The Lebesgue differentiation theorem holds for any finite Borel measure on Riemannian manifolds and locally compact separable ultrametric spaces. I'm interested to understand how it fails on somewhat more general spaces. In particular:

* Can the theorem fail on a compact separable metric space? Can you provide a counterexample?

* (EDIT) Is there a compact Polish space $X$ s.t. the theorem can fail for *any* metrization of $X$?