Analysis 1 - Reading

Venue: TBA

Class Timings:  TBA

First Meeting: TBA

Course Syllabus: 

• Measure Theory: Sigma-algebras, measures, outer measures, completion, construction and properties of the Lebesgue measure, non-measurable sets, measurable functions, pointwise 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: Hölder 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.
• Complex measures and Differentiation of measures.

Course Outcome:

• Understand the foundations of measure theory, integration, and Lp-spaces.
• Apply convergence theorems and decomposition results to measure and integration problems.
• Utilize the Riesz representation theorem in functional analysis contexts.