Operators and the space of integrable scalar functions with respect to a Frechet-valued measure

We consider the space L-1 (v, X) of all real functions that are integrable with respect to a measure v with values in a real Frechet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l(infinity) in the space of all linear continuous operators from a complete DF-space Y to a Frechet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure v and the space X that imply that L-1 (v, X) has the Dunford-Pettis property. Applications of these results to Frechet AL-spaces and Kothe sequence spaces are also given.