Lectures log

Week 1: Definitions of elementary, Jordan, and Lebesgue measure. Motivation for the introduction of the Lebesgue measure. Examples.

Week 2: Standard properties of elementary and Jourdan measure: monotonicity, subadditivity, additivity for disjoint sets. Jordan measurability of elementary sets. Subadditivity and countable subadditivity for outer Lebesgue measure. See, Ex. 1.1.5, 1.1.6, 1.2.1, and 1.2.3 from the book.

Week 3: Outer regularity lemma. The Lebesgue measure of elementary sets, Lebesgue measure of countable union of disjoint boxes. Additivity of Lebesgue measure for separated sets. Representing an open set in Rd as a countable union of almost disjoint boxes. Cantor set and its properties. See Lemma 1.2.12, Ex. 1.2.6, Lemma 1.2.6, Lemma 1.2.9, Ex. 1.2.9

Week 4: Lebesgue measurability of closed sets. Measurability of countable unions and intersections of Lebesgue measurable sets, and measurability of complement of a Lebesgue measurable set, see Lemma 1.2.13. Further properties of Lebesgue measurable sets, see Ex. 1.2.7. Countable additivity of Lebesgue measurable sets, see Lemma 1.2.15.

Week 5: Upward and downward monotone convergence theorem for measurable sets, see Ex.1.2.11. Pointwise limit of Lebesgue measurable sets, see Ex. 1.2.13.

Week 6: Equivalence relations. An example of a Lebesgue non-measurable set, Proposition 1.2.18, and Ex. 1.2.26. Uniqueness of Lebesgue measure, see Ex. 1.2.23.

Week 7: Revisit of Riemann integral, see p.15-17., and its relation to Jordan measure. Unsigned simple functions and their properties. Simple integral. Unsigned measurable functions. Examples. See, Lemma 1.3.4, Ex. 1.3.1.

Week 8: Properties of unsigned measurable functions, Lemma 1.3.9, Ex. 1.3.3, 1.3.4, 1.3.5

Week 9: Unsigned Lebesgue upper and lower integrals. Unsigned Lebesgue integral. Additivity of Lebesgue integral. See Ex. 1.3.10, 1.3.11, Cor. 1.3.14, Ex. 1.3.12, Ex. 1.3.15 (Markov's inequality), Lemma 1.3.15

Week 10: Absolute integrability, Lebesgue integral, L1-spaces, linearly of Lebesgue integral, see Ex. 1.3.19. Triangle inequality, see Lemma 1.3.19, Ex. 1.3.21. Monotone convergence theorem in R.

Week 11: Fatou's Lemma, Dominated convergence theorem and their applications, see Theorem 1.4.44, Cor. 1.4.46, Ex. 1.4.43, Ex.1.4.44, Cor. 1.4.47, Theorem 1.4.49. All proofs are done for the case X=R equipped with the Lebesgue measure. Littlewood's three principles. Approximations of L1-functions, see Theorem 1.3.20. Egorov's theorem, see Theorem 1.3.28.

Week 12: Modes of convergence: pointwise, uniform, locally uniform, L_p, see Examples 1.3.23, 1.3.24, 1.3.25. Lusin's theorem

Week 13: Boolean algebras, sigma-algebras. Examples including Borel sigma-algebras. Abstract measure spaces. Main theorems of the course in case of abstract measure spaces. Review