Concept

Measurable function

Related lectures (32)

Analysis IV: Measurable Sets and Functions

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

Probability Theory: Lecture 2

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

Lebesgue Integration: Cantor Set

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

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.

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.

Riemann Integral: Properties and Characterization

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

Sub-sigma-fields and Random Variables

Explores sub-sigma fields, random variables, and Borel measurable functions in probability theory.

Sigma Field: Random Variables

Explores sigma fields generated by random variables and their connection to measurable functions.

Optimal Transport: Analysis IV

Explores optimal transport, integration theorems, proof strategies, and function applications in multiple dimensions.

The Riesz-Kakutani Theorem

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

Martingale Inequalities

Explores martingale inequalities, including Chebyshev's and Azuma's, with practical examples and applications.

Functional Calculus: Simple Functions

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

Probability Theory: Basics

Covers the basics of probability theory, including probability spaces, random variables, and measures.

Minkowski's Theorems: Lattices and Volumes

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

Measure Theory: Sets and Algebras

Covers measurability, independence of sets, sigma-algebras, cylinder sets, co-algebras, uniqueness, and extension of measures.

Analysis IV: Measurable Sets and Properties

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

Lebesgue Integral: Properties and Convergence

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

Lattice Theory: Minkowski's Theorems

Delves into lattice theory, emphasizing Minkowski's theorems and their implications on lattice structures.

Probability Measures: Fundamentals and Examples

Covers the fundamentals of probability measures, properties, examples, Lebesgue measure, and terminology related to probability spaces and events.