MAL521 (Measure Theory-26M)

Syllabus (Pg 175) 

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: (Class assessment + Two presentations) 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 some 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 Jordan measurable sets.

• Lecture 07 : Characterization of Jordan measurable sets.

• Lecture 08 : Jordan measure of a bounded set and its closure.

• Lecture 09 : Jordan measurability from topological boundary.

• Lecture 10 : Introduction to Lebesgue outer measure (LOM), LOM is countably subadditive.

• Lecture 11 : LOM extends the elementary and Jordan measure.

• Lecture 12 : LOM is not finitely additive, LOM is countably additive for almost disjoint boxes and finitely additive for separated sets (presentation  topic for two students), Outer regularity : we express LOM 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 13 : Definition and examples of (Lebesgue) measurable sets (empty sets, singletons, boxes, open sets, countable unions of measurble sets and hence elementary sets).

• Lecture 14 : The collection M of all measurable sets is closed under taking translations, countable intersections and complements, hence closed sets (in particular, Cantor set) are also measurable; Jordan measurable sets belong to M.

• Lecture 15 : Countable additivity of Lebesgue measure; Vitali set as example of a non-measurable set, Borel-Cantelli Lemma, upward and downward monotone convergence theorems, Dominated convergence theorem, Motivation and definitions of limit inferior and limit superiors of sets, Inner regularity of Lebesgue measure (90 minutes).

Quiz 1 (August 24, 2026)

• Lecture 16 : Definition of unsigned simple functions and their integration.

• Lecture 17 : Absolute integrability and integrabtion of complex-valued simple functions.

• Lecture 18 : Equivalent notions of measurability of an unsigned function.

• Lecture 19 : Equivalent notions of measurability of an unsigned function.

• Lecture 20 : Complex valued simple measurable functions; Definition and finite additivity of Lebesgue unsigned integral.

• Lecture 21 : Markov's inequality and its consequences, Complex valued measurable functions.

• Lecture 22 : Measurability of functions defined on subsets of ℝᵈ, Triangle inequality, Approximations in L¹(ℝᵈ) (presentation).

• Lecture 23 : Egorov's theorem and Lusin's theorem (presentations), Boolean algebra with examples.

Mid-semester examination (September 25); Mid-Sem Feedback (September 10-13).

• Lecture 24 : Intersection of algebras, algebra generated by a collection of sets, sigma algebra with examples including Borel and Lebesgue sigma algebras.

• Lecture 25 : Null and Borel sigma algeras are properly contained in the Lebesgue sigma algebra and every Lebesgue measurable set is union of a null and a Borel set.

• Lecture 26 : Measurable spaces; Inverse image of measurable sets under a mesurable function may not be measurable (using Cantor function) but the inverse image of Borel measurable sets is always Borel measurable, as a consequence, right and left slices of Borel measurable sets are Borel measurable; If E and F are Bore measurable then so is E × F.

• Lecture 27 : Finite and countably additive measures; Measure space and its completion, Completion of Borel measure space is Lebesgue measure space; Measurable functions on measurable space and integration of measurable functions on measure spaces

• Lecture 28 : Escapes to infinity; Monotone convergence theorem and its application. 

• Lecture 29 : Fatou's lemma; Dominated convergence theorem; Riesz-Fischer theorem.

• Lecture 30 : Different modes of convergence of sequence of functions on a measure space and the relations between these modes of convergence.

• Lecture 31 : Uniqueness of limits under mixture of seven modes of convergence (without proof), the case of step functions (presentation).

• Lecture 32 : Different modes of convergence of sequence of functions on a measure space and the relations between these modes of convergence, Uniqueness of limits under mixture of seven modes of convergence (without proof), the case of step functions (presentation).

• Lecture 33 : Outer measure, Caratheodary extension theorem (restricting an outer measure to obtain a complete measure). Hausdorff measure is an example of a metric outer measure.

• Lecture 34 : Pre-measures and proof that the elementary measure extended on elementary algebra is a pre-measure.

• Lecture 35 : Hahn-Kolmogorov theorem (extending a pre-mesure to measure), Lebesgue Steiltjes measure, Radon Measure.

Quiz 2 (November 03-06)

• Lecture 36 : Existence and uniqueness of product measure.

• Lecture 37 : Monotone class lemma (presentation).

• Lecture 38 : Tonelli's theorem (complete version).

• Lecture 39 : Fubini's theorem, Riesz representation theorem.

• Lecture 40 : Overview of both versions of fundamental theorem of calculus, Lebesgue differentiation theorem in R : Indefinite integral F of an absolutely integrable function f is absolutely continuous almost everywhere differentiable function with F' = f a.e.

• Lecture 41 (Partially Expository) : Hardy-Littlewood Maximal inequality in higher dimensions (HLMIHD); Lebesgue differentiation theorem in multi-dimensions as a consequence of HLMIHD (without proof); (Continuous) Monotone functions are differentiable almost everywhere as an application of the rising sun lemma; Upper bound for second fundamental theorem for MND functions as application of Fatou's lemma.

• Lecture 42 (Expository):

A.  Absolutely continuous functions in bounded intervals are of bounded variation.

B. A function is of bounded variation if and only if it is difference of two monotone functions, hence every element of BV[a,b] is differentiable a.e.

C. Second fundamental theorem for absolutely continuous functions (SFTACF).

D. A characterization of absolutely continuous functions as a corollary of SFTACF (F:[a,b] --> ℝ is absolutely continuous iff there is a constant c such that F-c is indefinite integral of an element of L¹(ℝ)).

E. 'SFT' fails for BV[a,b], which may be observed using Cantor function.

F. Examples to demonstrate that the set inclusions mentioned below are proper.

End-semester examination (November 24 - December 02)

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.