MEASURE THEORY AND INTEGRATION
Academic year and teacher
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- Versione italiana
- Academic year
- 2016/2017
- Teacher
- MICHELE MIRANDA
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The target of the lectures is the knowledge of the basic notions of abstract measure theory to be applied in the study of solutions of differential equations, both in the classical and stochastic setting. The target is also the introduction of some problems of calculus of variations and geometric measure theory.
Prerequisites
- Prerequisites of the lectures are minimal; it is required the knowledge of the aruments of Calculus 1, while knowledges coming from Calculus 2 and Geometry are advisable but not mandatory.
Course programme
- Riemann integral and its limitations. Algebras and sigma-algebras; measures and examples. Outer measures; measurable sets and Caratheodory criterion. Metric measures; examples and Hausdorff measures. Cantor sets. Complements. (8 ore)
Measurable functions; equivalent definitions. Measurable functions; properties and approximation by simple functions. Comparison between Borel and Lebesgue measurable sets. Integral of measurable functions. Chebychev inequality and consequences. Convergence theorems; monotone convergence, Beppo Levi, Fatou and Lebesgue theorems. Comparison between Riemman and Lebesgue integral (6 hours)
Dynkin families and pi-systems. Product measure. Fubini theorem . Convergences; in measure, almost everywhere, almost uniform, in mean. Comparisons among different convergences. Borell-Cantelli and Egorov theorems. Lusin theorem; introduction to Lebesgue spaces; definition and completeness. (6 hours)
Vector and signed measures; total variation measure and finiteness of vector measures. Jordan and Hahn decomposition. Radon-Nikodym and Besicovitch theorems. Besicovitch-Vitali and Besicovitch derivation theorems. Generalizations of the Besicovitch derivation theorem. Lebesgue points for integrable functions. (8 hours)
Space of measures; Banach structure and duality with continous functions. Weak convergence of measures. Distribution functions and monotone functions; almost everywhere differentiability and distributional derivative. Relations among monotone functions and positive measures. Finite variation functions; difference of monotone functions. Lipschitz functions and almost everywhere differentiability. Convex functions; definition and first properties. Fundamental theorem of integral calculus; Lipschitz and abosolutely continuos functions. Sobolev functions in dimension one. Properties of convex functions and proof of the distributional second derivative as a positive measure. (8 hours)
Hausdorff measures; comparison with Lebesgue measure. Isodiametric inequality e coincidence of Hausdorff measures with surface measures on regular surfaces. (4 hours)
Riemann-Stieltjes integral. Brownian motions and stochastic integral; Ito integral. Conditional expectation and martingales. Ito processes and Ito forumla; ordinary stochastic differential equations. Construction of the Brownian motion and of the Wiener measure. (8 hours) Didactic methods
- The lectures are theoretical and are based on front lecture in the classroom; in the lectures we shall try to show examples and applications of the gained knowledges.
Learning assessment procedures
- The exam is based an oral interview of approximatively 45 minutes; it is up to the student wheter to expose a topic in-depth to be agreed with the teacher, or two or three small topics in-depth to be agreed with the teacher or finally to answer to three questions on three different topics seen during the lectures.
Reference texts
- The reference book for the lectures is;
"Measure Theory", Donald Cohn.
Lecture notes are also available on the web site.
