Course outline The content of the present one-year course splits into three parts as follows, where an emphasis indicates review topics which will not be examined. Firstly, regarding general measure theory, we treat covering theorems (adequate families, covering with enlargement, centred ball coverings, and Vitali relations), derivatives (existence of derivatives, indefinite integrals, density and approximate continuity, and curves of finite length), and Carathéodory’s construction (the general construction, Hausdorff and spherical measure, relation to Riemann-Stieltjes integration, densities, Cantor sets, Steiner symmetrisation, equality of measures over Euclidean space, and Lipschitzian extensions of functions). Secondly, regarding Grassmann algebra, we cover tensor products, graded algebras, the symmetric algebra of a vector space, symmetric forms and polynomial functions, the exterior algebra of a vector space, alternating forms and duality, interior multiplications, simple m-vectors, and inner products. Thirdly, regarding rectifiability, we study differentials and tangents (differentiation, Whitney’s extension theorem, approximate differentiation, Rademacher’s theorem, pointwise differentiation of higher order, factorisation of maps near generic points, submanifolds of Euclidean space, tangent vectors, and relative differentiation) and area for Lipschitzian maps (Jacobians, area of maps of Euclidean space, rectifiable sets, approximate tangent vectors and differentials, area of maps of rectifiable sets, Cartesian products, equality of measures on rectifiable sets, and rectifiable sets and manifolds of class 1). The second term of this course is expected to begin with the topic the exterior algebra of a vector space.

Details of the course The main reference text will be the instructor’s weekly updated lecture notes written in LATEX, see [Men20]. They are based on and expand the relevant parts of Federer’s treatise [Fed69] and, regarding pointwise differentiation of higher order, the presentation of [Men19]. Grading is solely determined by individual oral examinations conducted in English.

Prerequisites We employ measures, measurable sets, Borel regular measures, measurable functions, Lebesgue integration, linear functionals, and product measures. Familiarity with the majority of these concepts is thus expedient.