Analysis I for Research Scholars

• Syllabus:
  1. Measure Theory: Sigma-algebras, measures, outer measures, completion, construction and properties of the Lebesgue measure, non-measurable sets, Measurable functions, point wise convergence, almost uniform convergence, convergence in measure.
     
     
  2. Integration: Lebesgue integration, limit theorems, comparison with the Riemann integral, relationship with differentiation, functions of bounded variation and absolute continuity.
     
     
  3. Signed Measures: Radon - Nikodym theorem, Lebesgue decomposition theorem, change of variable formula, Product Spaces, Fubini's theorem and applications.
     
     
  4. Lp-Spaces : Holder and Minkowski inequalities, completeness, convolutions, approximation by smooth functions, duality.
     
     
  5. Riesz representation theorem: Riesz representation theorem for positive linear functionals, Proof of the theorem, construction of the Lebesgue measure via this approach.
     
     
• Homework Sets
  
  Homework 1
  
  Homework 2
  
  Homework 3
  
  Homework 4
  
  Homework 5
  
  Homework 6
  
  Homework 7
  
  Homework 7 (contd)
  
  Homework 9
  
  Homework 10