Analysis II @KSoM

/fiveSyllabus

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.

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

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