A integral de Kurzweil-Stieltjes

Abstract

Neste traballo introduciremos a integral de Kurzweil-Stieltjes e exporemos como, cun sinxelo cambio na clásica definición de Riemann, obtense unha poderosa ferramenta que chega incluso a superar á de Lebesgue. Exporemos dun xeito practicamente completo a teoría que trae consigo esta integral, e retornaremos á definición de Kurzweil-Henstock para destacar algunhas importantes propiedades que herda neste caso. Principalmente faremos fincapé nos teoremas clásicos de converxencia que comparte con Lebesgue-Stieltjes e culminaremos demostrando que a integrabilidade de Kurzweil-Stieltjes abrangue á de Riemann e a de Lebesgue-Stieltjes. Centrarémonos na integración sobre intervalos pechados e limitados da recta real por ser a base da integración sobre unha variable real.

In this dissertation, we will introduce the Kurzweil-Stieltjes integral and explore how a simple change in the classic Riemann definition leads to a powerful tool that even surpasses Lebesgue’s. We will extensively delve into the theory behind this integral and return to the Kurzweil-Henstock definition to highlight some important properties it inherits in this case. Our main focus will be on the classical convergence theorems shared with Lebesgue-Stieltjes, and we will culminate by demonstrating that Kurzweil-Stieltjes integrability encompasses both Riemann’s and Lebesgue-Stieltjes’. We will primarily concentrate on integration over closed and bounded intervals of the real line, as it serves as the fundamental basis for integration over a real variable.