MATH GR6151

Probability and Analysis; the construction of measure space; the (magnificent) Strong Law of Large Numbers; Martingale theory;

measure theory, some basic functional analysis stuff like Banach spaces, Hilbert spaces. Probability: laws of large numbers, central limit theorems, Markov chains, Brownian motions

Measure Theory, Functional Analysis, Probability Theory

Measure Theory Lp spaces Probability basics

Measure theory; elements of probability; elements of Fourier analysis; Brownian motion.

Foundations of measure theory, convergence theorems in measure theory, total variations, regularity and Polish spaces, Lp spaces and associated inequalities, Banach spaces and Hilbert spaces, Riesz representation theorems, signed measure and Radon-Nikodym theorem, product measures; Independence, Kolmogorov's 0- 1 law, LLNs, weak convergence, CLT, conditional expectation, Gaussian variables, Markov chains and introductory Brownian motion.

I learned these things/ learned to do them better than before measure theory, proof-writing, probability theory, martingales, problem-solving, thinking, hard work

It is a very intense and thorough introduction to analysis and fundamental probability. 1 Columbia University: Arts & Sciences Fall 2022 Course: MATHGR6151_001_2022_3-ANALYSIS&PROBABILITYI : MATHGR6151_001_2022_3 - ANALYSIS & PROBABILITY I Instructor: Julien Dubedat

Mostly probability, with measure theory and a little ergodic theory.

The fundamentals of measure theory: Measure, Monotone Class Theorem (useful and important), Borel Measure and CDF, L^p space and its duality, Absolutely Continuous and the famous Radon-Nikodym Theorem, with applications to signed measure, Fubini's Theorem and Jordan-Decomposition. Basic Theory of Functional Analysis: Hahn-Banach Theorem, Hilbert space technique and Fourier technique. Theory of Probability: The continuation of measure theory, with an introduction to conditional expectation. Different versions of Lae of Large Number is presented in details. Notions of convergence is introduced, with an emphasis on convergence in distribution and characteristic functions technique. It ends up with the rigorous theory of Markov Chains. This is my 3-rd time learning real-analysis and measure theory, and yet I stilled learned a lot from the course, especially from the perspective of abstract measure: This time I'm more clear about how abstract measure is constructed (without being confined to Lebesgue measure in R^n), and the properties of measures in topological spaces (e.g., regularity of finite/sigma-finite Borel measure on Polish space), also the applications of Monotone Class Theorem. The beautiful Radon- Nikodym Theorem upgrades my understanding of absolute continuity (in both terms of functions and measures) and Riesz Representation theorem, which also makes one more clear ab…