Written material (not by me):
Patrick Billingsley, Probability and Measure. (Ridiculously expensive.)
James Norris, Lecture notes on probability.
Franco Vivaldi, Mathematical writing for undergraduate students.
N.B. AND GRADUATE STUDENTS.
Claudio Landim, Lectures on measure theory
Hao Wu, Martingales and Markov Processes.
Fematika, Measure theory lectures made by a high-school student from Ohio named Lucas.
Written material (by me):
Supplemental notes will be posted for some topics. I usually start the term optimistic that I will post lots of my own notes, and don't find as much time as I hope for writing notes. But I'll do my best.
- My course notes.
- My handwritten lecture notes (large files).
- Random walks and branching processes; a story to motivate the development of the theory.
Will be posted here.
Solution template (LaTeX).
Assignment 1 (Solutions)
Assignment 3 (.tex)
Assignment 4 (.tex)
Assignment 5 (.tex)
Assignment 6 (.tex)
Midterm (and Solutions).
- Lecture 1. Introduction; start of measure theory development. Reference: Outline, my course notes.
- Lecture 2. Carathéodory extension theorem. Reference: my course notes.
- Lecture 3. Dynkin's lemma and Dynkin's theorem; uniqueness of extension. Reference: my course notes.
- Lecture 4. Stieltjes functions and cumulative distribution functions. Existence of measures (in particular Lebesgue measure). Reference: my course notes.
- Lecture 5. Independent events, independent sigma-fields; Borel-Cantelli lemmas. Reference: my notes.
- Lecture 6. Random variables and measurable maps; countable operations with random variables give random variables. Generated sigma-fields, independence of random variables, distribution of a random variable. Reference: my notes.
- Lecture 7. Existence of independent random variables with given distributions. Kolmogorov 0-1 law and examples. Reference: my notes.
- Lecture 8. Types of convergence: almost sure convergence, convergence in probability, convergence in distribution. Implications between types of convergence. Couplings; Skorohod representation theorem for real random variables. Reference: my notes.
- Lecture 9. Defining the integral: simple functions; non-negative functions; L1 functions; linearity of expectation; monotone convergence theorem. Reference: my notes.
- Lecture 10. Inevitability of the definition of the integral. Almost everywhere equivalence. Fatou's lemma; dominated, bounded convergence theorem. Expectations and independence; factorization of expectations for products of independent random variables. Reference: my notes.
- Lecture 11. Monotone class theorem. Examples: the probabilistic method. Densities and the change of variables formula. Reference: my notes.
- Lecture 12. More change of variables examples; computations with random variables. Start of product spaces. Reference: my notes.
- Lecture 13. Product measures and Fubini's theorem; measurability of sections and marginals. Reference: my notes. Reference: my notes.
- Lecture 14. Sums of independent random variables; convolution of CDFs; Markov's inequality and its variants; the L2 weak law of large numbers. Reference: my notes.
- Lecture 15. Chernoff bound and concentration of measure. Weak law of large numbers for L1 random variables. Start of Lacunary strong law of large numbers. Reference: my notes.
- Lecture 16. Lacunary strong law of large numbers and extension to strong law of large numbers. Reference: my notes.
- Lecture 17. Convexity: Jensen's inequality, Hölder and Minkowski's inequalities. Monotonicity of Lp norms; relations between convergence in Lp and almost sure convergence/convergence in probability. Reference: my notes.
Student assessment in this class, like in all McGill classes, is governed by McGill's student assessment policy.
In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.
Conformément à la Charte des droits de l'étudiant de l'Université McGill, chaque étudiant a le droit de soumettre en français ou en anglais tout travail écrit devant être noté (sauf dans le cas des cours dont l'un des objets est la maîtrise d'une langue).
McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/students/srr/honest/ for more information).
L'université McGill attache une haute importance à l'honnêteté académique. Il incombe par conséquent à tous les étudiants de comprendre ce que l'on entend par tricherie, plagiat et autres infractions académiques, ainsi que les conséquences que peuvent avoir de telles actions, selon le Code de conduite de l'étudiant et des procédures disciplinaires (pour de plus amples renseignements, veuillez consulter le site www.mcgill.ca/students/srr/honest/).