arXiv subjects

Hugo Guadalupe Reyna-Castañeda

Publications and source records attributed to Hugo Guadalupe Reyna-Castañeda.

3 recordsLinked to original sources

arXiv2026

Introduction to Measure and Integration Theory

These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis. The manuscript develops the basic theory of measurable sets, measurable functions, measures on $σ$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory. The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.

math.HO↗

arXiv2025

An Analytic Construction of Random Variables in Lebesgue Spaces

This work develops, from a functional analytic perspective, the construction of random variables in Lebesgue spaces L^p. It extends classical notions of measurability, integrability, and expectation to L^p valued functions, using Pettis's theorem and the Riesz representation theorem to define the Bochner integral as a natural generalization of classical expectation.

math.PR↗

arXiv2025

A variational problem to calculate probabilities

In this paper, we prove the existence and uniqueness of the conditional expectation of an event $A$ given a $σ$-algebra $\mathcal{G}$ as a linear problem in the Lebesgue spaces $L^{p}$ associated with a probability space through the Riesz Representation Theorems. For the $L^{2}$ case, we state the Dirichlet's principle. Then, we extend this principle for specific values of $p$, framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools.

math.PR↗