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

Convergences in L^{1}_{X}(μ)Charles Castaing and Mohamed Guessous

Département de Mathématiques, Case 051, Université Montpellier II, F-34095 Montpellier cedex 5, France 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
Lof Bochner integrable functions over a complete probability space^{1}_{X}(μ)(Ω,F, μ)with values in Banach spaceXvia the convergence of its truncated subsequences as well as we give several characterizations of weak compactness and conditionally weak compactness inL. New results involving subsets in^{1}_{X}(μ)Lwhich are closed in measure are obtained and also the characterizations of the Banach space^{1}_{X}(μ)Xin terms of these modes of convergence.

- Compactness, Komlós, Talagrand, tight.