Metric Spaces and Measure Theory

The first course of the analysis sequence at UCR, taught as Análisis I, following Dr. Pedro Méndez Hernández’s lectures. It falls into two halves. The first builds analysis on a general metric space: norms on \(\mathbb{R}^{d}\), open and closed sets, interior and closure, continuity and its topological characterisations, compactness in both its sequential and open-cover forms, completeness and the completion of a metric space, the Arzelà–Ascoli theorem, connectedness and path-connectedness, and the Baire category theorems.

The second half develops measure theory: functions of bounded variation and the Jordan decomposition, the Riemann–Stieltjes integral, Lebesgue outer measure and measurability, the σ-algebra of Lebesgue measurable sets, the Cantor set and a non-measurable set, measurable functions, the theorems of Egorov and Lusin, convergence in measure, the Lebesgue integral first for non-negative functions and then in general, the monotone and dominated convergence theorems, and finally Lebesgue’s characterisation of Riemann integrability.

These are personal notes I took as a student. They were written in Spanish and translated into English; both versions are available and you can switch between them at any point. They are shared as-is, may contain errors, and are not affiliated with the instructor.