Introduction to measure theory

• Grant number: 11/13165-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2011
• End date: August 31, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: José Roberto Nogueira
• Grantee: Patrícia Yukari Sato Rampazo
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil

Abstract

Among all analysis topics, the study of measure theory is certainly the one of greater responsibility. It comprises fundamental theorems that ensure convergence properties that are the essential part of this area. All the functional analysis depends on it, on the other hand this theory has as its base just a little more than the study of infinite series and set theory build up all of its matter. Far from disappointing, the result is often what we might call an amazing engineering mathematics: definition to definition, theorem to theorem, the measure theory creates its topics by connecting them and structuring them logically, so it gives rise to a theory that virtually builds itself. As main objective we hope, through this project the student understand the introductory concepts on the measure theory, such as the Lebesgue integral, convergence in measure, a differentiation of an integral, absolute continuity, among other topics. As more specific objective, we expect that student acquire knowledge of abstract mathematical concepts to prepare her to enter in a master's degree in Mathematics or in Applied Mathematics with more maturity and determination. We remember also that this subject is not treated in any discipline of degree course in Mathematics, so it will be very important for this preparation.