Limitations on the extendibility of the Radon-Nikodym Theorem

Given two locally compact spaces $X,Y$ and a continuous map $r: Y \rightarrow X$ the Banach lattice $\text{\normalsize {$\mathcal{C}$ }}_0(Y)$is naturally a $\text{\normalsize {$\mathcal{C}$ }}_0(X)$-module. Following the Bourbaki approach to integration we define generalized measures as $\text{\normalsize {$\mathcal{C}$ }}_0(X)$-linear functionals $\mu : \text{\normalsize {$\mathcal{C}$ }}_0(Y) \rightarrow \text{\normalsize {$\mathcal{C}$ }}_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.