MATH 5453 - Real Analysis I, Section 001 - Fall 2008
TR 10:30-11:45 a.m., 809 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Mon 1:30-2:30 p.m., Wed 1:30-2:30 p.m., or by appointment.

Prerequisite: 4433 (Introduction to Analysis I) or permission of instructor.

Course catalog description: Lebesgue measure and integration theory, absolutely continuous functions, metric spaces. (F)

Text: G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley-Interscience, 1999.

Homework (solutions are deposited after the due date in the Chemistry-Mathematics Library, 207 PHSC):

• Homework 1, due Thu, Sep 4.
• Homework 2, due Thu, Sep 11.
• Homework 3, due Thu, Sep 18.
• Homework 4, due Thu, Sep 25.
• Homework 5, due Thu, Oct 2.
• Homework 6, due Thu, Oct 9.
• Homework 6a, NOT due Tue, Oct 14.
• Homework 7, due Thu, Oct 23.
• Homework 8, due Thu, Oct 30.
• Homework 9, due Thu, Nov 6.
• Homework 10, due Thu, Nov 13.
• Homework 10a, NOT due Thu, Nov 27.
• Homework 11, NOT due Thu, Dec 11.

Content of the lectures:

• Lecture 1 (Tue, Aug 26): Introduction: motivation - Riemann and Lebesgue integrals, idea of measure, incompatibility of conditions (Banach-Tarski paradox). (Sec. 1.1).
  σ-algebras: definition of a σ-algebra (σ-field), elementary examples of σ-algebras (page 21 of Sec. 1.2).
• Lecture 2 (Thu, Aug 28): σ-algebras (cont.): elementary properties of σ-algebras, σ-algebra generated by a subset, elementary concepts of topology (topological space, open and closed sets, compact sets), Borel σ-algebra and Borel sets (pages 21-22 of Sec. 1.2).
  Measures: definition, countable and finite additivity, measurable space, measure space, finite and σ-finite measures (pages 24-25 of Sec. 1.3).
• Lecture 3 (Tue, Sep 2): Measures: semifinite measures, σ-finiteness imlpies semifiniteness (but not conversely); examples of measures, counting measures, point mass (Dirac measure) at a point; basic properties of measures (Theorem 1.8): monotonicity, subadditivity, continuity from above and below; (μ-)null sets, (μ-)almost everywhere ((μ-)a.e.), for (μ-)almost every ((μ-)a.e.) point x; complete measures, completion of measures, theorem on unique completion of measures (Theorem 1.9) (only proof that the extended set is a σ-field) (Sec. 1.3).
• Lecture 4 (Thu, Sep 4): Measures (cont.): theorem on unique completion of measures (Theorem 1.9) (finishing the proof) (Sec. 1.3).
  Outer measure: idea - approximating a set A by "simple" sets containing A; definition of outer measure μ^*, construction of outer measures (Proposition 1.10) (pages 28-29 of Sec. 1.4).
• Lecture 5 (Tue, Sep 9): Outer measure (cont.): a digression on the dangers in rearranging conditionally convergent series and Riemann Rearrangement Theorem; μ^*-measurable sets, Caratheodory's Theorem on constructing σ-algebras and complete measuers from outer measures (pages 29-31 of Sec. 1.4).
• Lecture 6 (Thu, Sep 11): Outer measure (cont.): premeasures, using a premeasure μ₀ defined on an algebra to define an outer measure μ^*; Proposition 1.13: the outer measure μ^* is an extension of μ₀ and every set in the original algebra is μ^*-measurable; Theorem 1.14: extension of a premeasure μ₀ defined on an algebra to a measure μ defined on the σ-algebra generated by the original algebra, uniqueness properties of the measure μ (the restriction of the outer measure μ^* defined through μ₀) (pages 30-32 of Sec. 1.4).
• Lecture 7 (Tue, Sep 16): Borel measures on the real line: Borel σ-algebra, h-intervals, distribution functions, defining a premeasure on the algebra of finite disjoint unions of h-intervals from a distribution function (Proposition 1.15), connnection between Borel measures and the corresponding distribution functions (Theorem 1.16), Lebesgue-Stieltjes measures (pages 33-35 of Sec. 1.5).
• Lecture 8 (Thu, Sep 18): Borel measures on the real line (cont.): regularity properties of the Lebesgue-Stieltjes measures (Lemma 1.17), μ(E) as infimum of the measures of all open sets containing E and as supremum of the measures of all compact sets contained in E (Theorem 1.18), equivalent characterizations of Lebesgue-Stieltjes measurable sets (Theorem 1.19) (pages 35-37 of Sec. 1.5).
• Lecture 9 (Tue, Sep 23): Borel measures on the real line (cont.): more on approximating a sets by a finite union of open sets (Proposition 1.20); Lebesgue measure, Lebesgue measurable sets, invariance of the Lebesgue measure with respect to translations and simple behavior under dilations, Cantor sets - topological properties, Lebesgue measure of the middle-third Cantor set, Cantor sets of non-zero measure (pages 37-39 of Sec. 1.5).
  Measurable functions: measurable functions, criterion for measurability (Proposition 2.1), measurability of functions between metric or topological spaces (Corollary 2.2), Lebesgue and Borel measurable sets, measurability properties of a compositoin of Borel or Lebesque measurable functions (pages 43-44 of Sec. 2.1).
• Lecture 10 (Thu, Sep 25): Measurable functions (cont.): equivalent conditions for measurability (Proposition 2.3), equivalence of the measurability of a complex-valued function and its real and imaginary part (without proof), extended real number system, Borel σ-algebra on the extended real numbers, measurability of f+g, αf, and f+g, measurability of sup_j(f_j), inf_j(f_j), limsup(f_j), liminf(f_j), lim_j→∞(f_j), measurability of max(f,g) and min(f,g), positive and negative parts of a real-valued function, polar decomposition of a complex valued function, indicator functions, simple functions, standard representation of a simple function, existence (and construction) of an increasing sequence of simple functions converging to an arbitrary nonnegative measurable function f (pointwise convergence on X and uniform convergence on any set on which f is bounded) (Theorem 2.10a) (pages 44-47 of Sec. 2.1).
• Lecture 11 (Tue, Sep 30): Measurable functions (cont.): existence of an increasing in module sequence of simlpe functions that converge to an arbitrafy measurable complex-valued function f pointwise and converge to f uniformly on sets where f is bounded (Theorem 10b), measurable functions with respect to a measure and its completion (Proposition 2.11) (pages 47-48 of Sec. 2.1).
  Integration of nonnegative functions: the space L⁺ of all nonnegative measurable functions, integral of a simple function from L⁺ over the whole space and over a subset, properties of these integrals (pages 49-50 of Sec. 2.2).
• Lecture 12 (Thu, Oct 2): Integration of nonnegative functions: integral of a general function from L⁺, elementary properties, the Monotone Convergence Theorem (Theorem 2.14), additivity of the integral (Theorem 2.15), necessary and sufficient condition for an integral to be zero, the MCT for a.e. increasing function sequence from L⁺ (Corollary 2.17) (pages 50-51 of Sec. 2.2).
• Lecture 13 (Tue, Oct 7): Integration of nonnegative functions (cont.): Fatou's Lemma, a.e.-MCT (Corollary 2.19), measures of the pre-images of 0 and of the positive semi-axis of a function from L⁺ with finite integral (Proposition 2.20) (pages 51-52 of Sec. 2.2).
  Integration of complex functions: definition of an integral of a real-valued function f, real-valued integrable functions form a real vector space and the integral is a linear functional on it (Proposition 2.21), definition of an integral of a complex-valued function, equivalence of the integrability of f:X→C and the integrability of Re(f) and Im(f), integrable complex-valued functions form a vector space, the space L¹(μ), |∫gdμ|≤∫|g|dμ for any g∈L¹ (Proposition 2.22) (pages 52-53 of Sec. 2.3).
• Lecture 14 (Thu, Oct 9): Integration of complex functions (cont.): |∫gdμ|≤∫|g|dμ for any g∈L¹ (Proposition 2.22 - finishing the proof), σ-finiteness of the set where an L¹ function is non-zero, conditions equivalent to two functions being equal a.e. (Proposition 2.23), remarks about the space L¹ as a space of equivalence classes, distance and the concept of convergence in L¹, the Dominated Convergence Theorem (pages 53-55 of Sec. 2.3).
• Lecture 15 (Tue, Oct 14): Integration of complex functions (cont.): convergence of a series of L¹ functions and integrating such a series (Theorem 2.25); review for the exam (page 55 of Sec. 2.3).
• Lecture 16 (Thu, Oct 16): Hour exam 1.
• Lecture 17 (Tue, Oct 21): Integration of complex functions (cont.): density of the simple functions and of the compactly supported continuous functions in L¹ (in the L¹ metric) (Theorem 2.26), interchanging the order of integration and taking a limit or differentiation (Theorem 2.27); comparison of Lebesgue and Riemannian integrals: partition P of a compact interval [a,b], "sup" and "inf" sums S_Pf and s_Pf, Riemann integral, Riemann integrable functions, Lebesgue measurability and integrability of bounded Riemann integrable functions on a compact interval, necessary and sufficient condition for a bounded function f on a compact interval to be Riemann integrable in terms of the Lebesgue measure of the points of discontinuity of f (Theorem 2.28) (pages 55-57 of Sec. 2.3).
• Lecture 18 (Thu, Oct 23): Integration of complex functions (cont.): end of the proof of Theorem 2.28(a), discussion of the relationship of Riemann and Lebesgue integrability (pages 57-58 of Sec. 2.3).
  Modes of convergence: definitions of uniform convergence, pointwise convergence, convergence μ-a.e., L¹ convergence, convergence in measure, Cauchy in measure; an initial discussion of the relationships between different kinds of convergence, examples of sequences of functions that converge (or do not converge) in different senses (pages 60-61 of Sec. 2.4).
• Lecture 19 (Tue, Oct 28): Modes of convergence (cont.): convergence a.e. and |f_n|≤g∈L¹ implies convergence in L¹; L¹ convergence implies convergence in measure (Proposition 2.29); convergence in measure implies the existence of a subsequence converging a.e., also, f_n→f and f_n→g in measure implies that f=g a.e. (Theorem 2.30); Egoroff's Theorem (Theorem 2.33) (pages 61-62 of Sec. 2.4).
• Lecture 20 (Thu, Oct 30): Product measures: Cartesian product, projections, product σ-algebra, product measure, (measurable) rectangle, defining a premeasure on the algebra of rectangles and the corresponding outer measure, measure μ×ν on the product algebra, x-section E_x and y-section E^y of a set E⊂X×Y, x-section f_x and y-section f^y of a function f:X×Y→R (pages 64-65 of Sec. 2.5).
• Lecture 21 (Tue, Nov 4): Product measures (cont.): measurability of the x- and y-sections of a function measurable w.r.t. the product σ-algebra (Proposition 2.34), monotone class, the Monotone Class Lemma (pages 65-66 of Sec. 2.5).
• Lecture 22 (Thu, Nov 6): Product measures (cont.): main lemma on measurability of characteristic functions on product spaces (Theorem 2.36) (page 66-67 of Sec. 2.5).
• Lecture 23 (Tue, Nov 11): Product measures (cont.): Fubini-Tonelli Theorem, remarks about the importance of all conditions in the theorem, problems with the completeness of the product measure, Fubini-Tonelli Theorem for complete measures (pages 67-68 of Sec. 2.5).
• Lecture 24 (Thu, Nov 13): The n-dimensional Lebesgue integral (a sketch): n-dimensional Lebesgue measure mⁿ, regularity properties of mⁿ (Theorem 2.40), density of simple functions on product of intervals in L¹(mⁿ), density of compactly supported continuous functions in L¹(mⁿ) (Theorem 2.41), invariance properties of mⁿ (Theorem 2.42) (pages 70-71 of Sec. 2.6, without proofs).
  Signed measures: signed measures, continuity properties of signed measures (Theorem 3.1), positivity of all subsets of a positive sets and of the union of a countable family of positive sets (Lemma 3.2) (pages 85-86 of Sec. 3.1).
• Lecture 25 (Tue, Nov 18): Signed measures (cont.): the Hahn Decomposition Theorem (Theorem 3.3), Hahn decomposition for a signed measure ν, mutually singular measures (pages 86-87 of Sec. 3.1).
• Lecture 26 (Thu, Nov 20): Signed measures (cont.): the Jordan Decomposition Theorem (Theorem 3.4), positive and negative variations of a signed measure, total variation of a signed measure, integration with respect to a signed measure, finite and σ-finite signed measures (pages 87-88 of Sec. 3.1).
  The Lebesgue-Radon-Nikodym Theorem: signed measure that is absolutely continuous with respect to a positive measure, necessary and sufficient condition for absolute continuity of a finite signed measure with respect to a positive measure in ε-δ form (Theorem 3.5) (pages 88-89 of Sec. 3.2).
• Lecture 27 (Tue, Nov 25): The Lebesgue-Radon-Nikodym Theorem (cont.): corollary from Theorem 3.5 for L¹(μ) functions (Corollary 3.6), dichotomy lemma: either μ⊥ν or ∃ε>0 and ∃E such that μ(E)>0 and ν≥εμ on E (Lemma 3.7), the Lebesgue-Radon-Nicodym Theorem (Theorem 3.8) - first part of the proof (pages 89-90 of Sec. 3.2)
• Lecture 28 (Tue, Dec 2): The Lebesgue-Radon-Nikodym Theorem (cont.): the Lebesgue-Radon-Nicodym Theorem (Theorem 3.8) - finishing the proof, Lebesgue decomposition of a signed measure ν with respect to a positive measure μ (both σ-finite), Radon-Nicodym derivative, properties of Radon-Nicodym derivatives (Proposition 3.9, Corollary 3.10), absolute continuity of μ₁+μ₂+…+μ_n with respect to μ if each μ_j<<μ (Proposition 3.11) (pages 90-91 of Sec. 3.2).
• Lecture 29 (Thu, Dec 4): Hour exam 2.
• Lecture 30 (Tue, Dec 9): Complex measures: definition, L¹(ν) for a complex measure ν, integral w.r.t. a complex measure, the Lebesgue-Radon-Nicodym Theorem for complex measures (Theorem 3.12), total variation |ν| (page 93 of Sec. 3.3).
• Lecture 31 (Thu, Dec 11): Complex measures (cont.): basic properties of complex measures (Proposition 3.13), "triangle inequality" for complex measures (Proposition 3.14), another expression for |ν| (Exercise 3.3/21) (page 94 of Sec. 3.3).

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Quizzes: Short pop-quizzes will be given in class at random times; your lowest quiz grade will be dropped. Often the quizzes will use material that has been covered very recently (even in the previous lecture), so you have to make every effort to keep up with the material and to study the corresponding sections from the book right after they have been covered in class.

Homework: Weekly homework assignments will be announced in class, and they will be posted on the course web page. Shortly after the due date, I will post solutions in the Chemistry-Math Library. I expect you to work on the problems for yourself or together with other students when they are assigned. I'll be happy to discuss any aspect of the homework with you during office hours or by appointment. After having worked on the problems yourself, you should then compare your work with my suggested solutions to the homework problems when they are posted. All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. Please take the homework very seriously. Solving problems is an essential part of the learning process.

Exams: There will be two in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are September 30 and November 11.
The final is scheduled for December 19 (Friday), 8:00-10:00 a.m.
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

Grading: Your grade will be determined by your performance on the following coursework:

Academic calendar for Fall 2008.

Course schedule for Fall 2008.

Policy on W/I Grades : Through October 3, you can withdraw from the course with an automatic "W". In addition, From October 6 to December 12, you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after November 3 requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) CHECK THE DATES!!!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it! For more details on the University's policies concerning academic misconduct see http://www.ou.edu/provost/integrity/. See also the Academic Misconduct Code, which is a part of the Student Code and can be found at http://www.ou.edu/studentcode/.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

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