: Let E be an ideal of L 0 over a ?-finite measure ce (?, ? ,?) and let (X, ?.? X ) be a real Banach space. Let E(? ,X) be a suace of the space L 0 (? ,X) of ?-equivalence of all strongly ? -measurable functions ƒ : ? ?X and consisting of all those ƒ ? L 0 (? ,X) for which the scalar function ?ƒ(.)? X belongs to E. Let E(?,X) n ? tand for the order continuous dual of E(? ,X). In this paper we characterize both conditionally ?(E(? ,X),I)-compact and relatively ?(E(? ,X),I)-sequentially compact subset of E(? ,X) whenever I in an ideal of E(?,X) n ? . Az an application, we obtain a characterization of almost reflexivity and reflexivity of a Banach space X in terms of conditionally ?(E(? ,X),I)-compact and relatively ?(E(? ,X),I)-sequentially compact subsets of E(? ,X).