Lectures related to Change of Variables Theorem in the Plane

Change of Variables Theorem

Explores the change of variables theorem and examples of polar coordinates.

Analysis IV: Convolution and Hilbert Structure

Explores convolution, uniform continuity, Hilbert structure, and Lebesgue measure in analysis.

Analysis IV: Measurable Sets and Functions

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

Cylindrical Coordinates: Integrability and Volumes

Explores cylindrical coordinates, integrability, and volume calculations using examples.

A Conjecture of Erdös: Proof by Moreira, Richter and Robertson

Presents a short proof of a conjecture by Erdös, exploring related questions and detailed proof of the proposition.

Manifolds: Charts and Compatibility

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Advanced Analysis II: Jordan-Measurable Functions

Explores Jordan-measurable functions and double integrals for volume calculations in 3D space.

Interior Points and Compact Sets

Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.

Multivariable Integral Calculus

Covers multivariable integral calculus, including rectangular cuboids, subdivisions, Douboux sums, Fubini's Theorem, and integration over bounded sets.

Functional Calculus: Simple Functions

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

Analysis IV: Measurable Sets and Properties

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

Measure Spaces: Integration and Inequalities

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

Probability Theory: Integration and Convergence

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

The Riesz-Kakutani Theorem

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

Distributions and Derivatives

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

Lebesgue Measure: Properties and Existence

Covers the properties of the Lebesgue measure and its existence.

Coordinate Systems and Applications

Covers the definition and use of coordinate systems and applications in bases and linear equations.

Probability Measures: Fundamentals and Examples

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

General Manifolds and Topology

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

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.