(in Polish) Zaawansowana teoria miary 1000-1M20ZTM
We will introduce the concept of a Vitali relation and show important facts about the density of one measure with respect to another. We will introduce the notion of an approximate limit, continuity and differentiation. We will then characterise measurable functions using these concepts. We will show that functions of bounded variation (BV) are almost everywhere differential and we will take a moment to address absolutely continuous functions. Then we will move on to the Caratheodory construction, which allows define a (outer) measure from any non-negative function defined on subsets of a metric space. We will construct a k-dimensional Hausdorff measure in R^n, as well as some other measures useful in geometry. We will introduce the concepts of upper and lower Hausdorff densities and prove some simple facts resulting from bounds on these quantities. We will show estimates for integrals of Hausdorff measures of levelsets of Lipschitz function in a very general version. If time allows, we will focus for a moment on the isodiametric inequality and the Steiner symmetrisation. We will prove the Kirszbrun theorem about extending Lipschitz functions. Then we will move on to the study of rectifiable sets. We will prove the Rademacher theorem showing that Lipschitz functions are differentiated almost everywhere. We will discuss the existence of a partition of unity and Whitney's extension theorem for functions of class C^1. We will then introduce the concept of the approximate Jacobian and prove the area and coarea formulas for functions defined on an open subset of a Euclidean space. Then we will generalize the area and coarea formulas to functions defined on rectifiable sets. Finally, we will deduce some corollaries from these formulas: Steiner's formula, Cauchy's formula and the Besicovitch theorem characterising rectifiable sets by orthogonal projections.
Main fields of studies for MISMaP
Type of course
Mode
Classroom
Requirements
Prerequisites
Prerequisites (description)
Learning outcomes
* Knowledge of the conventions and notations used in Federer's book and, consequently, ease of use and the possibility to study the book on ones own.
* Ability to carry out precise and formally correct computations on geometrical objects (functions, measures, linear subspaces, manifolds, etc.) of any dimension and codimension without choosing a coordinate system.
* Knowledge of proofs of classical, although not covered by the programme of other courses, theorems from the geometric measure theory; in particular, area and coarea formulas.
* Knowledge of Caratheodory construction and ability to apply it, e.g., to define the Hausdorff or the Favard (integralgeometric) measure over any metric space. Knowledge of the basic properties of these measures.
* Ability to use the approximate concepts, e.g., of boundary, derivative, tangent vectors, etc.
* Understanding what rectifiable sets are and what role they play in geometry. Knowledge of their basic properties (e.g. the existence of an approximate tangent space at almost every point) and the theorems (some without proofs) characterising these sets.
Assessment criteria
There will be a list of topics to be presented by students in exercise sessions. At least one talk must be given by each student who wants to take the exam. Most likely the material will come from the book of Federer.
There will be a take-home exam followed by an oral presentation of solutions.
Bibliography
H. Federer "Geometric Measure Theory" (https://doi.org/10.1007/978-3-642-62010-2)
F. Maggi "Sets of finite perimeter and geometric variational problems".
L. Ambrosio, N. Fusco, D. Pallara "Functions of bounded variation and free discontinuity problems".
L. Evans, R. Gariepy "Measure theory and fine properties of functions".
P. Mattila "Geometry of sets and measures in Euclidean spaces".
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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