Registry
Module Specifications
Archived Version 2022  2023
 
Description This module covers the topic of Measure and Integration theory being fundamental for all subsequent courses on Probability and Stochastic Processes. While the general theory discussed in this course covers the concepts of sigmaalgebras, measures, Lebesgue Integral, and convergence theorems, particular examples and applications to probability theory are also provided. Students will attend lectures on the course material and will work, independently, to solve problems on topics related to the course material. The students will have an opportunity to review their solutions, with guidance, at weekly tutorials. Students receive complete and selfcontained lecture notes. Supplementary course material is provided in the reference section of the lecture notes.  
Learning Outcomes 1. Learn important mathematical concepts as used in subsequent courses in Probability Theory and Stochastic Processes 2. Apply theoretical concepts to particular modelling and computational problems related with realworld applications  
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml 

Indicative Content and
Learning Activities Set Theory Sigma  Algebras, Rings, Dynkin System Measure Theory additive and sigmaadditive measures, continuity of measures, construction of measures (Caratheodory theorem) Examples of measures Lebesgue measure, point measures, application to discrete random variables in Stochastics Measurable functions Concept of measurability, elementary functions Lebesgue Integral Construction of the Lebesgue Integral, basic properties of the integral, sets of measure zero, relation to Riemann integral Convergence Theorems BeppoLevi, lemma of Fatou, dominated convergence, convergence in Lp  
 
Indicative Reading List  
Other Resources None  
Programme or List of Programmes  
Archives: 
