Office hours: Wednesdays 11:30 - 12:30, Thursdays 12:30 - 1:30 pm.
The 2nd assignment will be posted in the middle of March and will be due April 10th, 2013.
Jan 7th: The Riemann integral was recalled and some issues that raise the need for a more flexible theory of integration were discussed. For example, it was discussed that the metric spaces that we obtain by using the Riemann integral are not complete, and also, it was shown by some examples that changing the order of integration and pointwise limit of a sequence of functions can easily fail. Then sets of measure zero in the Euclidean space were defined. In order to give an overview about the Lebesgue measure and Lebesgue integral, it was mentioned what the Lebesgue integral of the Dirichlet function is.
Jan 9th: The Fouries series and Parseval's identity were recalled. Then, it was discussed that the space of Riemann integrable functions with the L^2 norm is not a complete metric space whereas via the Fourier series it is isometrically imbedded in the space of square summable sequeces on the integers, which is a complete metric space. Then, abstract measure spaces were defined, an overview of the Lebegue integration was given and it was compared with the idea of Riemann integration.
Jan 10th: Counting measure and the Dirac measure were introduced and it was intuitively explained that the Lebegue integral with respect to the counting meausre gives the summation of sequences and with respect to the Dirac measure gives the evaluation of functions at a point. Then, the idea of Lebesgue measure in the Euclidean spaces was explained and its construction started by introducing the extrior measure of a set in R^d.
Jan 14th: Some basic properties of coverings by cubes and rectangles in Euclidean spaces were discussed. Then, the exterior measure of several examples, such as countable sets, the Cantor set, intervals, rectangles, and the whole Euclidean space, were computed. Then, the investigation of general properties of the exterior measure started.
Jan 16th: These important properties of the exterior measure were proved: monotonicity, sub-additivity, approximation from outside by open sets, finite additivity when the sets are dijoint and have strictly positive distance from each other, and additivity for almost disjoint union of cubes. At the end, the measurable subset of Euclidean spaces were defined.
Jan 17th: First, it was shown that open subsets and sets of exterior measure zero in the Euclidean space R^d are measurable. Then, it was proved that the set of measurable subsets of R^d satisfies the axioms of a \sigma-algebra, namely that, it is closed under taking complements, countable unions and countable intersections.
Jan 21st: A very important theorem was stated and proved. That is, it was shown that the Lebesgue measure, which is the restriction of the extrior measure to the measurable sets, is additive on countable disjoint unions of measurable sets.
Jan 23rd: Some corollaries of the theorem proved in the previous lecture were given. First, it was shown that the Lebesgue measure of a countable union of increasing measurable sets in R^d is the limit of their measures. A similar statement was proved for a decreasing sequence of measurable sets which, except finitely many of them, have finite measure. Then, the relation between Lebesgue measurable sets and open and closed subsets of R^d was discussed. That is, it was shown any measurable subset of R^d can be well approximated from outside by open sets and from inside by closed sets, and if the set has finite measure, it can be well approximated from inside by a compact set, and also it can be well approximated by the union of finitely many cubes.
Jan 24th: Invariance properties of the Lebesgue measure in R^d under translation, dilation and reflection were discussed. Then, the construction of a non-measurable subset of R was done. At the end, G_\delta and F_\sigma sets were defined and their relation with Lebesgue measurable subset of R^d was discussed.
Jan 28th: Measurable functions were defined and their basic properties were introduced.
Jan 30th: It was proved that any non-negative measurable function is the point-wise limit of an increasing sequence of simple funtions. A similar statement was also proved for general measurable functions.
Jan 31st: The Lebesgue integral of a non-negative measurable function was defined and its very basic properties were discussed.
Feb 4th: It was shown that any non-negative simple function on a measure space gives rize to a new measure on the space. Also it was proved that the Lebesgue integral is additive on simple functions. Then we had a discussion that these statements still hold when we replace the simple funtions with measurable functions and that one needs to use the Lebesgue monotone convergence theorem to justify that. Then, this theorem was stated and the idea of the proof was given.
Feb 6th: The Lebesgue monotone convergence theorem was proved and few of its corollaries were discussed. At the end, Fatou's lemma was proved.
Feb 7th: It wah shown that any non-negative measurable funtion on a measure space gives rise to a new measure on the space, which is absolutely continuous with respect to the original measure. Then the statement of the Radon-Nikodym theorem was briefly explained. At the end, the space L^1(X, \mu) of complex-valued integrable functions was introduced, and it was shown that the Lebesgue integral defines a complex linear functional on this space.
Feb 11th: Fatou's lemma was recalled and the Lebesuge dominated convergence theorem was proved, which is a very important theorem. Then it was shown that for any sequence of integrable functions with summable L^1-norms, one can change the order of summation and Lebesgue integration.
Feb 13th: A few theorems about the relation between the values that a measurable function attains almost everywhere and the values of its integral over measurable subsets was proved.
Feb 14th: First it was shown that if for a sequence of measurable susets, the sequence of their measures is summable, then almost every point of the space lies in at most finitely many of these sets. Then the coincidence of the Riemann integral and the Lebesgue integral for Riemann integrable funcitons was proved. At the end the statement of the Riesz theorem about represeting positive linear functionals by measures was explained.
Feb 25th: It was shown that the space L^1(X, \mu) of integrable functions on a measure space is a complete metric space.
Feb 27th: The space L^p(X, \mu), 1 \leq p \leq \infty, was introduced, and Holder's inequality and Minkowski's inequality were proved. Convex functions and Jensen's inequality were also discussed.
Feb 28th: It was proved that L^p(X, \mu), 1 \leq p \leq \infty, is a complete metric space, and as a corollary of this, it was shown that if a sequence of functions f_n convereges to a function f in L^p(X, \mu) then there is a subsequece f_{n_i} of f_n that converges to f pointwise almost everywhere. Also the Jensen inequality was stated (for the case when the measure of the total space is 1 and there is a convex function defined on the range of a measurable function on the space), and some of its applicaitons were mentioned.
March 4th: It was shown that simple functions, step functions, and continuous functions with compact support are dense in the space of integrable functions L^1(R^d, m) on the Eudlidean space equipped with the Lebesgue measure. At the end, continuity of translations with respect to the L^1(R^d, m) norm was discussed.
March 6th: Egorov's theorem and Lusin's theorem were proved.
March 7th: After having discussions about Lusin's theorem and recalling its proof, it was explained that L^p(R^d, m), 1 \leq p < \infty, is the completion of continuous functions with compact supporton R^d, C_c(R^d), with respect to L^p norm. Then the question of completion of C_c(R^d) with respect to the L^\infty norm was raised and disussed.
March 11th: It was shown that for any locally campact Hausdorff space X, the completion of C_c(X) with respect to the supremum norm is the space of continuous functions on X that vanish at infinity. Then some questions were raised: 1. Is L^p(R^d, m) separable? 2. Is there a dense subset of L^\infty(R^d, m) that consists of continuous functions? 3. Is there a relation between L^p spaces in general? At the end, some ideas about product measures and Fubini's theorem were discussed and a counterexample for changing the order of two integrals was given.
March 13th: The product \sgima-algebra on the product of two measure spaces was introduced, and it was shown it is the smallest monotone class that contains all elementary sets.
March 14th: \sigma-finite measure spaces were introduced, and the product measure on the product of two such spaces was given and it was proved that it defines a measure.
March 18th: The Fubini theorem was proved. Then it was explained that any measure space has a completion, and was shown that the completion of the product of Lebesgue measures on R^{d_1} and R^{d_2} is the Lebesgue measure on R^{d_1+d_2}. At the end, the convolution product was introduced and it was proved that the convolution of two functions in L^(R^d, m) belongs to this space.
March 20th: It was proved that L^2(R^d, m) is a separable metric space. Then inner products on complex vector spaces were introduced. At the end the Cauchy-Shwartz inequality and the Parallelogram law in inner product spaces were proved.
March 21st: The notion of a Hilbert space was introduced. It was shown that for any point x and any convex closed subset C in a Hilbert space, there exists a unique ponint in C which, among the points in C, attains the minimum distance from x. Using this, it was shown that any closed subspace of a Hilbert space has an orthogonal complement.
March 25th: Bounded linear functionals on Hilbert spaces were introduced and it was proved that they form the space of continuous linear functionals. Then the Reisz representation theorem was proved, namely that any bounded linear functional on a Hilbert space is represented by a vector of the Hilbert space via taking inner products.
March 27th: Signed measures were defined and some examples were given. Then, the total variation of a signed measure was introduced and it was shown that it gives a positive measure. At the end, the notion of \sigma finite signed measures and the notion of sets in which a signed measure is supported were introduced.
March 28th: Two notions were introduced: first, two signed measures being mutually singular, second, a signed measure being absolutely continuous with respect to a positive measure. Then, some related examples were discussed. A proposition was proved, which establishes the relation between the fact that a measure \nu is absolutely continuous with respect to another measure \mu, and the assertion that for any \epsilon > 0 there exists a \delta >0 such that \mu(E) < \delta implies \nu(E) < \epsilon. Then, the Lebesgue-Radon-Nikodym theorem was stated. At the end, the idea of a proof for this theorem, which is due to von Neumann and exploits basic facts from Hilbert space theory, was discussed.
April 1st: The Lebesgue-Radon-Nikodym theorem was proved.
April 3rd: The discussion of the relation between differentiation and Lebesgue integration was started. The fact that differentiation is the left inverse of integration (for example for continuous functions) raises the more general question: for a given integrable function on R^d, do the avarages of this function over balls around a point converge to the value of the function at the point, as the radii of the balls approach 0? In order to answer this question, the Hardy-Littlewood maximal function was introduced and some of its properties were given in a theorem.
April 5th: It was proved that for any integrable function on R^d, for almost every point x in R^d, the avarages of the function over balls around x converge to the value of the function at x, as the radii of the balls approach 0.
April 8th: The Cantor-Lebesgue function was constructed and it was explained that the fundamental theorem of calculus does not hold for this function. Then we found a necessary condition on functions for which the fundamental theorem of calculus holds. At the end, a theorem was discussed, which states that the following three conditions are equivalent for continuous non-decreasing functions: 1. It is absolutely continuous. 2. It sends sets of measure 0 to set of measure 0. 3. The fundamental theorem of calculus holds for the function.
April 10th: The last theorem stated in the previous lecture for non-decreasing continuous functions was proved. Then the total variation of a function was introduced, and was used to show that a function defined on an interval is absolutely continuous if and only if it satisfies the fundamental theorem of calculus.
April 11th: The notion of semi-continuous functions was introduced and a theorem about approximating integrable functions by bounded semi-continuous functions from below and above was proved. This was used to prove that the fundamental theorem of calculus holds for any function defined on an interval which is differentiable everywhere and its derivative is integrable.