MAL521 (Measure Theory)

Syllabus (Pg 162) 

There will be 42 lectures of 55 minutes each.

Evaluation Plan

• End-Semester Examination: 40 marks

• Mid-Semester Examination: 25 marks

• Quizzes: 2 quizzes, 10 marks each (total 20 marks)

• Assignments: 8 (lecture notes + problems + presentations) total 15 marks

Assignments

There will be two types of assignments in this course.

1. Each students is 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 (extra) 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 measure sets using singleton sets

• Lecture 02 : Sum over an uncountable set and related exercises

• Lecture 03 : Elementary measure and Jordan measure

• Lecture 04 : Characterizations of Jordan measure

• Lecture 05 : Introduction to Lebesgue outer measure, proof that the Lebesgue outer measure is not additive. 

• Lecture 06 : 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)

• Lecture 07 : Lebesgue outer measure is countably additive for almost disjoint boxes and finitely additive for separated sets. Definition and examples of Lebesgue measurable sets

• Lecture 08 : Measurability criteria and measurability of Cantor set, motivation and definitions of limit inferior and limit superiors of sets

• Lecture 09 : Jordan measurable sets are Lebesgue measurable

• Lecture 10 : Countable additiity of measure; upward and downward monotone convergence theorems; dominated convergence theorem

• Lecture 11 : Inner regularity of Lebesgue measure; Comparison between the ideas of Riemann and Lebesgue integrals

• Lecture 12 : Definitions and properties of unsigned and complex-valued simple functions and their integrals; Equivalent notions of measurability of an unsigned function

• Lecture 13 : Complex valued measurable functions; Definition and finite additivity of Lebesgue unsigned integral

• Lecture 14 : Markov's inequality and its consequences; triangle inequality; approximation of L^1 functions

Quiz 1

• Lecture 15 : Egorov's theorem; Lusin's theorem

• Lecture 16 : Boolean algebra; sigma algebra and measurable spaces, Finite and countably additive measure

• Lecture 17 : Measure space and its completemess

• Lecture 18 : Measurable functions on measurable space and integration of measurable functions on measure spaces

• Lecture 19 : Escapes to infinity; Monotone convergence theorem

• Lecture 20 : Fatou's lemma; Dominated convergence theorem

• Lecture 21 : Riesz-Fischer theorem

Mid-semester examination

• Lecture 22 : Different modes of convergence of sequence of functions on a measure space and the relations between these modes of convergence

• Lecture 23 : Uniqueness of limits under mixture of seven modes of convergence and the case of step functions.

• Lecture 24 : Overview of both versions of fundamental theorem of calculus

• Lecture 25 : 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 26 : Proof of the Hardy-Littlewood Maximal inequality in higher dimensions (HLMIHD) using Vitali-type covering lemma

• Lecture 27 : Lebesgue differentiation theorem in multi-dimensions as a consequence of HLMIHD

• Lecture 28 : The rising sun lemma

• Lecture 29 : Upper bound for second fundamental theorem for MND functions as application of Fatou's lemma

• Lecture 30 : Absolutely continuous functions in bounded intervals are of bounded variation

• Lecture 31 : A function is of bounded variation if and only if it is difference of two monotone functions

• Lecture 32 : Second fundamental theorem for absolutely continuous functions (without proof) and characterization of absolutely continuous functions as a corollary

• Lecture 33 : (Continuous) monotone functions are differentiable almost everywhere as an application of the rising sun lemma

Quiz 2

• Lecture 34 : Outer measure

• Lecture 35 : Caratheodary extension theorem (restricting an outer measure to obtain a complete measure)

• Lecture 36 : Pre-measures and proof that the elementary measure extended on elementary algebra is a pre-measure.

• Lecture 37 : Hahn-Kolmogorov theorem (extending a pre-mesure to measure)

• Lecture 38 : Existence and uniqueness of product measure.

• Lecture 39 : Monotone class lemma

• Lecture 40 : Tonelli's theorem (complete version)

• Lecture 41 : Fubini's theorem, Riesz representation theorem

• Lecture 42 : Riesz representation theorem as way of constructing Radon (in particular, Borel) measures.

End-semester examination

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.