Analysis II @KSoM26

Syllabus

Evaluation Plan

• End-Semester Examination: 50 marks

• Mid-Semester Examination: 25 marks

• Quizzes: 2 quizzes, 5 marks each (total 10 marks)

• Assignments: 15 (4/5 lecture notes + 11/10 'problems') assignments, total 15 marks

Assignments

There will be two types of assignments in this course.

1. Students are required to prepare written lecture notes for four/five lectures, providing complete and well-structured details of arguments and proofs that may be omitted during class.

2. Students will be assigned specific problems. For each assigned problem, students must present their solution in class and submit a written version of the solution.

For both types of assignments, all submissions are expected to be clearly organized, mathematically rigorous, and carefully written.

Grading: The final assignment score will be based on the average of all assignment marks. (Percentage of marks obtained × 0.15)

Topics covered in

• Lecture 01: Intuitive 'integration' of the Dirichlet's function; what issues arise when we try to find measure of sets using singleton sets; sum over an uncountable set and related exercises.

• Lecture 02: Elementary measure and Jordan measure.

• Lecture 03: Characterizations of Jordan measure.

• Lecture 04: Introduction to Lebesgue outer measure, proof that the Lebesgue outer measure is not additive.

• Lecture 05: Lebesgue outer measure extends the elementary measure, outer regularity : we experess Lebesgue outer measure of an arbitrary set E in terms of Lebesgue outer measure of open covers of E (which may be found using almost disjoint diadic cubes), Lebesgue outer measure is countably additive for almost disjoint boxes and finitely additive for separated sets.

• Lecture 06: Definition and examples of Lebesgue measurable sets, measurability criteria and measurability of Cantor set, motivation and definitions of limit inferior and limit superiors of sets.

• Lecture 07: Jordan measurable sets are Lebesgue measurable; countable additiity of measure; upward and downward monotone convergence theorems; dominated convergence theorem; inner regularity of Lebesgue measure.

• Lecture 08: Comparison between the ideas of Riemann and Lebesgue integrals, definitions and properties of unsigned and complex-valued simple functions and their integrals.

• Lecture 09: Equivalent notions of measurability of an unsigned function; Complex valued measurable functions.

• Lecture 10: Definition and finite additivity of Lebesgue unsigned integral; Markov's inequality and its consequences; triangle inequality; approximation of L^1 functions.

• Lecture 11: Egorov's theorem; Lusin's theorem; Boolean algebra; sigma algebra and measurable spaces.

• Lecture 12: Finite and countably additive measure, measure space and its completemess, measurable functions on measurable space and integration of measurable functions on measure spaces.

Mid-semester exam.

• Lecture 13: Escapes to infinity; Monotone convergence theorem; Fatou's lemma; Dominated convergence theorem.

• Lecture 14: Riesz-Fischer theorem; Different modes of convergence of sequence of functions on a measure space and the relations between these modes of convergence.

• Lecture 15: Uniqueness of limits under mixture of seven modes of convergence and the case of step functions.

• Lecture 16: Overview of both versions of fundamental theorem of calculus; Lebesgue differentiation theorem in R and its proof using the Hardy-Littlewood Maximal inequality in higher dimensions: Indefinite integral F of an absolutely integrable function f is absolutely continuous almost everywhere differentiable function with F' = f a.e.

• Lecture 17: Proof of the Hardy-Littlewood Maximal inequality in higher dimensions (HLMIHD) using Vitali-type covering lemma.

• Lecture 18: Lebesgue differentiation theorem in multi-dimensions as a consequence of HLMIHD; the rising sun lemma, upper bound for second fundamental theorem for MND functions as application of Fatou's lemma.

• Lecture 19: Absolutely continuous functions in bounded intervals are of bounded variation;(Recall: A function is of bounded variation if and only if it is difference of two monotone functions); Second fundamental theorem for absolutely continuous functions (without proof) and characterization of absolutely continuous functions as a corollary.

• Lecture 20: (Continuous) monotone functions are differentiable almost everywhere as an application of the rising sun lemma.

• Lecture 21: Outer measure; Caratheodary extension theorem (restricting an outer measure to obtain a complete measure); Pre-measures and proof that the elementary measure extended on elementary algebra is a pre-measure.

• Lecture 22: Hahn-Kolmogorov theorem (extending a pre-mesure to measure); Existence and uniqueness of product measure.

• Lecture 23: Monotone class lemma (presentation for two students); Tonelli's theorem.

• Lecture 24: Tonelli's theorem (complete version); Fubini's theorem, Riesz representation theorem as way of constructing Radon (in particular, Borel) measures (only statement); Signed measure.

• Lecture 25: Radon-Nikodym theorem for finite measures (Assuming Hahn and Jordan decompositions for signed measures).

References

• Terence Tao, An introduction to measure theory, AMS Publication.

• Gerald B. Folland, Real analysis, second ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.

• Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005, Measure theory, integration, and Hilbert spaces.

• Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.