1st semester 
20 credits / 30 lectures

In this module, we will study measure theory and integration. A concept of a measure in mathematics is a formalization and a generalization of a geometrical measure (length, area, volume). Besides their great importance within mathematics, measures play a fundamental role in probability theory: probabilities of events are formally defined as measures. Moreover, measures have far reaching applications in physics, in particular in quantum physics. A notion of a measure is a basis for the modern theory of an integration.

Syllabus:          

• Families of sets: semirings, rings, sigma-algebras.
• Measures. Examples and properties. Carathéodory-measurability.
• The Lebesgue measure on R. Existence of non-measurable sets in R.
• Measurable functions.
• Lebesgue integral.
• Chebyshev's inequality. Null sets, almost everywhere properties.
• Fatou's lemma. Lebesgue's dominated convergence theorem.
• Product measures and Fubini's theorem. The Lebesgue measure on R^d.
•  L^p-norms and L^p-spaces. Inequalities of Young, Hölder and Minkowski. Properties of L^p.
• Signed measures. Hahn decompositions. The Radon-Nikodym theorem.