MATH7706 Measure Theory and Integration

2016

4.5

3 x 1-hour lectures weekly
1 x 2-hour tutorial weekly

1 2 of MATH2701, MATH2711, MATH2014, MATH2111
1a ENGR2702 - Electrical Circuits and Machines
1b 2 of MATH2111, MATH1141
2 Admission into GCEB-Graduate Certificate in Engineering (Biomedical)
2a Admission into GDPEB-Graduate Diploma in Engineering (Biomedical)
2b Admission into MEB-Master of Engineering (Biomedical)
2c Admission into GCESI-Graduate Certificate in Engineering (Smart Instrumentation)
2d Admission into GDPESI-Graduate Diploma in Engineering (Smart Instrumentation)
2e Admission into MESI-Master of Engineering (Smart Instrumentation)
2f Admission into GCEE-Graduate Certificate in Engineering (Electronics)
2g Admission into GDPEE-Graduate Diploma in Engineering (Electronics)
2h Admission into MEE-Master of Engineering (Electronics)
2i Admission into HBIT-Bachelor of Information Technology (Honours)
2j Admission into HBSC-Bachelor of Science (Honours)
Must Satisfy: (((1 or 1a or 1b)) or ((2 or 2a or 2b or 2c or 2d or 2e or 2f or 2g or 2h or 2i or 2j)))

MATH4706 has been successfully completed

Students undertaking the one year honours programs should check to make sure they have the appropriate background from their undergraduate degree/s.

Lebesgue measure and integration: intervals and volume, outer measure, measurable sets, measurable functions, the integral, repeated integrals, transformations of integrals. General measure and integration: sigma-rings and sigma-algebras, measures, measures generated from outer measures, outer measures on metric spaces, applications (Lebesgue, Hausdorff, spherical, Lebesgue-Stieltjes measures), measurable functions, the integral, repeated integrals, connections with probability theory.

This topic develops the idea of measure as a basis for integration. Such a theoretical development not only permits a robust theory of the ordinary integral, but also allows extensions into more general concepts of measure.

At the completion of the topic, students are expected to be able to:

1. Have a sound knowledge of the measure-theoretical basis of integration
2. Derive general properties of integration
3. Construct general measures and their corresponding integrals
4. Understand the most important applications of general measures