LFU:online

Students of this module have been given insights into topic areas that are covered by the content of modules 1-22.

Real-valued functions on the real line admit a Riemann  integral if they are bounded and ‚almost everywhere’ continuous.  In analysis and in stochastics one meets functions for which these conditions do not apply, but for which one still desires to have some sort of integral at hand. (Think e.g. of a function on the real line  with value 1 on each rational number and with value 0 on each irrational number.) In this context the so-called Lebesgue integral is still available. It coincides with the Riemann integral where this is defined and, as a special feature and under proper conditions, it allows interchange between limiting processes and integration. In probability spaces Lebesgue integration furnishes the expectation of a random variable, in functional analysis it allows the definition of special function spaces (as e.g.  L^p-spaces).

Lecture, assessment is based on a single examination at the end of the course.

Course examination according to § 7, statute section on "study-law regulations"

Tentative content of this course:

§1 The Lebesgue measure

§2 Measurable sets

§3 s-algebras

§4 Measures

§5 Measurable functions

§6 Convergence of function sequences

§7 The integral of non-negative functions

§8 Integrable functions

§9 Rieman- and Lebesgue-Integral

§10 L^p-spaces

§11 Fubini’s theorem