Foundations of Analysis

The modern concept of integral was introduced on April 29, 1901, in a short article by Henri Lebesgue. That article opened a new chapter of analysis. The flaws of the Riemann Integral will be pointed out, and the Lebesgue integral will be introduced to remedy the situation. Key concepts include measure theory including sigma-algebras, measurable spaces, measurable functions, outer measures, construction of the Lebesgue measure, and product measures. The course also covers the classical convergence theorems, Fubini's theorem, functions of bounded variation, and the fundamental theorem of integral calculus. Fundamentals of probability theory including expectation, probability distribution and independence, will be discussed.

1. Knowledge: The student masters basic concepts from measure theory, including sets of measure zero, measurable functions, the Lebesgue integral and Lebesgue spaces. The student has an overview of the central results of the theory of Lebesgue integration, including convergence theorems and Fubini's theorem. Moreover, the student is familiar with applications of measure theory to probability theory.

2. Skills: The student is able to perform operations using the Lebesgue integral and Lebesgue spaces. Moreover, the student is able to apply integration theory in one or several variables to formulate and solve problems in mathematics and technology, including problems involving discontinuous data.

Lectures and exercises.