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. Representation of 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, 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. Equivalence relations. An example of a Lebesgue non-measurable set, see Ex. 1.2.14 and Proposition 1.2.18.

Week 6: Uniqueness of Lebesgue measure, see Ex. 1.2.23. Pointwise limit of Lebesgue measurable sets, see Ex. 1.2.13. Revisit of Riemann integral, see p.15-17., and its relation to Jordan measure.

Week 7: 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, see Lemma 1.3.9, Ex. 1.3.3, Ex. 1.3.5, and Remark 1.3.10 (very unexpected statement)

Week 9: Upper and lower unsigned Lebesgue integral and their properties, see Ex 1.3.10. Unsigned Lebesgue integral. Finite additivity of the Lebesgue integral, see Cor. 1.3.14. Markov's inequality, see 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.

Week 11. Littlewood's three principles. Approximations of L1-functions, see Theorem 1.3.20. Modes of convergence: pointwise, uniform, locally uniform, L_p, see Examples 1.3.23, 1.3.24, 1.3.25. Lusin's theorem, Egorov's theorem, see Theorem 1.3.26, Theorem 1.3.28.

Week 12. Monotone convergence theorem, 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^d equipped with the Lebesgue measure.

Week 13. Bolean algebras, sigma-algebras. Examples including Borel sigma-algebras. Absract measure spaces. Main theorems of the course in case of abstract measure spaces.

Week 14. Review