Registry

Module Specifications

Archived Version 2022 - 2023

Module Title
Module Code
School
Online Module Resources
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

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 sigma-algebras, 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 self-contained 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 real-world applications

*Type*	*Hours*	*Description*
Workload	Full-time hours per semester
Lecture	36	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	12	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent Study	78	Corresponding to each lecture students will devote approximately one additional hour of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three and a half hours to solving homework problems which are to be discussed in that tutorial.
Total Workload: 126

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 sigma-additive 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
Beppo-Levi, lemma of Fatou, dominated convergence, convergence in Lp

Assessment Breakdown
Continuous Assessment	%	Examination Weight	%

Type	Description	% of total	Assessment Date
Course Work Breakdown

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
Unavailable

Indicative Reading List

Other Resources

None

Programme or List of Programmes

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