Operators and the Space of Integrable Scalar Functions with Respect to a Fréchet-valued Measure

We consider the space L1.1; X/ of all real functions that are integrable with respect to a measure 1 with values in a real Fréchet space X . We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of `1 in the space of all linear continuous operators from a complete DF-space Y to a Fréchet 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 1 and the space X that imply that L1.1; X/ has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 46A04, 46A40, 46G10, 47B07.