Syllabus
*1. Basic notions of measure theory. a) Sigma-albegra and related structures, measure b) Measurable functions *2. Construction of the integral a) Integral on a measure space b) Monotone convergence theorem c) Linearity of the integral *3. Constructions of measures a) Abstract outer measure b) Carathéodory theorem c) Construction of the Lebesgue measure *4. Lebesgue integral a) Lebesgue integral on the real line b) Convergence theorems c) Integrals depending on a parameter *5. Measure theory a) Dynkin systems, uniqueness results b) Premeasures, the Hopf theorem c) Signed measures d) Lebesgue decomposition and Radon-Nikodým theorem e) Sequences of measurable functions, Jegorov theorem f) Measurable mappings and push-forward of a measure *6. Multiple integrals a) Product of measures, the Fubini theorem b) Change of variables c) Polar and spherical coordinates *7. L^p spaces a) Basic definitions, equivalence classes b) Hölder and Minkowski inequalities c) Completeness *8. Lebesgue-Stieltjes integral a) Regularity of measures b) Lebesgue-Stieltjes measures and distribution functions c) Integration by parts d) Absolutely continuous and discrete cases
Annotation
Introductory course on measure theory and integration. Required course for bachelor's programs General
Mathematics and Information Security.