Adv. Math. Econ. Volume 1, pp.17-37 (1999)

Convergences in L1X(μ)

Charles Castaing1 and Mohamed Guessous2
1 Département de Mathématiques, Case 051, Université Montpellier II, F-34095 Montpellier cedex 5, France
2 Université Hassan II Mohamedia, Faculté des Sciences Ben M'sik, Département de Mathématiques, BP 7955 Ben M'sik, Casablanca, Maroc
We present new modes of convergences for bounded sequences in the space L1X(μ) of Bochner integrable functions over a complete probability space (Ω, F, μ) with values in Banach space X via the convergence of its truncated subsequences as well as we give several characterizations of weak compactness and conditionally weak compactness in L1X(μ). New results involving subsets in L1X(μ) which are closed in measure are obtained and also the characterizations of the Banach space X in terms of these modes of convergence.
Key Words:
Compactness, Komlós, Talagrand, tight.