Lectures related to Measure Spaces: Integration and Inequalities

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

Covers normed spaces, Banach spaces, and Hilbert spaces, as well as dual spaces and weak convergence.

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

Covers the Theoreus Chain Role for Lipschitz functions and its practical applications.

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

Covers the properties of the Lebesgue measure and its existence.

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

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

Introduces Lp spaces, covering norms, inequalities, and integrability of functions.