Lectures related to Martingale Inequalities

Analysis IV: Measurable Sets and Properties

Covers the concept of outer measure and properties of measurable sets.

Probability Theory: Integration and Convergence

Covers topics in probability theory, focusing on uniform integrability and convergence theorems.

Lebesgue Integral: Definition and Properties

Explores the Lebesgue integral, where functions self-select partitions, leading to measurable sets and non-measurable complexities.

Probability Theory: Lecture 2

Explores toy models, sigma-algebras, T-valued random variables, measures, and independence in probability theory.

Analysis IV: Measurable Sets and Functions

Introduces measurable sets, functions, and the Cantor set properties, including ternary development of numbers.

The Riesz-Kakutani Theorem

Explores the construction of measures, emphasizing positive functionals and their connection to the Riesz-Kakutani Theorem.

Lebesgue Integration: Cantor Set

Explores the construction of the Lebesgue function on the Cantor set and its unique properties.

Lebesgue Measure: Properties and Existence

Covers the properties of the Lebesgue measure and its existence.

Riemann Integral: Properties and Characterization

Explores the properties and characterization of the Riemann integral on different sets and measurable sets.

Analysis: Measure and Integration

Introduces the course on measure and integration, focusing on developing a new theory to overcome the limitations of the Riemann integral.

Advanced analysis II

Covers Jordan-measurable sets, Riemann-integrability, and function continuity on compact sets.

Measurable Sets: Countable Additivity

Explores the countable additivity of measurable sets and the properties of sigma algebra, highlighting the significance of understanding measurable functions in analysis.

Lebesgue Integral: Properties and Convergence

Covers the Lebesgue integral, properties, and convergence of functions.

Distributions and Derivatives

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Independence and Products

Covers independence between random variables and product measures in probability theory.

Minkowski's Theorems: Lattices and Volumes

Explores Minkowski's theorems on lattices, volumes, and set comparisons.

Measure Spaces: Integration and Inequalities

Covers measure spaces, integration, Radon-Nikodym property, and inequalities like Jensen, Hölder, and Minkowski.

Infinite Coin Tosses: Independence

Explores independence in infinite coin tosses, covering sets, shifts, and T-invariance.

Basic Properties of Conditional Expectation

Covers basic properties of conditional expectation and Jensen's inequality in probability theory.

Functional Calculus: Simple Functions

Covers the extension of functional calculus to simple functions and the concept of *-homomorphism.