Limitations on the extendibility of the Radon-Nikodym Theorem

Abstract:

Given two locally compact spaces $X,Y$ and a continuous map $r: Y \rightarrow X$ the Banach lattice $\mathcal {C}_0(Y)$ is naturally a $\mathcal {C}_0(X)$-module. Following the Bourbaki approach to integration we define generalized measures as $\mathcal {C}_0(X)$-linear functionals $\mu : \mathcal {C}_0(Y) \rightarrow \mathcal {C}_0(X)$. The construction of an $L^1(\mu )$-space and the concepts of absolute continuity and density still make sense. However we exhibit a counter-example to the natural generalization of the Radon-Nikodym Theorem in this context.