We define Hausdorff measure, which allows us to extend the notion of length and area to sets other than curves and surfaces, and generalizes these notions for non-integral dimensional sets. We also introduce Frostman measures, which are key for estimating dimension. Finally, we introduce self-similar sets, a convenient way of constructing sets of any specified dimension.
- Notes: Hausdorff Measure, Frostman Measures, and Dimension, Self-Similar Sets (to cut down on reading, it is not necessary to read all the proofs on self-similar sets, just familiarize yourself with the results.
Alternatively: read Chapters 4 and 5 and Sections 8.3 and 8.4 of Mattila, and Falconer 8.3)
- Videos: Hausdorff Measure, Hausdorff Dimension, Frostman Measures, Self-Similar Sets
- Exercises (prepare to discuss problems 1, 2, 8, 9)