Advanced Real Analysis: Course Outline (MAT-623)
Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.
- General Measure spaces; their properties and construction: Measure and their properties, Signed Measure, Hahn and Jordan Decomposition theorems. Caratheodory Measure and outer measure.
- Integration over General Measure: Measurable function, integration of general measurable function and their properties. The Radon-Nikodym Theorem. The Nikodym metric space; Vitali-Hahn-Saks Theorem.
- General Lp spaces: completeness, Riesz representation for dual of Lebesgue spaces; Weak Sequential Compactness in L (X, µ),
- Measure and Topology: Locally Compact Topological Spaces; The Construction of Radon Measures; The Representation of Positive Linear Functionals on C (X); The Riesz Representation Theorem for the Dual of C (X); Regularity Properties of Baire Measures.