Measure-theoretic Probability - M1
This is a core course for first-year M.Math students.
The following syllabus (prone to minor changes and not strictly same as the official syllabus) is only indicative of the topics to be covered. The handwritten notes uploaded so far give an idea of the topics covered
in the class.
Class timings : 9.55 - 10.55 and 11.10 - 12.10 on Tuesdays and Thursdays. Venue : G25.
Syllabus: Review of discrete probability (by Prof. Parthanil Roy) ; Review of sequences and series (Assignment 1) ; Introduction to measure theory (Note 1) ; Sigma-algebras (Note 2) ; Measure (Note 3) ; Outer-measures and Caratheodory extension theorem (Note 4) ; Semirings, Extension, Approximation and Regularity (Note 5); Completion of measures (Note 6) ; Measurable functions (Note 7) ; Integration - Definition and Properties (Note 8) ; Fatou's Lemma, Monotone and Dominated Convergence Theorems (Note 9) ; Notions of Convergence and Littlewood's three principles (Note 10) ; Product measures and Fubini-Tonelli theorem (Note 11) ; Radon- Nikodym theorem (Note 12);
Probability measures, independence, examples and Kolmogorov's consistency theorem (Note 13) ; Borel-Cantelli Lemma, Kolmogorov's 0-1 law and applications to Bond percolation (Note 14); Weak and Strong Laws of Large Numbers (Note 15) ; Weak convergence and central limit theorem (Note 16).
The complete set of hand-written notes are uploaded here.
Additional Topics (not discussed in the class but highly recommended that students study on their own).
1. Generating sigma-fields. See end of Section 2 in Billingsley.
2. Non-existence of purely atomic measures (was discussed in Tutorial 3) or translation invariant measures on the power-set of [0,1]. See end of Section 3 of Billingsley.
3. Cardinality of Lebesgue sigma-algebra and existence of non Lebesgue measurable sets. See Chapter 3 of Dudley or Exercise 12.4 of Billingsley. Or read about Vitali sets.
4. Lebesgue measurable but not Borel measurable sets.
5. Supremum of uncountablly many measurable functions need not be measurable.
6. Lebesgue measurable functions are a.e. equal to Borel measurable functions.
7. Measurability of a Function Almost Equivalent to a Measurable Function
9. A very interesting article by Shafer and Vovk on the world of probability up to Kolmogorov's definitive introduction of the axiomatic foundations.
10. The Mathematics of election forecasting - R. L. Karandikar.
Prerequisites : Basic real analysis (at least some idea of Chapters 1-7 of Rudin's Principles of Mathematical Analysis), discrete probability (say Chapters 1 - 4 of this book or appropriate chapters from these notes). Some topology will be helpful but not more than initial parts of Topology - I.
References :
For my own notes, I shall be referring to the following books.
Patrick Billingsley Probability and measure.
Richard Dudley Real analysis and probability.
Krishna B. Athreya and Soumendra N. Lahiri Measure Theory and Probability Theory.
Terence Tao Introduction to measure theory.
Siva R. Athreya and V. S. Sunder Measure and Probability.
Manjunath Krishnapur Probability Theory - Measure theoretical framework. (Lecture Notes)
Davar Khoshnevinsan Probability.
John B. Walsh Knowing the Odds.
Additional notes or links will be uploaded here.
Assignments : See the attached files below.
Grading :
Assignments : 20 (average of 4 or 5 assignments)
Quizzes : 10 ( max of 2 quizzes).
Mid-sem : 20
End-Sem : 50
For mid and End-sem exam schedule, see here.