Resolution of constant sheaf by $L^2$ function sheaves

Question

Let $X$ be a compact Hausdorff space equipped with a Radon measure of full support. Then $U\mapsto L^2(U)$ is a fine sheaf, hence can be taken for a first step in an acyclic resolution of the constant sheaf $\mathbb C$. My question is, whether it can be completed to an acyclic resolution consisting of sheaves which look like $L^2$ or tensor/exterior powers thereof.