Lack of metric projectivity, injectivity, and flatness for modules $L_p$

In this paper, we demonstrate that for a locally compact Hausdorff space $S$ and a decomposable Borel measure $\mu$, metric projectivity, injectivity, or flatness of the $C_0(S)$-module $L_p(S,\mu)$ implies that $\mu$ is purely atomic with at most one atom.