[efr-1A5R]

Let p: E \to X be a continuous map between locally compact Hausdorff spaces. Then CO_0(E) acquires the structure of a CO_0(X) module in an obvious way, given by (f \cdot u)(e) = f(p(e))u(e). Moreover, \rho (u)(x) = \sup _{e \in p^{-1}(x)}u(x) equips CO_0(E) with a Frechet module structure.

The bounded homomorphisms CO_0(E) \to CO_0(X) are in bijection with families of signed, real-valued Radon measures \mu _x \in CO_0(E_x)^* which vary continuously, in the sense that the resulting map X \to CO_0(E)^* (given by pushing forward along the inclusion E_x \hookrightarrow E) is weak-* continuous, and which are uniformly bounded in