ADVANCED REAL ANALYSIS-I (Optional Paper)

Ordinary number: Order types, Well-ordered sets, Transfinite induction, Ordinal numbers, Comparability of ordinal numbers, Arithmetic of ordinal numbers, First uncountable ordinal ohm  .

Descriptive properties of sets: Perfect sets, Decomposition of a closed set in terms of perfect sets of first category, 2^nd category and residual sets, Characterization of a residual set in a compete metric space, Borel sets of class alpha, ordinal alpha<ohm,  Density point of a set in R, Lebesgue density theorem.

Functions of some special classes: Borel measurable functions of class alpha (alpha<ohm,) and its basic properties, Comparison of Baire and Borel functions, Darboux functions of Baire class one.

Continuity: Nature of the sets of points of discontinuity of Baire one functions, Approximate continuity and its fundamental properties, Characterization of approximate continuous functions.

Henstock integration on the real line: Concepts of d-fine partition of the closed interval [a, b] where d is a positive function on [a, b], Cousin’s lemma, definition of Henstock integral of a functions over the interval [a, b] and its basic properties, Saks-Henstock lemmas and its applications, Continuity of the indefinite integral, Fundamental theorem, Convergence theorems, Absolute Henstock integrability, Characterization of Lebesgue integral by absolute Henstock integral.