Lectures related to Analysis IV: Measurable Sets and Properties

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

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

Introduces Cartesian product and induction for proofs using integers and sets.

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

Covers the properties of the Lebesgue measure and its existence.

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

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

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

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

Covers conditional expectation, Fubini's theorem, and their applications in probability theory.

Explores optimal transport in heat equations and metric spaces.

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

Explores the composition of applications in mathematics and the importance of understanding their properties.

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

Covers the Fourier inversion formula, exploring its mathematical concepts and applications, emphasizing the importance of understanding the sign.

Explores countable and uncountable sets, Cantor set, Mandelbrot set, and Box dimension in nonlinear dynamics and complex systems.

Covers the basics of analysis, including real numbers, proofs, sets, and operations.

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Explores mapping functions, surjections, injective and surjective functions, and bijective functions.