Syllabus
1. Differentiation of measures
- covering theorems (Vitali, Besikovich, perhaps also Whitney)
- maximal operator
- application to absolutely continuous functions and to functions of bounded variation
- mutual differentiation of two Radon measures
- Lebesgue points of locally integrable functions
- Rademacher theorem, relationship of Lipschitz functions and W^{1,\infty} 2. Hausdorff measure and dimension
- outer Hausdorff measure
- Hausdorff measure
- Hausdorff dimension
- connections to Lebesgue measure
- area formula (without a proof)
Annotation
Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master studies. Content: differentiation of measures, absolutely continuous functions, fuctions of bounded variation,
Lipschitz function, Hausdorff measure and dimension.