Math 55003
Theory of Functions of a Real Variable I
Fall 2024

Information on this syllabus is subject to change. Any updates will be posted to this site. In particular, any required changes in course delivery, such as transition to remote delivery only, may necessitate significant modifications to this syllabus.

Homework assignments and important dates

DateEvent
Monday, Aug. 19First day of class
Friday, Aug. 23Last day to add a course
Friday, Aug. 30Last day to drop a course
Monday, Sep. 2Labor day (no class)
HW 1Tuesday, Sep. 3, 11:59 p.m.

(HW 1.1) Let \(\mathcal{F}\) be a family of sets.

  1. Show that \(\subseteq\) is a partial ordering on \(\mathcal{F}\).
  2. When is \(A\in\mathcal{F}\) a maximal element with respect to the relation \(\subseteq\)?
  3. If \(A\in\mathcal{F}\) and \(\mathcal{E}\subseteq\mathcal{F}\), when is \(A\) an upper bound for \(\mathcal{E}\) under the relation \(\subseteq\)?
  4. The relation \(\supseteq\) is also a partial ordering on \(\mathcal{F}\). When is \(A\in\mathcal{F}\) a maximal element with respect to the relation \(\supseteq\)?
  5. If \(A\in\mathcal{F}\) and \(\mathcal{E}\subseteq\mathcal{F}\), when is \(A\) an upper bound for \(\mathcal{E}\) under the relation \(\supseteq\)?

(HW 1.2) Let \(\mathbb{F}\) be an ordered field and let \(\mathbb{N}_{\mathbb{F}}\) be the natural numbers in \(\mathbb{F}\) (in the sense of being the intersection of all inductive sets). Show that if \(n\in\mathbb{N}_{\mathbb{F}}\) and \(n\lt f \lt n+1\), then \(f\notin \mathbb{N}_{\mathbb{F}}\). Use the definition of natural numbers in terms of inductive sets; do not invoke known facts regarding the natural numbers constructed from the Peano axioms. Hint: One approach is to start by showing that if \(n\in\mathbb{N}_{\mathbb{F}}\) and \(n\neq 1\), then \(n-1\in\mathbb{N}_{\mathbb{F}}\). (This is Problem 1.9 in your book but if you use it, I would like you to prove it.)

(HW 1.3) [revised]

  1. \(F=\mathbb{Z}/7\mathbb{Z}\) is a field. Let \(\varphi:F\to\mathbb{R}\) be a ring homomorphism, that is, satisfy \(\varphi(a+b)=\varphi(a)+\varphi(b)\) for all \(a\), \(b\in F\). Show that \(\varphi(a)=0\) for all \(a\in F\).
  2. Let \(F\) and \(R\) be two ordered fields. If \(n\in\mathbb{N}_F\), let \(A_n=\{k\in\mathbb{N}_F:k\leq n\}\). Let \(S=\{n\in\mathbb{N}_F:\)there is a unique injective function \(\varphi_n:A_n\to\mathbb{N}_R\) such that \(\varphi_n(1_F)=1_R\) and \(\varphi_n(k+1_F)=\varphi_n(k)+1_R\) whenever \(k\), \(k+1\in A_n\}\). Show that \(S=\mathbb{N}_F\).
  3. Use the functions \(\varphi_n\) of part (b) to show that there exists a unique injective function \(\varphi:\mathbb{N}_F\to\mathbb{N}_R\).
  4. Show that \(\varphi:\mathbb{N}_F\to\mathbb{N}_R\) is surjective.

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 2Sunday, Sep. 8, 11:59 p.m.

(HW 2.1) Prove that the collection of Borel sets \(\mathcal{B}\) is the smallest \(\sigma\)-algebra that contains \(\{(-\infty,a):a\in\mathbb{R}\}\). (To prove this, you need to prove not only that \(\{(-\infty,a):a\in\mathbb{R}\}\subseteq\mathcal{B}\), but also that no smaller \(\sigma\)-algebra contains \(\{(-\infty,a):a\in\mathbb{R}\}\).)

(HW 2.2) Let \((X,\mathcal{M})\) be a measurable space and let \(\mu\) be a measure on \((X,\mathcal{M})\). Suppose that \(\{E_k\}_{k=1}^\infty\) is a sequence of elements of \(\mathcal{M}\). We do not make any assumptions about disjointness or nondisjointness of the sets \(E_k\). Prove that \(\mu\left(\bigcup_{k=1}^\infty E_k\right)\leq \sum_{k=1}^\infty \mu(E_k)\).

(HW 2.3) Let \(A\), \(B\subset\mathbb{R}\) be bounded. Suppose that \(r=\inf\{|a-b|:a\in A,b\in B\}\gt 0\). Show that \(m^*(A\cup B)=m^*(A)+ m^*(B)\).

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 3Sunday, Sep. 15, 11:59 p.m.

(HW 3.1)

  1. Let \(A\), \(B\subseteq\mathbb{R}\). Assume that \(\sup A\leq \inf B\). Show that \(m^*(A\cup B)=m^*(A)+m^*(B)\).
  2. Let \(E\subseteq\mathbb{R}\) satisfy \(m^*(E)\lt\infty\). Define \(f:\mathbb{R}\to [0,\infty)\) by \(f(a)=m^*(E\cap (-\infty,a))\). Show that \(f\) is continuous.

(HW 3.2) Let \(E\subseteq\mathbb{R}\). Suppose that \(m^*(E)\gt0\).

  1. Show that there is a bounded set \(D\) with \(D\subseteq E\) and with \(m^*(D)>0\).
  2. Suppose that in addition \(m^*(E)<\infty\) and that \(\varepsilon>0\). Show that there is a bounded set \(D\) with \(D\subseteq E\) and with \(m^*(D)>m^*(E)-\varepsilon\).

(HW 3.3) Let \(E\subseteq\mathbb{R}\). Suppose that \(m^*(E)\lt\infty\).

  1. Show that there is a \(G_\delta\) set \(G\) with \(E\subseteq G\) and with \(m^*(G)=m^*(E)\), even if \(E\) is not measurable.
  2. Suppose that there is a \(F_\sigma\) set \(F\) with \(F\subseteq E\) that satisfies \(m^*(F)=m^*(E)\). Show that \(E\) is measurable.

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 4Sunday, Sep. 22, 2024, 11:59 p.m.

(HW 4.1) Let \(E\subseteq\mathbb{R}\). Suppose that, for all \(a\), \(b\in\mathbb{R}\) with \(a\lt b\), we have that \[b-a=m^*((a,b)\cap E)+m^*((a,b)\sim E).\] Show that \(E\) is measurable.

(HW 4.2) Let \(f:\mathbb{R}\to\mathbb{R}\) be continuous. Let \(B\) be a Borel set. Show that \(f^{-1}(B)\) is also a Borel set.

(HW 4.3) Let \(f:\mathbb{R}\to\mathbb{R}\) be continuous and strictly increasing. Let \(B\) be a Borel set. Show that \(f(B)\) is also a Borel set.

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 5Sunday, Sep. 29, 2024, 11:59 p.m.

(HW 5.1) Let \(f:[0,1]\to[0,1]\) be Lipschitz; that is, there exists a constant \(c<\infty\) such that if \(x\), \(y\in [0,1]\) then \(|f(x)-f(y)\leq c|x-y|\). Suppose that \(A\subseteq [0,1]\) is measurable. Show that \(f(A)\) is measurable. Hint: Start with the cases where \(A\) has measure zero and where \(A\) is a \(F_\sigma\)-set and then use Theorem 2.11.

(HW 5.2) Let \((X,\mathcal{A})\) be a measurable space (that is, let \(X\) be a set and let \(\mathcal{A}\) be a \(\sigma\)-algebra over \(X\)). Let \(\mu\) be a function \(\mu:\mathcal{A}\to[0,\infty]\) such that

  1. \(\mu(\emptyset)=0\),
  2. \(\mu(A\cup B)=\mu(A)+\mu(B)\) for all \(A\), \(B\in\mathcal{A}\) with \(A\cap B=\emptyset\),
  3. If \(\{A_k\}_{k=1}^\infty\) is a sequence of sets in \(\mathcal{A}\) such that \(A_k\subseteq A_{k+1}\) for all \(k\geq 1\), then \(\mu\Bigl(\bigcup_{k=1}^\infty A_k \Bigr) =\lim_{n\to\infty}\mu(A_n)\).
Show that \(\mu\) is a measure, that is, is countably additive.

(HW 5.3) Let \(E\subseteq\mathbb{R}\) be measurable and let \(f:E\to\mathbb{R}\) be a measurable function. Let \(B\subseteq\mathbb{R}\) be a Borel set. Show that \(f^{-1}(B)\) is measurable.

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 6Sunday, Oct. 6, 2024, 11:59 p.m.

(HW 6.1) Let \(E\subseteq \mathbb{R}\) be measurable. Find a necessary and sufficient condition on \(E\) such that, whenever \(f\), \(g:E\to\mathbb{R}\) are continuous functions that satisfy \(f=g\) almost everywhere on \(E\), then in fact \(f=g\) everywhere on E.

(HW 6.2) Let \(\{f_n\}_{n=1}^\infty\) be a sequence of measurable functions from \(E\) to~\(\mathbb{R}\), where \(E\subseteq\mathbb{R}\) is measurable. Let \(E_0=\{x\in E:\{f_n(x)\}_{n=1}^\infty\) converges\(\}\). Is \(E_0\) measurable? If so, prove it; if not, give a counterexample. Hint: Write \(\{x\in E:\{f_n(x)\}_{n=1}^\infty\) is Cauchy\(\}\) using unions and intersections.

(HW 6.3) Let \(I\subseteq\mathbb{R}\) be a closed, bounded interval and let \(E\subseteq I\) be measurable. Show that, for each \(\varepsilon>0\), there exists a step function \(h:I\to\mathbb{R}\) and a measurable set \(F\subseteq I\) such that \(h=\chi_E\) on \(F\) and such that \(m(I\sim F)\lt\varepsilon\).

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

Monday, Oct. 14Fall break (no class)
HW 7Tuesday, Oct. 15, 2024, 11:59 p.m.

(HW 7.1) Prove the converse of Tietze's extension theorem in \(\mathbb{R}\). That is, suppose that \(E\subseteq\mathbb{R}\) and that, if \(f:E\to\mathbb{R}\) is continous, then there exists a \(g:\mathbb{R}\to\mathbb{R}\) that is continuous and satisfies \(f=g\) on \(E\). Show that \(E\) is closed.

(HW 7.2) Let \(K\subseteq\mathbb{R}\) be compact (closed and bounded), and let \(f_n\), \(f:K\to\mathbb{R}\). Suppose that \(f\) and each \(f_n\) is continuous, that \(f_n\leq f_{n+1}\) for all \(n\), and that \(f_n\to f\) pointwise on \(K\). Show that \(f_n\to f\) uniformly.

(HW 7.3) Let \(E\subset\mathbb{R}\) be measurable with \(m(E)\lt\infty\), and let \(f:E\to[-M,M]\) be a bounded measurable function. Let \(A\subset E\) be measurable. Show that \(\int_A f=\int_E f\chi_A.\) (Note that we used this problem in the proof of Corollary 4.6, and so you may not use Corollary 4.6 in your solution.)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 8Sunday, Oct. 20, 2024, 11:59 p.m.

(HW 8.1)

(HW 8.2)

(HW 8.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 9Sunday, Oct. 27, 2024, 11:59 p.m.

(HW 9.1)

(HW 9.2)

(HW 9.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 10Sunday, Nov. 3, 2024, 11:59 p.m.

(HW 10.1)

(HW 10.2)

(HW 10.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 11Sunday, Nov. 10, 2024, 11:59 p.m.

(HW 11.1)

(HW 11.2)

(HW 11.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

Friday, Nov. 15 Last day to withdraw from a course.
HW 12Sunday, Nov. 17, 2024, 11:59 p.m.

(HW 12.1)

(HW 12.2)

(HW 12.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 13Sunday, Nov. 24, 2024, 11:59 p.m.

(HW 13.1)

(HW 13.2)

(HW 13.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

Nov. 27–29Thanksgiving break (no class)
Date TBA Complete the online course evaluation on or before the due date. If at least 80% of the class completes the course evaluation before the deadline, (or if the registrar cancels course evaluations in the event of low attendance), I will drop your 2 lowest homework scores; otherwise, I will drop your 1 lowest homework score.
Wednesday, Dec. 4Penultimate day of class
HW 14Sunday, Dec. 8, 2024, 11:59 p.m.

(HW 14.1)

(HW 14.2)

(HW 14.3)

A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.

HW 15TBAAttend class at the date and time reserved for our final exam (as indicated on UAConnect or the registrar’s website) and make a good faith effort to present at least one problem. This is the only day that you are required to attend class and also the only day you are required to present your problem. Any presentation points earned may in addition be applied to the problems on HW 14.

Instructor: Ariel Barton

Contact information:
Email: aeb019@uark.edu
Office: SCEN 348
Email will be reviewed within one business day.

Class time and location: Monday, Wednesday, and Friday, 9:40–10:30 a.m., August 19–December 4, 2024, Kimpel Hall 116.

Office hours: Office hours will be held at the following times and places:

These office hours are on a drop in basis; at these times, you do not need an appointment.

If the selected times do not work for you, or if you wish to speak to me privately, please email me and we will schedule an appointment at another time.

Course Description: Real number system, Lebesque measure, Lebesque integral, convergence theorems, differentiation of monotone functions, absolute continuity and the fundamental theorem of calculus Lp spaces, Holder and Minkowski inequalities, and bounded linear functionals on the Lp spaces.

Prerequisites: MATH 45203 or MATH 52203 (Advanced Calculus II), and graduate standing in mathematics or statistics, or departmental consent.

Course Objectives: During this course, students should:

Textbook: Real Analysis, fifth edition, by Halsey L. Royden and Patrick M. Fitzpatrick, ISBN 9780137906529/9780136853541. Students may also wish to consult Measure, Integration & Real Analysis by Sheldon Axler.

Inquiry-based learning: Students will be supplied with a list of lecture problems (PDF or HTML). This list of problems constitutes the notes for this course. Written homework assignments will be posted to this web page or Blackboard.

Students may choose to either present solutions to lecture problems during class, or to turn in written homework assignments.

Specifically, a student who presents a complete solution at the board during class will be granted 2–24 presentation points (depending on the problem and on originality of their solution). Students may view their remaining presentation points on Blackboard.

If a student does not turn in a written assignment or turns in an assignment with fewer than 3 problems worked, and has remaining presentation points to spend, then 12 presentation points will be deducted per unworked problem and the assignment score will be recorded as though the student had received full credit on the missing problem or problems.

Homework: Homework assignments will be posted to Blackboard. I expect to have 14 regular assignments throughout the semester. I expect to drop your 2 lowest homework scores; your course grade will be determined by your 12 highest homework scores. You may work together on the homework assignments, but each student must write up their work in their own words and submit their own work for grading. Graded homework will be returned via email.

Homework rewrites: You can expect to have three problems on each of your weekly homework assignments. On each homework assignment, you are permitted to rewrite one problem (not two problems or all three problems) whose grade you are unhappy with. The rewrite’s due date will be included in the email you receive returning your homework to you and will generally be at least five days later. Homework may be uploaded to the same location in Blackboard as the original assignment.

Homework formatting: Homework submission is generally required to follow the following rules. If you feel that they place an undue burden on you (for example, if you do not own a computer), talk to me and we will arrange an exception or accommodation.

Exams and course grades: There will be no exams in this course unless the majority of students request that exams be held. Your course grade will be based entirely on your homework scores (including presentation points as indicated above). This means that we will hold an (otherwise as usual) two-hour class on Wednesday, December 13, 3:00–5:00 p.m., or at the date and time reserved for our final exam as indicated on the registrar’s website.

Electronic devices: Cell phones, tablets, laptops, and other electronic devices may be used in class. The expectation is that these devices will be used for taking notes, routine calculations (i.e., calculator apps), accessing course materials, and other course-related uses only. Please do not text or play games in class!

Inclement weather policy: Class will meet unless the University is closed. On-campus students are expected to be present. Off-campus students should make their own decisions in the best interest of personal safety. Off-campus students will not be penalized for being absent on those days the Fayetteville Public Schools are closed due to weather, but are encouraged to attend by Zoom if possible.

If the University is closed due to inclement weather, office hours will be held over Zoom unless the instructor is without electricity or internet access. Class will either be also held over Zoom or cancelled; you will be notified of which by email as soon as possible.

If attendance is severely affected by weather, deadlines and exam dates may be adjusted. Please do not call the Department of Mathematical Sciences with weather-related inquiries. You may contact me for information.

Communication: The best way to communicate with me (beyond coming to my office hours) is by email. You can expect a reply within 24 hours, excluding weekends and holidays. Students are expected to monitor their uark email address regularly and consult the course Blackboard site for important announcements.

Accommodations: Under University policy and federal and state law, students with documented disabilities are entitled to reasonable accommodations to ensure the student has an equal opportunity to perform in class. If any member of the class has such a disability and needs special academic accommodations, please report to the Center for Educational Access (CEA). Reasonable accommodations may be arranged after the CEA has verified your disability.

All accommodations requested by the Center for Educational Access will be fulfilled to the best of the instructor’s ability. It is the student’s responsibility to discuss the implementation of the accommodations with the instructor, either in person or via email.

Certain accommodations will be granted only at the CEA’s request. Some accommodations must be fulfilled using the CEA’s resources. Other accommodations are available to any student who makes the request in a timely fashion. Please consult with your instructor in all such cases.

Classroom Etiquette: Students and instructors each have an important role in maintaining a classroom environment optimal for learning, and are expected to treat each other with respect during class, using thoughtful dialogue, and keeping disruptive behaviors to a minimum. Both students and faculty perceive abusive language directed towards others as the most disruptive behavior. Other behaviors that can be disruptive of in person classes are chatting and whispering during class, the use of smartphones or laptops for texting or in other ways unrelated to the course, preparing to leave before class is over, and consistently arriving late to class. Inappropriate behavior in the classroom may result in a request to leave the class and/or a referral to the Office of Academic Integrity and Student Conduct.

Recording of Class Lectures: The lecture will be recorded on the room’s camera and recordings will be posted to Blackboard for student reference. These recordings are provided “as is” as a hopefully-useful supplement, but their availability and quality is not guaranteed and no make-up recordings or materials will be posted in the event of technical issues (such as the room's camera recording the wrong wall).

Students who need or particularly desire high-quality reliable recordings are allowed to record lectures with their own devices for their own use.

In addition, if the University is open but the weather is severe, a Zoom meeting will be opened during class so that students may participate remotely in real time.

By attending this class, the student understands the course may be recorded and consents to being recorded for official university educational purposes. Be aware that incidental recording may also occur before and after official class times. Recordings may include personally-identifiable comments submitted to the chat stream during class.

Unauthorized Use of Class Recordings: These recordings may be used by students only for the purposes of the class. Students may not download, store, copy, alter, post, share, or distribute in any manner all or any portion of the class recording, (e.g. a 5- second clip of a class recording sent as a private message to one person is a violation of this provision). This provision may protect the following interests (as well as other interests not listed): faculty and university copyright; FERPA rights; and other privacy interests protected under state and/or federal law. Unauthorized recording, or transmission of a recording, of all or any portion of a class is prohibited unless the recording is necessary for educational accommodation as expressly authorized and documented through the Center for Educational Access with proper advance notice to the instructor. Unauthorized recordings may violate federal law, state law, and university policies. Student-made recordings are subject to the same restrictions as instructor-made recordings. Failure to comply with this provision will result in a referral to the Office of Student Standards and Conduct for potential charges under the Code of Student Life. In situations where the recordings are used to gain an academic advantage, it may also be considered a violation of the University of Arkansas’ academic integrity policy.

Grade Disputes: The instructor is committed to keeping students informed of their standing in the class. Scores on all graded items will be posted in a timely manner. Students are expected to bring any possible errors to the attention of the instructor within one week of posting. This maintains an accurate grade record throughout the semester. All scores posted before Dead Day will be deemed accurate unless a possible error is brought to the attention of the instructor before the scheduled final exam.

Incompletes: Only given in extreme circumstances, and only when the student has satisfactorily completed all but a small portion of the work in the course. Students must make prior arrangements with the professor well before the end of the semester.

Academic Integrity: As a core part of its mission, the University of Arkansas provides students with the opportunity to further their educational goals through programs of study and research in an environment that promotes freedom of inquiry and academic responsibility. Accomplishing this mission is only possible when intellectual honesty and individual integrity prevail. Each University of Arkansas student is required to be familiar with and abide by the university’s ‘Academic Integrity Policy’ at http://honesty.uark.edu/policy/index.php. Students with questions about how these policies apply to a particular course or assignment should immediately contact their instructor.

Unauthorized Websites or Internet Resources: There are many websites claiming to offer study aids to students, but in using such websites, students could find themselves in violation of our University’s Academic Integrity and Code of Student Life policies. These websites include (but are not limited to) Quizlet, Bartleby, Course Hero, Chegg, and Clutch Prep. The U of A does not endorse the use of these products in an unethical manner. These websites may encourage students to upload course materials, such as test questions, individual assignments, and examples of graded material. Such materials are the intellectual property of instructors, the university, or publishers and may not be distributed without prior authorization. Furthermore, paying for academic work to be completed on your behalf and submitting it for academic credit is considered ‘contract cheating’ per the Academic Integrity Policy. Students found responsible for this type of violation face a grading penalty of ‘XF’ and a minimum one-semester academic suspension per the University of Arkansas Sanction Rubric. Please let us know if you are uncertain about the use of a website.

Intellectual Property: Notes, review material, exams, quizzes, videos or other learning material used in this class are the intellectual property of the instructor. Selling or freely sharing this content in electronic or written form is a violation of intellectual property rights and also constitutes a violation of the University’s academic integrity policy. Your continued enrollment in this class signifies your understanding of and your intent to abide by this policy. There are severe consequences for sharing class content online.

Emergency Procedures: Many types of emergencies can occur on campus; instructions for specific emergencies such as severe weather, active shooter, or fire can be found at https: //safety.uark.edu/emergency-preparedness/.

Health and wellness: https://catalog.uark.edu/generalinfo/studentaffairs/#healthcentertext