- Syllabus:
- 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.
- Integration:
Lebesgue integration, limit theorems, comparison with the Riemann
integral, relationship with differentiation, functions of bounded
variation and absolute continuity.
- Signed Measures:
Radon - Nikodym theorem, Lebesgue decomposition theorem, change of
variable formula, Product Spaces, Fubini's theorem and applications.
- Lp-Spaces : Holder and Minkowski inequalities, completeness,
convolutions, approximation by smooth functions, duality.
- 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
|