Explorando aspectos del análisis desde un enfoque probabilista: el Teorema de densidad de Lebesgue como una aplicación de la Teoría de Martingalas

Abstract

Within Mathematical Analysis, the Lebesgue Density Theorem is frequently presented as a corollary of the Lebesgue Differentiation Theorem, which requires the Vitali Covering Theorem for its proof. In this paper, a proof is constructed for the Lebesgue Density Theorem without using the Vitali Covering Theorem. For it, a brief review of some results of Measure Theory and Probability Theory is given, emphasizing specially the Martingale Convergence Theorem, through which the necessary tools are built to provide a probabilistic proof of both the Radon-Nikodym Theorem, as the Lebesgue Density Theorem.