MAL521 (Measure Theory)

Syllabus (Pg 162) 

There will be 42 lectures of 55 minutes each.

Office Hours: Wednesday, 3:00–4:00 PM. If you have another class during this time, feel free to suggest an alternative time that works for everyone. Other appointments may also be scheduled by email.

Evaluation Plan

• End-Semester Examination: 40 marks

• Mid-Semester Examination: 25 marks

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

• Continuous assessment: (10 - 15  problems + 2 presentations + Class assessment) 15 marks

Topics covered in

• Lecture 01 : Intuitive 'integration' of the Dirichlet's function; Comparison between the ideas of Riemann and Lebesgue integrals with examples (several properties are preserved under limits in the Lebesge theory of integration); why do we need to 'measure sets'.

• Lecture 02 : What are the geometrically intuitive properties that the idea of a 'measure' have.

• Lecture 03 : Issues that arise when we try to measure sets using singleton sets; Sum over an uncountable set and related exercises, length of an interval as a limit.

• Lecture 04 : The same 'limiting formula' will not give us a translation invariant function defined on the power set of ℝ, Elementary sets may be written as disjoint union of intervals.

• Lecture 05 : Definition of elementary measure and its (geometrically intuitive) properties.

• Lecture 06 : Jordan measure and its characterizations.

• Lecture 07 : Jordan measurability and topological boundary.

• Lecture 08 : Introduction to Lebesgue outer measure, proof that the Lebesgue outer measure is not additive, Lebesgue outer measure extends the elementary measure.

• Lecture 09 : 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. Definition and examples of Lebesgue measurable sets.

• Lecture 10 : Measurability criteria and measurability of Cantor set, motivation and definitions of limit inferior and limit superiors of sets, Jordan measurable sets are Lebesgue measurable

• Lecture 11 : Countable additiity of measure; upward and downward monotone convergence theorems; dominated convergence theorem.

• Lecture 12 : Inner regularity of Lebesgue measure.

Quiz 1

• Lecture 13 : Definitions and properties of unsigned and complex-valued simple functions and their integrals; Equivalent notions of measurability of an unsigned function.

• Lecture 14 : Complex valued measurable functions; Definition and finite additivity of Lebesgue unsigned integral.

• Lecture 15 : Markov's inequality and its consequences; triangle inequality; approximation of L^1 functions.

• Lecture 16 : Egorov's theorem; Lusin's theorem

• Lecture 17 : Boolean algebra; sigma algebra and measurable spaces, Finite and countably additive measure

• Lecture 18 : Measure space and its completemess

• Lecture 19 : Measurable functions on measurable space and integration of measurable functions on measure spaces

• Lecture 20 : Escapes to infinity; Monotone convergence theorem

• Lecture 21 : Fatou's lemma; Dominated convergence theorem, 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.