1. Introduction. This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29]. Given a Banach space X and an algebra j/of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on se with J-valued sup

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... norm continuous linear operators on the space S(s/) of j/simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology T on S(sf). It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space S(s/) T to yield results on strong and weak compactness in spaces of vector measures. These measure-operator identifications are stated in Section 2. The main theorem on strong compactness is stated and proved in Section 3 as Theorem 3.1. It gives necessary and sufficient conditions that a subset of X-valued strongly bounded, countably additive (here called strongly countably additive) measures with relatively compact ranges on an algebra of sets be relatively compact in the strong topology of the semi-variation norm. Related results have recently been found by Brooks and Dinculeanu [9]. Among the several equivalent conditions given in Theorem 3.1, condition (3.1.6) seems, from the point of view of necessity, stronger than any heretofore known. It is the generality of the measure-theoretic setting (strongly countably additive measures on algebras of sets) of Theorem 3.1 which permits many applications in Section 4. These include strong compactness results for spaces of Pettis-integrable functions, spaces of unconditionally convergent series, spaces of compact-range bounded additive measures on Boolean