Week 7: Tangent Measures, Part I Tangent Measures are measures generated by zooming in on the support of a measure. Their appeal is that sometimes properties that hold a.e. for a measure improve or hold uniformly for their tangent measures. To demonstrate their versatility, we prove Marstrand’s Theorem, which says any measure whose s-dimensional densities exist must have s an integer.
- Notes (Alternatively Chapter 14 of Mattila).
- Videos: Densities of Hausdorff Measure, Tangent Measures, Uniform Measures, Tangent Measures and Densities, Marstrand’s Theorem.
- Exercises (TODO: Need to add solutions) (prepare 1, 2, 3, 5, 6 ,7)