Syllabus

Title Of Paper: Real Analysis

Unit-I: Open Sets, Closed Sets and Borel Sets, Lebesgue Outer Measure, The sigma algebra of

Lebesgue Measurable Sets, Countable Additivity, Continuity and Borel-Cantelli Lemma, Non measurable Sets. 15 Lectures

Unit- II: Sums, Product and Composition of Measurable Functions, Sequential Pointwise limits and Simple Approximation. Littlewood’s Three Principles, Egoroff’s Theorem and Lusin’s Theorem, Lebesgue Integration of a Bounded Measurable Function, Lebesgue Integration of a Non-negative Measurable Function.                     15 Lectures

Unit–III: The General Lebesgue Integral, Characterization of Riemann and Lebesgue Integrability, Differentiability of Monotone Functions, Lebesgue’s Theorem, Functions of Bounded Variations: Jordan’s Theorem.                                                        15 Lectures

Unit – IV: Absolutely Continuous Functions, Integrating Derivatives: Differentiating Indefinite Integrals, Normed Linear Spaces, Inequalities of Young, Holder and Minkowski, The Riesz-Fischer Theorem.                                                           15 Lectures

Unit-I: Open Sets, Closed Sets and Borel Sets, Lebesgue Outer Measure, The sigma algebra of

Lebesgue Measurable Sets, Countable Additivity, Continuity and Borel-Cantelli Lemma, Non measurable Sets.                                                                15 Lectures

Unit- II: Sums, Product and Composition of Measurable Functions, Sequential Pointwise limits and Simple Approximation. Littlewood’s Three Principles, Egoroff’s Theorem and Lusin’s Theorem, Lebesgue Integration of a Bounded Measurable Function, Lebesgue Integration of a Non-negative Measurable Function.                     15 Lectures

Unit–III: The General Lebesgue Integral, Characterization of Riemann and Lebesgue Integrability, Differentiability of Monotone Functions, Lebesgue’s Theorem, Functions of Bounded Variations: Jordan’s Theorem.                                                        15 Lectures

Unit – IV: Absolutely Continuous Functions, Integrating Derivatives: Differentiating Indefinite Integrals, Normed Linear Spaces, Inequalities of Young, Holder and Minkowski, The Riesz-Fischer Theorem.                                                           15 Lectures

Unit V: Examples, seminars, group discussions on above four units.                            15 Lectures