📔 Measure Theory
Graduate Course, IISc, 2023
📌 Logistics: These notes have been compiled from coursework undertaken at IISc.
- Lec-1: Introduction to Measure.
- Lec-2: Definition of Measures.
- Lec-3: Properties of $\sigma$-algebra.
- Lec-4: Outer Measure.
- Lec-5: Continuation of Outer Measure
- Lec-6: Introduction to Lebesgue Measure
- Lec-7: Lebesgue $\sigma$-algebra
- Lec-8: Topological perspective on Lebesgue $\sigma$-algebra
- Lec-9: Proof on Lebesgue $\sigma$-algebra
- Lec-10: Measurable functions
- Lec-11 & Lec-12: Continuation of Measurable functions
- Lec-13: Intergration
- Lec-14: Continuation of Integration
- Lec-15: Monotone Convergence Theorem & Fatou’s Lemma
- Lec-16 & Lec-17: Dominated Convergence Theorem (DCT)
- Lec-18 & Lec-19: Monotone Approximation Theorem
- Lec-20 & Lec-21 and Lec-22: Product Measures
- Lec-23 & Lec-24: Tonelli and Fubini’s Theorem
- Lec-25: Introduction to $L^p$-spaces
- Lec-26: Holder’s and Minkowski’s Inequalities
- Lec-27: Properties of $\mathbb{L}^p(\mu,\mathbb{F})$ spaces
- Lec-28: Reisz-Fisher Theorem
- Lec-29 & Lec-30: Change of Variables formulae
- Lec-31: Signed and Complex Measures
- Lec-32 & Lec-33: Hahn and Jordan Decomposition
- Lec-34: Lebesgue Decomposition
- Lec-35 and Lec-36: Radon-Nikodym Theorem and Applications
- Lec-37: Reisz Representation Theorem