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.