*Conducted in term:*2024/25Z

*ECTS credits:*6

*Language:*English

*Organized by:*Faculty of Mathematics and Computer Science

# (in Polish) Measure Theory 1000-OG-EN-TM

Lectures

Constructions and extensions of measures

The Lebesgue integral

The Radon–Nikodym theorem

Images of measures under mappings; change of variables

Disintegration theorem; Young measures

The Fourier transform, Bochner theorem, Bernstein functions

Hausdorff measures, Hausdorff dimension, Choquet capacities,

Differentiation and the Integration, Egoroff theorem, Vitali Covering Theorem, Hardy and Littlewood inequality

The area and coarea formulas

Density points and Lebesgue points

Uniform integrability, Komlos theorem, Dunford-Pettis theorem, Lebesgue–Vitali theorem

Convergence of measures; Kantorovich–Rubinshtein metric, Wasserstein metric, Levy–Prohorov metric, biting lemma

Lorentz spaces, Marcinkiewicz spaces, BMO spaces

The Skorochod space, Wiener measure, Gaussian measure, Hermite polynomials

Classes

Applications of abstract theorems in practice.

## Total student workload

## Learning outcomes - knowledge

## Learning outcomes - skills

## Learning outcomes - social competencies

## Teaching methods

## Prerequisites

## Course coordinators

## Assessment criteria

Oral examination.

## Additional information

Additional information (*registration* calendar, class conductors,
*localization and schedules* of classes), might be available in the USOSweb system: