18.125. Measure Theory and Analysis.
Spring 2019.

GENERAL INFORMATION.
This course will cover fundamentals of measure theory with applications in probability theory.
Prerequisite: 18.100A, 18.100B, 18.100P, or 18.100Q.

ADMINISTRATION.
Instructor: Nike Sun (nsun at ##). Office hours Monday 3:00-4:00 and Friday 9:30-10:45 in 2-432.
TA: Ao Sun (aosun at ##). Office hours Tuesdays 3:00-4:00 outside 2-390.
## = mit dot edu. Please include "18.125" in the subject line of all emails.
Homework will be announced here and posted on Stellar.

REFERENCES. References marked * are available electronically through libraries.mit.edu.
References for analysis and probability:
[online] T. Tao, An Introduction to Measure Theory. AMS, 2011.
[online] *D. W. Stroock, Essentials of Integration Theory for Analysis. Springer, 2011.
[online] A. Dembo, lecture notes in probability theory.
[online] *O. Kallenberg, Foundations of Modern Probability. Springer, 1997.
E. H. Lieb and M. Loss, Analysis. AMS, 2001.
References for random walk, the heat equation, and related topics:
E. M. Stein and R. Shakarchi, Fourier Analysis. Princeton UP, 2003.
G. Lawler, Random Walk and the Heat Equation. AMS, 2010.
[online] G. Lawler and V. Limic, Random Walk: A Modern Introduction. Cambridge UP, 2010.
[online] R. Lyons and Y. Peres, Probability on Trees and Networks. Cambridge UP, 2016.
M. Barlow, Random Walks and Heat Kernels on Graphs. Cambridge UP, 2017.

GRADING. Homework (20%), midterm (30%), final (45%), class participation (5%). The midterm will be in class on Wednesday March 20, and the final will be in the time slot assigned by the registrar. No makeup exams will be given for any scheduling conflict, including other classes. See the course syllabus (posted on Stellar) for the full grading policy.

SCHEDULE.

  • Lecture 1 (02/06) Introduction: some basic "problems of measure."
    A guessing game on an infinite sequence with axiom of choice (example due to Persi Diaconis).
    Volume, elementary measure, and Jordan measure (Tao 1.1.1-2). Please read Tao 1.1.3.
  • Lecture 2 (02/11) Definition of sigma-field (sigma-algebra); axioms of an abstract measure space.
    Vitali example of a subset incompatible with the natural length measure on the real line.
    Definition of Lebesgue outer measure (beginning of Tao 1.2).
  • Lecture 3 (02/13) Lebesgue outer measure is well behaved on open sets, and is outer regular.
    Definition of Lebesgue measurability by comparison with open sets.
    Proof that the Lebesgue measurable sets form a sigma-field. Tao 1.2.1.
  • Lecture 4 (02/19) Assignment 1 due.
    Characterization of Lebesgue measurability in terms of approximation by open or closed sets.
    Proof that Lebesgue outer measure restricts to a measure on the Lebesgue sigma-field.
    Characterization of Lebesgue measurability in terms of G-delta, F-sigma, and null sets.
    A Borel set that is neither G-delta nor F-sigma (Baire category theorem).
    A Lebesgue set that is not Borel (Cantor function & Vitali construction). Tao 1.2.2, 1.2.3.
  • Lecture 5 (02/20) A very brief review of Riemann integration.
    Integration of nonnegative simple functions.
    Characterization of nonnegative Lebesgue measurable functions.
    (Lower/upper) Lebesgue integral for nonnegative functions. Tao 1.3.1-1.3.3.
  • Lecture 6 (02/25) The Lebesgue integral; the space L^1; the L^1 triangle inequality.
    Approximation of L^1 functions by step functions, or by continuous and compactly supported functions.
    Littlewood's three principles; Lusin's theorem; Egorov's theorem. Tao 1.3.4.
  • Lecture 7 (02/27) Some basic objects of probability: binomial random variables and approximation by gaussians; simple random walk and approximation by Brownian motion (informal discussion).
    Integration of measurable functions on a general measure space; random variables and expectation.
    Markov's inequality, Chebychev's inequality. Tao 1.4.
    For more on random variables see Dembo lecture notes (link above) 1.2-1.3.
    If you are entirely unfamiliar with probability, it would be a good idea to browse an introductory book such as the ones by Sheldon Ross or Jim Pitman.
  • Lecture 8 (03/04) Given Unif[0,1] random variables, generate random variable X with P(X <= x) = F(x)?
    Is the law of a (real-valued) random variable X uniquely determined by its cdf F(x)?
    Theorem: corresponding to each Stieltjes measure function F is a unique measure on the real line.
    Existence: we already constructed Lebesgue measure, so can define X by a pushforward construction.
    Uniqueness: pi-systems, lambda-systems, and Dynkin's pi-lambda theorem.
    Basic integral convergence theorems (assuming sigma-finite measure space): bounded convergence theorem, Fatou's lemma, monotone convergence theorem, dominated convergence theorem.
    Tao 1.4, Dembo 1.1-1.2 (for pi-lambda theorem see Dembo 1.1).
  • Lecture 9 (03/06) Change of variables formula, proved using monotone convergence theorem.
    Comparison with calculus change of variables for random variables with smooth density. Dembo 1.3.
  • Lecture 10 (03/11) A set function that is finitely additive and countably subadditive on a semialgebra extends to a pre-measure on an algebra. A pre-measure on an algebra has a unique extension to a measure on a sigma-algebra (Carathéodory extension theorem). Introduction to product measures. Tao 1.6.
  • Lecture 11 (03/13) Product of (finitely many) measure spaces. Fubini and Tonelli theorems. Tao 1.7 and Dembo 1.4.3.
  • Lecture 12 (03/18) Product of arbitrarily many probability spaces. Kallenberg Chapter 5. Midterm review.
  • Lecture 13 (04/01) Independence of sigma-algebras and random variables.
    L^2 weak law of large numbers. De Moivre--Laplace theorem (binomial central limit theorem).
    Dembo 1.4.1, 1.4.2, 2.1.1.
  • Lecture 14 (04/03) Jensen's inequality, L^p monotonicity. Dembo 1.3.2.
    Moment-generating function (Laplace transform) and characteristic function (Fourier transform).
    A simple case: Fourier transform on the discrete cube {0,1}^n.
  • Lecture 15 (04/08) Fourier transform on {-1,+1}^n. Arrow's theorem. O'Donnell Ch. 2.
  • Lecture 16 (04/10) Schwarz inequality for L^2(R). Convolution with a kernel. Lieb and Loss Ch. 2.
  • Lecture 17 (04/17) The L^2(R) Fourier transform: isometry and unitarity (Fourier inversion formula). Lieb and Loss Ch. 5.
  • Lecture 18 (04/22) Fourier inversion formula for probability measures on the real line. Dembo 3.3.
  • Lecture 19 (04/24) Weak convergence, and relation to convergence of characteristic functions. Tightness, Prohorov theorem, Skorohod representation theorem, portmanteau theorem (on real line). Dembo 3.2.
  • Lecture 20 (04/29) Multidimensional Fourier transform and central limit theorem. Lieb and Loss Ch. 5. Introduction to the heat equation: separation of variables method; discretization and simple random walk interpretation.
  • Lecture 21 (05/01) Dirichlet boundary value problem, harmonic interpolation, Dirichlet energy, Poisson problem on finite graphs. See above for references.
  • Lecture 22 (05/06) Poisson problem on finite graphs. Green's function as inverse of (negative, normalized) Laplacian. Time reversal identity for Green's function.
  • Lecture 23 (05/08) Local central limit theorem, asymptotics of Green kernel on Z^d for d at least 3, Green kernel on R^d. Lawler and Limic Ch. 4; Lieb and Loss Ch. 6.
  • Lecture 24 (05/13) Basic theory of electrical networks: voltage and current, Kirchhoff node law and Kirchhoff cycle law, effective conductance and effective resistance. Lyons and Peres Ch. 2.
  • Lecture 25 (05/15) Review.

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