(in Polish) Zaawansowana teoria miary

mathematics

elective monographs

Linear algebra and geometry II 1000-112bGA2a
Mathematical analysis II.2 1000-114bAM4a
Topology I 1000-113bTP1a

Mathematical analysis II.2* 1000-114bAM4*
Measure Theory 1000-135TM

One should understand the notions of an embedded manifold, abstract (outer) measure, and the Lebesgue integral. One should know what are Borel sets and measures. One should not confuse the gradient with the derivative of a function. It might be profitable to understand what is the tensor product of vector-spaces. Helpful might be knowing classical covering theorems of Vitali and Besicovitch. The lecture will be supplemented by student talks in exercise sessions covering gaps in the material.

A well-established understanding of the material of Analysis II.2 is an essential minimum. Sometimes I will refer to certain facts from functional analysis and measure theory. All unproven facts will be formulated with precise references in the literature.

If a student has not previously attended a lecture entitled "Measure Theory" (1000-135TM), this is best done in the winter semester.

Blended learning
Classroom

I plan to present excerpt from the theory covered in cahpters 2.9, 2.10 and 3.2 of the excellent book "Geometric Measure Theory" by H. Federer (https://doi.org/10.1007/978-3-642-62010-2).

The following topics will be mentioned:

- area and coarea formulas

- rectifiable and purely unrectifiable sets

- approximate differentiability

- Caratheodory construction

- Federer-Besicovitch characterisation of rectifiable sets via orthogonal projections

- Steiner formula

- Minkowski content in relation to the Hausdorff measure

- integralgeometric mesure and realted formulas

- general Morse-Sard Theorem

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.

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".

* 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.

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.