# MAT-61757 Measure and Integration, 5 cr

#### Lisätiedot

After the academic year 2018-2019 the course will be lectured every second year.
Suitable for postgraduate studies.

Janne Kauhanen

#### Opetus

 Toteutuskerta Periodi Vastuuhenkilö Suoritusvaatimukset MAT-61757 2018-01 4 Janne Kauhanen Final exam, weekly exercises, and weekly written exercises.

#### Osaamistavoitteet

After the completion of the course the student knows the main concepts and results of the measure and integration theory. The student is capable of defining the main concepts precisely. The student is capable of writing justifications and applying results in simple proofs and calculations. The student is able to verify the most important results. The student can apply the concepts and results in advanced studies and applications in the area of analysis and stochastics. The exact mathematical reasoning is emphasized during the course.

#### Sisältö

 Sisältö Ydinsisältö Täydentävä tietämys Erityistietämys 1. Unions and intersections with general index sets, countable and uncountable sets. The limit superior and the limit inferior. Sigma algebra and measure. Outer measure and measurable sets. The Axiom of Choice 2. Lebesgue outer measure. Sigma-algebra and measure, measure spaces. Lebesgue measure and Borel sets. An example of the non-Lebesgue-measurable set. 3. Measurable functions. The integral with respect to a measure: the integral of a simple function, the integral of a nonnegative function, the Monotone Convergence Theorem, the integral of a measurable function, and Dominated Convergence Theorem. The connections between the Lebesgue integral and the Riemann integral. L^p spaces. The Lebesgue integral in R^n and Fubini's Theorem.

#### Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali Book Measure, Integration & Real Analysis (Preliminary Edition) Sheldon, Axler No

#### Esitietovaatimukset

 Opintojakso P/S Selite MAT-60206 Mathematical Analysis Mandatory

#### Vastaavuudet

 Opintojakso Vastaa opintojaksoa Selite MAT-61757 Measure and Integration, 5 cr MAT-61756 Measure and Integration, 7 cr

 Päivittäjä: Kauhanen Janne, 17.12.2018