This is a continuation of an earlier paper [Inventiones Math., 8 (1969), 175-221]. It is assumed that a space W and a sheaf H over W are given, such that the pair ( W , H ) satisfies the Brelot axioms and also satisfies, locally, the additional hypotheses of the theory of adjoint sheaves. The following subjects are considered: 1) Extension of the adjoint-sheaf theory to the case where ( W , H ) does not admit a global potential (in particular, the case where W is compact). 2) Construction of a new fine resolution O → H → R → L → O of the sheaf H , in which L is a (complete pre-)sheaf of measures on W . 3) Construction of a natural duality between the flux functional corresponds to a distinguished positive element of H W * .