# MAT-61757 Measure and Integration, 5 cr

#### Additional information

The course will be lectured every second year.

Suitable for postgraduate studies.
The implementation will not be executed during the academic year 2019-2020.

#### Person responsible

Janne Kauhanen

#### Lessons

Implementation | Period | Person responsible | Requirements |

MAT-61757 2019-01 | - |
Janne Kauhanen |
Final exam and weekly exercises. |

#### Learning Outcomes

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. The exact mathematical reasoning is emphasized during the course.

#### Content

Content |
Core content |
Complementary knowledge |
Specialist knowledge |

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. |

#### Study material

Type |
Name |
Author |
ISBN |
URL |
Additional information |
Examination material |

Book | Measure, Integration & Real Analysis (Preliminary Edition) | Sheldon, Axler | No |

#### Prerequisites

Course | Mandatory/Advisable | Description |

MAT-60206 Mathematical Analysis | Mandatory |

#### Correspondence of content

Course | Corresponds course | Description |

MAT-61757 Measure and Integration, 5 cr | MAT-61756 Measure and Integration, 7 cr |