MEASURE THEORY AND INTEGRATION
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- MICHELE MIRANDA
- Credits
- 6
- Didactic period
- Secondo 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 and 2 and. Geometry.
Course programme
- FIRST PART
Riemann integral and limitations. Algebras and sigma-algebras; measures and examples. Outer measure; measurable sets and Caratheodory criterion. Metric measures; examples and Hausdorff measures. Cantor sets. (5 hours)
Measurable functions; equivalent definitions. Properties of measurable functions and approximation by simple functions. Comparison between Borel and Lebesgue measurable sets. Integral of measurable function. Chebychev and consequences. Convergence theorems; monotone, Beppo Levi, Fatou and Lebesgue. Comparison between Riemann and Lebesgue integral; Riemann-Stjielties integral. (5 hours)
Dynkin families and pi-systems. Ntroduction to product measure. Fubini Theorem and disintegration of measure. Conergences; in measure, almost everywhere, almost uniform, in mean. Comparisons among different convergences; Borell-Cantelli and Egorov Theorem. Lusin theorem; introduction to Lebesgue spaces; definition and completeness. (5 hours)
Vector measures and measures with sign; total variation measure and finiteness of vector measures. Banach structure of the set of measures and duality with continuous functions; weak convergence of measures. Jordan and Hahn decomposition of measures. Radon-Nykodym theorem and Besicovitch-Vitali derivation theorem. (6 hours)
SECOND PART
Continuous functions (5 hours)
- modulus of continuity of a continuous function
- families of continuous functions.
- properties of Lipschitz and convex functions over open sets of $R^N$
Convex analysis (5 hours)
- convex functions on normed vector spaces
- Fenchel-Legendre transform
- sub-gradients and sub-differential calculus
- Fenchel-Young inequality
- Fenchel-Moreau theorem
Monge-Kantorovich Problems (9 hours)
- introduction to the problem: Monge's classical problem
- transport maps
- examples and counter-examples
- transport plans
- Kantorovich's reformulation
- existence of a solution for the Kantorovich problem
- c-concave functions and c-transform
- dual problem
- existence for the dual problem
- Kantorovich potentials
- duality theorem
- primal-dual optimality conditions
- the Ambrosio-Sudakov theorem
- the Brenier-McCann theorem
Some applications (2 hours)
- the isoperimetric inequality: the proof by Knothe-Gromov 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; during the oral exam the candidate has to
answer to two questions, one related to the first
part of the course, the second on the second part. Reference texts
- The reference books for the first part of the course are;
"Measure Theory", Donald Cohn,
Birkhäuser, Boston, Mass., 1980.
"Real analysis and probability", Richard Dudley,
The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove,1989.
whether for the second part the reference
book is:
"Topics in optimal transportation", Cedric Villani,
Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.
If possible, we shall diistribute partial notes of the course, available on the web site.
