Elements of Real Analysis II – Homework 4
Prof. John Chiarelli
Due Date: April 13, 2022

Problem 1.

Let $E$ be a measurable set with $m(E)<\infty$, and $\psi: E \rightarrow \mathbb{R}$ be a simple function with canonical form $\sum_{i=1}^{n} a_{i} \chi_{E_{i}}$. Show that $\psi$ is Lebesgue integrable, and that (using the definition of the Lebesgue integral for all bounded and measurable functions over $E), \int_{E} \psi=\sum_{i=1}^{n} a_{i} m\left(E_{i}\right)$.

Problem 2.

Suppose that the bounded function $f$ on $[a, b]$ is Lebesgue integrable over $[a, b]$. Show that there is a sequence $\left{\eta_{n}\right}_{n=1}^{\infty}$ of finite measurable partitions of $[a, b]$ (i.e. $\eta_{n}=\left{E_{n, i}\right}_{i=1}^{k_{n}}$, and their disjoint union over $i$ is $E$ ) and simple functions $\phi_{n}, \psi_{n}: E \rightarrow \mathbb{R}$ such that:

• for all $n \in \mathbb{N}$ and $x \in E, \phi_{n}(x) \leq f(x) \leq \psi_{n}(x)$.
• for all $n \in \mathbb{N}, i \in\left{1,2, \ldots, k_{n}\right}, \phi_{n}$ and $\psi_{n}$ are constant over $E_{i, n}$.
• $\lim {n \rightarrow \infty} \int{E} \psi_{n}-\phi_{n}=0$.

Problem 3.

Using the definition of $\int_{E} f$, where $E$ is a set of finite measure and $f$ is a bounded measurable function defined over $E$, show that $f(t)=t^{2}+\chi_{\mathbb{Q}}$ is measurable over $E=[0,1]$, and compute $\int_{E} f$.

Problem 4.

Let $E$ be a measurable set such that $m(E)<\infty$, and $f: E \rightarrow \mathbb{R}$ be a bounded, measurable function. Assume $g$ is bounded and $f=g$ almost everywhere on $E$. Show that $\int_{E} f=\int_{E} g$.

Problem 5.

Let $f: E \rightarrow \mathbb{R}$ be a nonnegative and Lebesgue integrable function such that $\int_{E} f<\infty$, and $\epsilon>0$. Show that there is a simple function $\sigma: E \rightarrow$ such that $m({x \in E: \sigma(x) \neq 0})<\infty, 0 \leq \sigma \leq f$ over $E$, and $\int_{E} f-\sigma<$ $\epsilon$.

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Announcements

• Final Exam: will take place 3-6pm on Tuesday, June 7, in AP&M B402A. Please be on time. Bring your own Blue Books in which to write the exam. You may bring two double-sided sheets of notes (that you prepare yourself). No calculators or other electronic devices will be allowed. Here is a Summary of Key Facts for the Final; and here again are the summaries for Exam 1 and Exam 2. Here are 4 Final Practice Problems focusing on material from later in the quarter. Solutions have been posted on Piazza.

Course Information

• Textbook: The required textbook for this course is
Principles of Mathematical Analysis, 3rd edition, by Walter Rudin; ISBN 9780070542358We will cover parts of chapters 4-8 of the text. All posted reading and homeowork assignments will refer to the chapter and section numbers of this textbook. We will also post lecture notes as we go.
• Coursework: There will be weekly homework assignments due on Fridays; they are posted below. There will be two evening midterm exams and a final exam; dates, times, and locations posted below.
• Piazza: We will use Piazza, an online discussion board. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, exams, etc. The instructors will also monitor and post to Piazza regularly. You can sign up here if you are not signed up.
• TritonEd: We will use TritonEd (formerly known as Ted) to disseminate grades. It will not be used for anything else: all course resources are hosted here on the main webpage.

Instructional Staff

Our office hours can be found in the following calendar.Calendar
Class Meetings

Welcome to Math 140B: the second course (of three) introducing the foundations of real analysis (i.e. the rigorous mathematical theory of calculus). According to the UC San Diego Course Catalog, the topics covered are differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions. Most of the material for these topics will be taken from chapters 5-8 of the text by Rudin. (Students may not receive credit for both Math 140B and Math 142B.)

Prerequisite:  Math 140A.

Lecture:  Attending the lecture is a fundamental part of the course; you are responsible for material presented in the lecture whether or not it is discussed in the textbook. You should expect questions on the exams that will test your understanding of concepts discussed in the lecture.

Homework:  Homework assignments are posted below, and will be due at 5:00pm on the indicated due date.  Please turn in your homework assignments in the dropbox in the basement of AP&M.  Late homework will not be accepted. Your lowest two homework scores will be dropped.  It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Regrades:  Homework and midterm exams will be returned in the discussion section.  If you notice an error in the way your homework/exam was graded, you must return it immediately to your TA.  Regrade requests will not be considered once the homework/exam leaves the room.  If you do not retrieve your homework/exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.

Grading: Your course grade will be determined by your cumulative average at the end of the quarter. The letter grades assigned will depend on the performance of the class.

Your cumulative average will be the best of the following two weighted averages:

• 20% Homework,  20% Exam I,  20% Exam II,  40% Final Exam
• 20% Homework,  20% Best Midterm Exam,  60% Final Exam

In addition,  you must pass the final examination in order to pass the course.  Note also: there are no makeup exams, if you miss an exam for any reason then your course grade will be computed with the second two option. There are no exceptions; this grading scheme is intended to accommodate emergencies that require missing an exam.

Academic Integrity:  UC San Diego’s code of academic integrity outlines the expected academic honesty of all studentd and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers respectfully in class. In addition, here are a few of our expectations for etiquette in and out of class.

• Entering/exiting class: Please arrive on time and stay for the entire class/section period. If, despite your best efforts, you arrive late, please enter quietly through the rear door and take a seat near where you entered. Similarly, in the rare event that you must leave early (e.g. for a medical appointment), please sit close to the rear exit and leave as unobtrusively as possible.
• Noise and common courtesy: When class/section begins, please stop your conversations. Wait until class/section is over before putting your materials away in your backpack, standing up, or talking to friends. Do not disturb others by engaging in disruptive behavior. Disruption interferes with the learning environment and impairs the ability of others to focus, participate, and engage.
• Electronic devices: Please do not use devices (such as cell phones, laptops, tablets, iPods) for non-class-related matters while in class/section. No visual or audio recording is allowed in class/section without prior permission of the instructor (whether by camera, cell phone, or other means).
• E-mail etiquette: You are expected to write as you would in any professional correspondence. E-mail communication should be courteous and respectful in manner and tone. Please do not send e-mails that are curt or demanding.

Accommodations:

Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. Students are required to present their AFA letters to Faculty (please make arrangements to contact the instructor privately) and to the OSD Liaison in the department in advance (by the end of week 2, if possible) so that accommodations may be arranged. For more information, see here.

Weekly homework assignments are posted here. Homework is due by 5:00pm on Friday, in the dropbox in the basement of AP&M. Late homework will not be accepted.

Lecture Notes

Here are Lecture Notes, continuing from Math 140A, most recently updated on June 1.

Here are self-contained notes on the Weierstrass Function, which is continuous but nowhere differentiable.