Lectures related to Lebesgue integration

Analysis IV: Convergence Theorems and Integrable Functions

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

Lebesgue Integral: Criteria and Analysis

Explores the concept of Lebesgue integrability and the criteria for Lebesgue integrability, emphasizing the importance of upper and lower integrals.

Lebesgue Integration: Cantor Set

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

Lebesgue Integration: Simple Functions

Covers the Lebesgue integration of simple functions and the approximation of nonnegative functions from below using piecewise constant functions.

Lebesgue Integral: Comparison with Riemann

Explores the comparison between Lebesgue and Riemann integrals, demonstrating their equivalence when the Riemann integral exists.

Lebesgue Integral: Properties and Convergence

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

Integration: Taylor Approximation & Convex Functions

Covers Taylor approximation, convex functions, and integrable properties.

Lebesgue Integral: Definition and Properties

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

Cylindrical Coordinates: Integrability and Volumes

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

Analysis IV: Convolution and Hilbert Structure

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

Negligible Sets: Integrable and Almost Everywhere

Explores negligible sets, integrable sets, and almost everywhere concept in mathematical analysis.

Inverse Limits and Topologies

Explores inverse limits, profinite completions, and Hausdorff topologies in group theory and topology.

Lebesgue Measure and Fourier Analysis

Explores Lebesgue measure, Fourier analysis, PDE applications, and optimal transport in PDEs.

Change of Variables: Integrability and Fubini's Theorem

Explores changing variables in double integrals and applying Fubini's theorem in R² for simplifying calculations.

Continuous Functions: Integrability and Examples

Explores continuous functions and integrability through practical examples.

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Integration on H_pxH and Arithme

Covers integration on H_pxH and arithmetic topics, focusing on the Hensel Lemma and the process of finding R and S such that R.S-P=0.

Existence of Solutions for Poisson-Dirichlet Problem

Covers the existence of solutions for the Poisson-Dirichlet problem, focusing on showing that certain conditions hold for locally bounded and Hölder continuous functions.

Probability Measures: Fundamentals and Examples

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

Lebesgue Measure: Properties and Existence

Covers the properties of the Lebesgue measure and its existence.