A space of vector-valued measures and a strict topology

Let X be a completely regular Hausdorff space and let E be a real locally convex Hausdorff space. Katsaras 2] has studied the topologies ₀, , and ₁, for the vector-valued case on C_rc(X,E), the space of all continuous E-valued functions on X with relatively compact range. The corresponding dual spaces are the spaces M_t (B,E'), M (B,E'), and M (B,E') of all t-additive, all -additive, and all -additive members of M(B,E'), the dual space of C_rc (X,E') under the uniform topology. In this paper we study the subspace M_e(B,E') of M(B,E'). A locally convex topology _e is defined on C_rc(X,E) that yields M_e (B,E') as a dual space. It is proved that if E is strongly Mackey then (C (X,E),_e) is strongly Mackey.The author is grateful to Professor J. Schmets for useful suggestions.