In order to pass the course the student should be able to
use the concepts of measurable set and measurable function;
state and explain the construction of the Lebesgue integral and use it;
apply the theorems of monotone and dominated convergence and Fatou's lemma;
describe the construction of product measure and to apply Fubini's theorem;
state and explain properties of Lp spaces;¨
define absolute continuity and singularity of measures;
apply Lebesgue decomposition and the Radon-Nikodym theorem.
Measure and outer measure. Lebesgue measure in one and higher dimensions. Measurable functions. Lebesgue integral and convergence theorems. The connection with the Riemann integral. Product measure and Fubini's theorem. L^1 and L2 theory. Hilbert space. Fourier series and Fourier integrals. Convergence of Fourier series in L2 norm. Convergence in measure, almost everywhere and in Lp. Lp as a normed space. Hölder's and Minkowski's inequalities. Absolute continuity and singularity of measures. Lebesgue decomposition and the Radon-Nikodym theorem. Radon-Nikodym derivative.
Lectures and ptoblem solving sessions.
Written examination at the end of the course in combination with homework assignments according to instructions given at the course start.
The course cannot be included in higher education qualification together with Measure theory and Integration theory I and II or equivalent courses.
week 30, 2015
Measure theory and probability