10 - Rectifiable Sets, II

1 min read

We continue our discussion of rectifiable sets and prove Marstrand?s Rectifiability Criterion, which says that any measure whose lower and upper densities are positive and finite a.e. and whose tangent measures a.e. are all flat must be rectifiable. Combining this with the results we covered on tangent measures, we can finally establish Preiss? Theorem: a measure is m-rectifiable if and only if its m-dimensional densities exist almost everywhere.