Lebesgueov integral

master's thesis

• Author: Ivana Galić
• Mentor: Ivan Ivec (mentor)
• Committee member: Ivan Ivec (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada jest prikazati konstrukciju Lebesgueovog integrala i prednosti u odnosu na Riemannov integral. U tu svrhu smo, nakon kratkog uvoda, definirali izmjerive funkcije te naveli neka njihova osnovna svojstva. Zatim smo definirali mjeru i naveli njezina svojstva te smo konstruirali mjeru iz vanjske mjere. Nakon toga smo pogledali Lebesgueovu mjeru na euklidskom prostoru. Potom smo konstruirali Lebesgueov integral za jednostavne nenegativne funkcije, nenegativne funkcije, realne i kompleksne funkcije. Zatim smo proučili konvergenciju Lebesgueovog integrala. Na kraju smo naveli još neka dobra svojstva Lebesgueovog integrala u odnosu na Riemannov integral.

Abstract (english)

The aim of this paper is to show the construction of the Lebesgue integral and its advantages over the Riemann integral. For this purpose, after a short introduction, we have defined measurable functions and listed some of their basic properties. After that, we defined the measure and listed its properties and constructed the measure from an outer measure. Then we constructed Lebesgue measure in Euclidean space. We then constructed the Lebesgue integral for simple nonnegative functions, nonnegative functions, real and complex functions. Next, we investigated the convergence of the Lebesgue integral. Finally, we have listed some other good properties of the Lebesgue integral in relation to the Riemann integral.

• Keywords: Lebesgueov integral; Riemannov integral
• Keywords (english): Lebesgue integral; Riemann integral
• Language: croatian
• URN:NBN: urn:nbn:hr:217:467578
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-10 13:44:31