Real Analysis
A.Y. 2026/2027
Learning objectives
The aim of the course is to provide the basic elements of the theory of Radon measures on locally compact spaces with a countable base, including some covering theorems and the theory of the differentiation of Radon measures on locally compact and separable metric spaces, as well as some of the most relevant interaction between Lipschitz maps and Hausdorff measures in Euclidean spaces, up to the area and coarea formulae.
Expected learning outcomes
At the end of the course, the students should be able to: Take advantage of the double representation of Radon measures as set functions and as linear functionals. Determine the vague limit of simple sequences. Discuss the structure of a Radon measure with respect to another one in terms of the Lebesgue decomposition and of the differentiation of measures. Interpret, with the aid of the area formula, the usual notions of volume for submanifolds of the Euclidean spaces.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Recap on abstract measure theory: outer measures on a set. Measurable sets in the sense of Carathéodory. \sigma-algebras and measures. Borel outer measures on a metric space. Carathédory's criterion. Borel-regular outer measures. Outer and inner regularity for finite Borel measures on a metrizable space. Measurable functions and integration theory. Densities. Image measures. Product Measures.
2. Radon measures: Radon measures on a locally compact space with a countable base. Lusin's theorem. Approximation of measurable functions with continuous functions. Egoroff's theorem. The space of continuous functions with compact support and its topology. Riesz's representation theorem. Characterization of positive Radon measures. The vague topology. Characterization of vaguely compact sets. Characterization of the vague convergence for positive measures. Vague limit and support. Lower semi-continuity of the total variation. The Radon-Riesz property for measures.
3. Differentiation of Radon measures: Vitali's covering lemma. Doubling measures. Weak (1,1) estimate for the (centred) Hardy-Littlewood maximal function for doubling measures. The Besicovitch covering theorem. The Vitali-Besicovitch covering theorem. Absolutely continuous and singular measures. The Lebesgue decomposition of a measure. The Radon-Nikodym derivative and Lebesgue points. Density of a set with respect to a measure. Functions of bounded variation and absolutely continuous functions.
4. Recap on Hausdorff measures: construction of Hausdorff measures. Borel-regularity of Hausdorff measures. The Hausdorff dimension. Comparison between Lebesgue measure and the n-dimensional Hausdorff measure on R^n: the isodiametric inequality. The k-dimensional density of a measure and its relations with the k-dimensional Hausdorff measure.
5. Lipschitz functions and the area and coarea formulae: Lipschitz maps. Lipschitz constant. McShane's extension lemma. Mention of the Kirszbraun extension theorem. Characterization of the Lipschitz condition via inclusions for the graph. Rademacher's theorem. Estimates on the Hausdorff measure of a Lipschitz image. Hausdorff dimension of a Lipschitz graph. Derivative of a Lipschitz function on a level set. Polar decomposition of a linear mapping. Jacobian of a linear mapping. The area formula. Mention of the coarea formula. A version of the Morse-Sard theorem for mappings of class C^1.
2. Radon measures: Radon measures on a locally compact space with a countable base. Lusin's theorem. Approximation of measurable functions with continuous functions. Egoroff's theorem. The space of continuous functions with compact support and its topology. Riesz's representation theorem. Characterization of positive Radon measures. The vague topology. Characterization of vaguely compact sets. Characterization of the vague convergence for positive measures. Vague limit and support. Lower semi-continuity of the total variation. The Radon-Riesz property for measures.
3. Differentiation of Radon measures: Vitali's covering lemma. Doubling measures. Weak (1,1) estimate for the (centred) Hardy-Littlewood maximal function for doubling measures. The Besicovitch covering theorem. The Vitali-Besicovitch covering theorem. Absolutely continuous and singular measures. The Lebesgue decomposition of a measure. The Radon-Nikodym derivative and Lebesgue points. Density of a set with respect to a measure. Functions of bounded variation and absolutely continuous functions.
4. Recap on Hausdorff measures: construction of Hausdorff measures. Borel-regularity of Hausdorff measures. The Hausdorff dimension. Comparison between Lebesgue measure and the n-dimensional Hausdorff measure on R^n: the isodiametric inequality. The k-dimensional density of a measure and its relations with the k-dimensional Hausdorff measure.
5. Lipschitz functions and the area and coarea formulae: Lipschitz maps. Lipschitz constant. McShane's extension lemma. Mention of the Kirszbraun extension theorem. Characterization of the Lipschitz condition via inclusions for the graph. Rademacher's theorem. Estimates on the Hausdorff measure of a Lipschitz image. Hausdorff dimension of a Lipschitz graph. Derivative of a Lipschitz function on a level set. Polar decomposition of a linear mapping. Jacobian of a linear mapping. The area formula. Mention of the coarea formula. A version of the Morse-Sard theorem for mappings of class C^1.
Prerequisites for admission
Good knowledge of metric spaces and locally compact spaces with a countable base; normed spaces and the norm of bounded operators between normed spaces; Hilbert spaces; basic measure theory, including Lebesgue and Hausdorff measures.
Some basic constructions and result in functional analysis (uniform boundedness principle, metrizability lemmas for weak topologies, basic properties of locally convex spaces) may shed some light on some constructions presented during the course, but are not strictly necessary.
Some basic constructions and result in functional analysis (uniform boundedness principle, metrizability lemmas for weak topologies, basic properties of locally convex spaces) may shed some light on some constructions presented during the course, but are not strictly necessary.
Teaching methods
Lectures.
Teaching Resources
- M. Calzi, Lectures notes (in inglese), available on the MyAriel page of the course.
- L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015.
- F. Maggi, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, Volume 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012.
For further reading:
- M. Calzi, Lectures notes (extended version), available on the MyAriel page of the course.
- L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Science Publications, 2000.
- L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Oxford University Press, 1983.
- N. Bourbaki, Integration I (Elements of Mathematics), Springer, 2004.
- N. Bourbaki, Integration II (Elements of Mathematics), Springer, 2004.
- L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Textbooks in Mathematics. CRC Press, Boca Raton, FL, revised edition, 2015.
- F. Maggi, Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory, Volume 135 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2012.
For further reading:
- M. Calzi, Lectures notes (extended version), available on the MyAriel page of the course.
- L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Science Publications, 2000.
- L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Oxford University Press, 1983.
- N. Bourbaki, Integration I (Elements of Mathematics), Springer, 2004.
- N. Bourbaki, Integration II (Elements of Mathematics), Springer, 2004.
Assessment methods and Criteria
The exam consists of an oral discussion on the topics presented during the lectures. The purpose of the discussion is to verify that the student knows and understands the content of the lectures, can establish relationships among the various topics, and can effectively apply the techniques presented in concrete situations.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
The exam is passed upon successful completion of the oral examination. A final mark in the range 0-30 (with 18 being the minimum passing grade) is given and communicated immediately at the end of the oral examination.
MATH-03/A - Mathematical Analysis - University credits: 6
Lessons: 42 hours
Professor:
Calzi Mattia
Shifts:
Turno
Professor:
Calzi MattiaProfessor(s)