Summary of lectures
- Lecture 1, 20/1: Chapter 2 in Schilling. We covered some operations of sets, and defined how one can compare the cardinalities of different sets using injections, surjections, and bijections. The main takeaway was that the cardinality of the rational number Q is countable, and so too is Q^d for any natural number d.
- Lecture 2, 21/1: Chapter 3 in Schilling. We motivated what a measure is (more on that next week), and defined sigma-algebras, the power set of X. For a family of sets G \subset X, we defined sigma(G), the sigma-algebra generated by G. We looked at the Borel sigma-algebra on R^n, and showed that the sigma-algebra generated by the family of half-open rectangles is equal to the Borel sigma-algebra.
- Lecture 3, 27/1: Chapter 4 in Schilling. We covered the properties of measures in Proposition 4.3, gave some examples of measures and presented Lemma 4.8: for additive functions with mu(emptySet) = 0, mu is a measure iff it is continuous from below.
- Lecture 4, 28/1: Chapter 5 in Schilling. We looked at Dynkin systems proved connections between Dynkin systems and sigma algebras generated by a family of subsets calligraphic \subset Powerset(X). And we proved Theorem 5.7 on uniqueness of measures.
- Lecture 5, 3/2: Chapter 6 in Schilling. We talked about semi-rings, extensions of pre-measures and proved that an outer-measure extension indeed is a measure on the sigma-algebra \mathcal{A}^*.
- Lecture 6, 4/2: Chapter 6 in Schilling. We proved that the semi-ring \mathcal{S} is contained in \mathcal{A}^*, stated the Caratheodory extension theorem and proved it using the smaller results on extensions, Lemmas 1-3, that we had developed before that. Then we used Caratheodory to prove that the Lebesgue measure is well-defined on R^d, and showed that it is translation invariant, and up to a constant, the only translation invariant sigma-finite measure on R^d.
- Lecture 7, 11/2: Chapter 7 in Schilling. We defined and studied measurable mappings, showed that continuous mappings on are measurable with respect to Borel sigma-algebras, and defined the pushforward measure.
- Lecture 8 12/2: Chapter 8 in Schilling. We studied measurable functions, introduced simple functions and proved the Sombrero lemma.
- Lecture 9 17/2: Chapter 8 in Schilling. We decomposed measurable functions into positive and negative parts, extended the Sombrero lemma to obtain pointwise convergence of a sequence of simple functions to measurable functions, and looked at measurability properties of various pointwise limits of sequences of measurable functions.
- Lecture 10 18/2: Chapter 9 in Schilling. We defined the integral of non-negative measurable functions, and proved the Beppo-Levi theorem.
- Lecture 11 24/2: Chapter 9 in Schilling. We studied properties of the integral of non-negative functions, looked at applications of the Beppo-Levi theorem and proved Fatou's lemma.
- Lecture 12 25/2: Chapter 10 in Schilling. We extended the integral to so-called mu-integrable signed integrands (a subset of measurable functions denoted by caligraphic L^1(\mu)), and studied properties of such integrals.
- Lecture 13 3/3: Chapters 11 and 12 in Schilling. We studied mu-null sets, properties that hold mu-almost everywhere, showed that mu-null sets do not affect the integral of a function, and covered the Markov inequality and Lebesgue dominated convergence theorem (DCT).
- Lecture 14 4/3: Chapter 12 in Schilling. We relaxed the constraints in Lebesgue dominated convergence theorem to hold almost everywhere rather than everywhere, used DCT to study the regularity and differentiability of parameter-dependent integrals, proved the Montone Convergence theorem, and studied how one can evaluate Lebesgue integrals in some settings through Riemann integration.
- Lecture 15 10/3: Chapter 12 in Schilling. We showed that for measurable and Riemann-integrable functions on intervals [a,b], the Riemann integral equals the Lebesgue integral.
- Lecture 16 11/3: We introduced calligraphicLp function spaces, studied properties of their semi-norm || f ||_p, proved the H?lder inequality and the Minkowski inequality, and shortly discussed the benefits of working on the related quotient space Lp.
- Lecture 17 17/3: We studied convergence on Lp spaces and showed that for any p in [0, \infty], Lp is a complete normed vector space.
- Lecture 18 18/3: We studied continuity and representation properties of convex functions and proved Jensen's inequality.
- Lecture 19 We studied the product sigma algebra, proved the existence and uniqueness of a product measure and covered Theorem 14.5, which is the precursor to Tonelli's theorem, showing that Tonelli's theorem holds for indicator functions.
- Lecture 20 We proved Tonelli's theorem and Fubini's theorem, defined integration of complex-valued functions and studied some applications of the mentioned theorems.
- Lecture 21 We proved the transformation theorems 15.1 and 16.1 for integrals in Schilling (the only parts of those chapters we will cover). We also looked at parts of Chapters 1 and 2 in Rynne and Youngson: normed linear spaces, linear subspaces, Banach spaces, and showed that, unlike for finite-dimensional vector spaces, not every linear subspace of an infinite-dimensional vector space is closed (Example 2.21).
- Lecture 22 Topics from chapters 2 and 3 in Rynne and Youngson. We looked at Riesz lemma (Thm 2.25), showed how that result could be used to generate a sequence of linearly independent vectors with Gram-Schmidt orthogonalization, showed that for an infinite-dimensional normed vector space, the unit ball is not compact (Thm 2.26). We introduced inner products, inner product spaces and Hilbert spaces, studied properties of orthogonal complements (Lem 3.29) and showed that for a Hilbert space H with a closed, non-empty, convex set A \subset H, any point p \in H has a closest point in A (Thm 3.32).
- Lecture 23 Topics from chapters 3 and 4 in Rynne and Youngson: Orthogonal decomposition of vectors (Thm 3.34), orthonormal (ON) sequences in inner product spaces. Bessel's inequality and convergence of ON sequences in Hilbert spaces. ON basis and separability of Hilbert spaces (Thm 3.52). Chapter 4: Space of linear transformations between normved vector spaces U and V, L(U,V), and the linear subspace of bounded linear transformations B(U,V). Lemma 4.1 on continuity of linear operators in B(U,V).
- Lecture 24 Topics from chapters 4 and 5 in Rynne and Youngson: Invertibility of bounded linear operators. Isometric isomorphisms, sufficient conditions for when B(U,V) is complete (Thm 4.27). Dual spaces, and linear functionals (Def 4.28), Riesz-Frechet theorem (Thm 5.2).
- Lecture 25 This lecture was devoted to proving the Hahn-Banach theorem for general real vector spaces, as stated in Theorem 5.13 in Rynne and Youngson. For that purpose, we first introduced sublinear functionals, partial ordering and Zorn's lemma, and proved a "mini Hahn-Banach" lemma for iterative extensions of linear functionals to linear subspaces that are one dimension higher than the base space.
- Lecture 26
Published Jan. 20, 2025 5:26 PM
- Last modified Apr. 22, 2025 2:16 PM