Measurement theory and Lebesgue integral

• Grant number: 25/12869-9
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2025
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Rodrigo da Silva Rodrigues
• Grantee: João Paulo Araujo Milan
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

This Scientific Initiation project aims to introduce the student to the main concepts of Measure Theory, with a focus on the construction of the Lebesgue integral and the understanding of its theoretical foundations. The research will begin with a historical and conceptual analysis of the development of the notion of the integral, highlighting the limitations of the Riemann formulation and motivating the need for a more general approach. Subsequently, the formal foundations of Measure Theory will be studied, such as $\sigma$-algebras, measurable functions, and outer measures, culminating in the construction of the Lebesgue measure based on Carathéodory's Theorem. The final stage will be dedicated to the main convergence theorems - such as the theorems of Beppo Levi, Fatou, and the Dominated Convergence Theorem - emphasizing how these results significantly expand the applicability of integration. By the end of the project, the student is expected to have developed both technical and conceptual mastery of the theory's foundations, understanding the Lebesgue integral as a powerful and necessary generalization in light of the limitations of the Riemann model. (AU)