Analysis IV: Measurable Sets and Functions

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

Lebesgue Integral: Definition and Properties

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

Compact Subsets of R^n

Explores compact subsets of R^n, convergence theorems, and set properties.

Geometric Considerations in Rn

Covers the concept of intervals in Rn using geometric balls and defines open and closed sets, interior points, boundaries, closures, bounded domains, and compact sets.

Preliminaries in Measure Theory

Covers the preliminaries in measure theory, including loc comp, separable, complete metric space, and tightness concepts.

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Hausdorff Dimension and Brownian Motion

Explores the Hausdorff dimension of Brownian motion zeros, demonstrating a dimension of 1/2 with certainty.

Properties of Closed Sets

Covers the properties of closed sets and their convergence within subsequences.

Endomorphisms and automorphisms of totally disconnected locally compact groups I

Covers the definitions and basic results of endomorphisms and automorphisms of totally disconnected locally compact groups.