MAT3105: Real Analysis II

This is a pure mathematics course covering topics needed for physical, life and social science disciplines.  Topics covered include: Lebesgue Measure; Lebesgue Integration; and Convergence and the Lebesgue Integral. 
 
Course Objectives   
On successful completion of this course unit, the learners should be able to: 

• Describe the concept of measure theory in terms of measurable sets and their various types and properties. 
• Describe various types of measurable sets including -sigma algebras, Borel -sigma algebras, and cantor sets.  
• Describe the concept of Lebesgue measure for bounded and unbounded sets. 
• Define and state properties of measurable functions, as well as the concept of preservation of measurability of functions and simple functions. 
• Describe and work with the concept of Lebesgue integral as applies to bounded measurable functions, the elementary properties of the integral, and integral for unbounded functions. 
• Describe the concept of convergence and the Lebesgue integral and the convergence theorem. 
• State and apply the necessary and sufficient condition for Riemann integrability as well as the Ergoff’s and Lusin’s theorem.  

Expected Learning Outcome  
This course unit is meant: 

• To discuss the basic competence in the concepts, principles, analysis and procedures necessary to develop the learners’ habit of critical thinking in pure Mathematics.  
• To encourage orderliness, speed and accuracy in the presentation of mathematics.  
• To help learners acquire the analytic skills of expression in proper mathematical language and using mathematical symbols correctly.  
• To provide instructions that contributes to the learners’ abilities to think critically and solve real life problems, to reason mathematically and apply computational skills.  
• To build a strong foundation in measure theory as preparation for subsequent courses in mathematics and other sciences.  
• To Lay a foundation for postgraduate study in pure Mathematics