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.
- Notes Note: The proof of Lemma 4.4 is incorrect, been meaning to fix it. Also note that my approach for proving Marstand’s Rectifiability Criterion is a bit different from what others do. For the usual way, see Chapter 5 of De Lellis. For another source on $\alpha$-numbers, see Tolsa’s paper where they are introduced. There he assumes his measures are Alhfors regular. For an alternative source on why the densities of rectifiable sets are 1 a.e, see Theorem 16.2 in Mattila)
- Videos: alpha numbers, Marstrand’s Rectifiability Criterion, Rectifiability, Densities, and Preiss' Theorem.
- Exercises and Partial Solutions(prepare problems 1, 2b, 3bc).