained by M. A. Khan and M. Majumdar [ J. Math. Anal. Appl. 114 (1986), 569–573] and Z. Artstein [ J. Math. Econ. 6 (1979), 277–282], and they can be used to obtain dominated convergence results for the Aumann integral. Our results have useful applications in Economics and Game Theory.