# Lebesgue Integration Theory

This is a classic class in integration theory that allows students to understand what is behind probability theory. The class starts by defining and studying measurable spaces, by concentrating on sigma algebras – including Borel sigma algebras. Then, measurable functions and Borel functions are introduced. Within the study of measures, sigma additivity and sigma subadditivity are examined together with the Lebesgue measure and the Carathéodory condition.

All these ingredients being prepared, we arrive at the construction of Lebesgue integrals – after providing reminders on the Riemann integral viewed as a limit of Darboux sums. Fundamental theorems are obtained and illustrated: the Beppo-Levi monotone convergence theorem, the Lebesgue dominated convergence theorem – Fatou’s lemma being examined in passing. Then, we derive theorems for integrals that depend on a parameter and for multiple integrals (Fubini-Tonelli’s theorem in the latter case). Convolution is also studied.