Measure and Integration

Code: MAT14220L

6 ECTS

Duration: 15 weeks/156 hours

Scientific Area: Mathematics

Teaching languages: Portuguese

Languages of tutoring support: Portuguese, English

Regime de Frequência: Presencial

Presentation

In this mandatory subject, taught in the 6th semester, it is intended that students learn the common principles and universal methods of Measure Theory, as well as its application to the problems of geometry, physics and probability theory.

Sustainable Development Goals

Educação de qualidade - Assegurar a educação inclusiva, e equitativa e de qualidade, e promover oportunidades de aprendizagem ao longo da vida para todos.

Learning Goals

Learning of the general principles and universal methods of the Measure Theory. Applications to various geometric, physic and probability problems

Jordan and Lebesgue measure in a finite dimensional space. Measurable sets. Abstract measure. Extension and completion. Measurable functions. Lebesgue integral. Convergence almost everywhere and by measure. Convergence theorems. Product of measures. Fubini theorem. Sign measures. Radon-Nikodym theorem. Differential inequalities and extremal solutions. Continuous dependence of initial conditions. Applications.

Teaching Methods

Teaching methodologies:
- Structured exposition, examples with emphasis on applications and on solving exercises.
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam. Students who achieve a grade of 18 or above may have to do an extra oral exam. For these students the final grade is the maximum between 17 and the simple average of the previous grade and the oral exam.