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. 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). Definition of Lebesgue measurability by comparison with open sets. Proof that the Lebesgue measurable sets form a sigma-field. Tao 1.2.1. 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. 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. 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. 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.
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). Comparison with calculus change of variables for random variables with smooth density. Dembo 1.3. 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. 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.
{-1,+1}^n. Arrow's theorem.
O'Donnell Ch. 2.
L^2(R). Convolution with a kernel. Lieb and Loss Ch. 2.L^2(R) Fourier transform: isometry and unitarity (Fourier inversion formula).
Lieb and Loss Ch. 5.Z^d for d at least 3, Green kernel on R^d.
Lawler and Limic Ch. 4; Lieb and Loss Ch. 6.
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