**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**

- Teacher: Kishor Kucche Maths