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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.

Course Outline

  • 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.

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