#### Lectures log

`First week`

- Tuesday: Practicalities and introduction to the main topics of the course, which main be found in these slides. We then started talking about signed measures.

- Thursday: Hahn and Jordan decomposition theorems; the total variation \(|\mu|\) of a measure \(\mu\); the normed space \( (\mathscr{M} (\Omega), || \ ||)\), where \(\mathscr{M} (\Omega)\) is the space of all finite signed measures on \(\Omega\) and \(||\mu||:=|\mu|(\Omega)\).

Eugenia's notes on signed measures can be found here, while Tao's blog notes on the same topic (and on the next week's topic) are here.

`Second week`

- Thursday: The proof of the the Lebesgue-Radon-Nikodym theorem and its consequences ––Radon-Nykodym theorem and Lebesgue's decomposition theorem. Here are the slides shown in class, summarizing the relevant definitions and theorems. We still have a few things left to cover on this topic, we will do it next week.

`Third week`

- Tuesday: We finished up the discussion of the Lebesgue-Radon-Nikodym theorem and its consequences: the Radon-Nikodym theorem and the Lebesgue decomposition theorem. The latter also required defining continuous measures and pure point (or discrete) measures. A summary is in theses slides.
- Thursday: We defined the distribution function of a Borel measure on the real line and studied its basic properties. We presented several results relating properties of a measure with those of its distribution function. These topics, and those for next class, are treated in Eugenia's notes (start on page 4), although we will offer more details in class, so take good notes.

`Fourth week`

- Tuesday: Positive linear functionals on the space of continuous functions. The Riesz-Markov-Kakutani representation theorem.
- Thursday: We continued the proof of Riesz-Markov-Kakutani representation theorem. These slides contain the statement of the theorem and the main steps in the proof. I have been following Tao's blog notes on this topic (scroll down to Section 2, Theorem 8). He presents a more general result, for continuous functions on a "locally compact Hausdorff space \(X\)". To avoid dealing with topological concepts you might not be familiar with, our space \(X\) is just the interval \([0, 1]\).

`Fifth week`

- Tuesday: Partition of unity and the completion of all the remaining details in the proof of the Riesz-Markov-Kakutani representation theorem, most importantly, the countable sub-additivity of the extension.

`Sixth week`

- Tuesday: We continued with Tao's Notes 1. We defined the cumulative distribution function (CDF), the probability density function (PDF) of a random variable and gave examples of important distribution functions, namely the uniform distribution, the (negative) exponential distribution (related to a Poisson process) and the standard normal distribution.
- Thursday: We studied the expectation of a random variable, convergence theorems for random variables and basic inequalities for random variables.

`Seventh week`

- Tuesday: We studied the change of variables formula and its application to computing the expectation, moments, and variance of a random variable. We then began discussing product measures and independence, following Tao's Notes 2–more precisely, we formulated and sketched the proof of the existence and uniqueness of the product measure plus Tonelli's and Fubini's theorems.
- Thursday: We studied independent events, independent random variables and independent \(\sigma\)-algebras, continuing with the same notes.

`Eight week`

- Tuesday: The tail \(\sigma\)-algebra, tail events, tail random variables; Kolmogorov's zero-one law and some consequences.
- Thursday: The weak and the strong law of large numbers (LLN), using the moments method. We followed Tao's notes 3.

`Ninth week`

- Monday (make-up lecture for the canceled class on March 16): Some consequences of the LLN: the infinite monkey theorem; Bernstein's probabilistic proof of the Weierstrass approximation theorem of continuous functions by polynomials. Same notes 3 of Terry Tao. The second (or converse) Borel-Cantelli lemma.
- Tuesday: The weak law of large numbers by means of the truncation method (we remove the finite second moment assumption). Begin the central limit theorem, on which we will spend some time.
- Thursday: Preparations for understanding the statement of the central limit theorem, following notes 4 of Terry Tao. More precisely, we have seen that—assuming mean \(0\) and standard deviation \(1\)—the process \(\frac{S_n}{\sqrt{n}}\) is typically bounded, but it does
*not*converge either almost surely or in probability. It will converge in*distribution*though. We discussed this concept thoroughly.

`Tenth week`

- Tuesday: More on convergence in distribution, namely proving that it is equivalent with the pointwise convergence (at all but an at most countable set) of the corresponding cumulative distribution functions. We then formulated the CLT, and discussed the fact that it is a
*universal law*. Finally, we defined the characteristic function of a random variable and gave the idea of the Fourier analysis proof of the CLT.

`Eleventh week`

- Thursday: We have completed the proof of the CLT using Fourier analysis. The crucial part of the argument was one direction in Lévy's continuity theorem relating convergence in distribution with convergence of the characteristic functions. We established this fact using the Fourier inversion formula. We used Terry Tao's notes 4—scroll down to Section 2, "The Fourier-analytic approach to the central limit theorem".

`Twelfth week`

- Tuesday: We began to study topics in dynamical systems and ergodic theory, following the notes of Omri Sarig. We covered section 1.1 on the Poincaré's recurrence theorem and the birth of cool, i.e. of ergodic theory.
- Thursday: We described the abstract, measure theoretical setup of ergodic theory, defined measure preserving dynamical systems and reformulated Poincaré's recurrence theorem in this setting. We described the probabilistic point of view of ergodic theory. We introduced the concept of ergodicity and described equivalent formulations.

`Thirteenth week`

- Tuesday: We proved the equivalence of several formulations of ergodicity, then we defined mixing and explained its intuitive meaning as a weak form of independence.
- Thursday: We began the study of the fundamental examples of measure preserving dynamical systems (MPDS): the torus translation and the Bernoulli shift.

`Fourteenth week`

- Tuesday: We finished with the basic properties of the Bernoulli shift, then we briefly described the doubling map, Arnold's cat map, skew translations and linear cocycles. We formulated the pointwise ergodic theorem of Birkhoff and related it to the law of large numbers.
- Thursday: We proved (as the summit of all our hard work this semester) the pointwise ergodic theorem of Birkhoff following the paper of I. Katznelson and B. Weiss, "A simple proof of some ergodic theorems", published in the Israel Journal of Mathematics 42 (1982), no. 4, 291–296. Here is a copy of the paper.

`Fifteenth week`

- Tuesday: We described some simple applications of Birkhoff's ergodic theorem: a quantitative version of Poincaré's recurrence theorem; the strong law of large numbers; the fact that almost every real number between 0 and 1 is normal relative to any base; the distribution of the first digits of the powers of 2 and Benford's law. We concluded our course with the formulation of Szemerédi's theorem (also proven by Furstenberg using ergodic theory) and that of Green-Tao's theorem on the existence of arbitrarily long arithmetic progressions in the primes.