- May 24, 2020

1 TAKE-HOME EXAMINATIONS APRIL – JUNE 2020 Please read through all instructions carefully before you start your exam Papers are available from 10:00am UK time on the day of your take-home exam; please refer to your exam timetable for the dates of your take-home exams. Papers will be in Blackboard in the Take Home Exams area. In most cases, the take-home exam will be available for 23 hours and your answer(s) must be submitted before 9.00am UK time the day after your take-home exam opens, unless you have been given a different deadline from your School. Completing your take-home exam Take-home exams require either submission of written work or completion of a Blackboard Test. For written work you need to download the paper and any accompanying documents from Blackboard and your answers can be written in a Word document as you would for coursework (unless you have been given specific instructions from your School). Further instructions for completing Blackboard Tests are in the section below. You are advised only to work on your answers for the duration of the time stated on the front page of the take-home exam paper. You are not expected to work for 23 hours on your answer. Please read your exam paper thoroughly before you start to ensure that you understand what you need to do. Do not exceed the specified word limits where they are stated. You should use 12pt font size, Arial and 1.5 line spacing for written submissions. Please write your candidate number, module code and question answer on the top of each piece of work that you submit. Save your work regularly as you are working on it. You are responsible for the content of the work you upload and for the academic integrity of your answer(s). Complete and upload your answer(s) to the submission point(s) in Blackboard or Turnitin. You are responsible for organising your time and should aim to submit your answer(s) as early as possible to ensure your work is submitted prior to the deadline specified for your take-home exam paper. Full guidance can be found in the Take Home Exams area of this Blackboard course. 2 Completing Blackboard Tests (Multiple Choice exams) Exam papers which consist of Multiple Choice questions will be delivered through an online Blackboard Test. For these exams 1. You will not need to download the exam paper – you will access the questions through a Blackboard Test. 2. The test will normally have a specific time limit, and a timer will display on screen as you take the test. 3. Your test will be accessible for 23 hours, as with other exams, but you can take the test only once. Submission 1. Please check the front of your take-home exam paper and Blackboard instructions for any specific instructions for uploading your work to submission point(s) for each paper. 2. You can submit multiple times for most exams (unless you have been told there is only one submission possible by your School) but you should ensure that your final version is uploaded before the deadline. This does not apply to Multiple Choice examinations, where only one attempt will be allowed. 3. Your Turnitin submission title must include your candidate number and module code. 4. You are responsible for ensuring that you have uploaded the correct document to the correct submission point. Some exams require submission of answers to different questions to different submission points. 5. You are responsible for ensuring that your file has been uploaded successfully. 6. You will not be able to submit anything past the deadline so please allow yourself plenty of time to upload and submit your answer(s). 7. We will not be emailing reminders from the University ahead of exam papers and submission points opening, or non-submissions after the take-home exam submission has closed. 8. Guidance on how to submit files for your exams can be found at https://rdg.ac/takehomeexam Submission instructions for students who are submitting hand written work (e.g. for maths and science-based work, ancient and non-European languages, diagrams and sketches). If your school have specified that the whole of your take home exam should be handwritten and submitted to Blackboard / Turnitin: 1. Include your candidate number, module code and question answer at the top of each piece of your take home exam; 2. Number each page carefully; 3 3. Refer to the TEL Guide on Submitting handwritten exams to Blackboard or Turnitin: Submit scanned/handwritten work (depending on the submission point for your take home exam) for help with your submission. If you need to include hand written sketches, diagrams or images into your typed take home exam, please refer to the TEL Guide on Capturing and inserting images into a document for online submission. Full guidance can be found in the Take Home Exams area of this Blackboard course. Where to get support For technical issues (Blackboard and IT) contact us on +44 (0) 118 378 7049 or raise a ticket via the DTS Self Service Portal. Please note that ‘live’ support is available from 8:00am-6:00pm UK time Monday to Friday, and will be available for limited hours 8:00am-9.30am UK time on Saturday morning during the exam period. For other non-technical queries, please check the exams FAQs on Essentials or email [email protected] Emails need to be sent from your University email account and you should provide your candidate number. Please note that ‘live’ support is available from 8:00am-6:00pm UK time Monday to Friday and will be available for limited hours 8:00am-9.30am UK time on Saturday morning during the exam period. DAS registered students 1. If you have been provided with a green sticker, please attach this to the front page of your answer(s), as you would do for coursework submissions. 2. If you have any additional arrangements including extra time, you should consider the necessary requirements prior to the start of your exam and read the advice in the exam FAQs on Essentials. 3. If you still have queries please contact the Disability Advisory Service (DAS) or email [email protected] Emails need to be sent from your University email account and you should provide your candidate and student number. IMPORTANT – You must read this before you start your exam Academic Integrity We are treating this online examination as a time-limited open assessment. This means that: 1. You are permitted to refer to published materials to aid you in your answers. 2. Published sources should be referenced in the usual manner. 4 3. Over-reliance on published sources is likely to be penalised as poor academic practice. Apart from appropriate referencing, you must ensure that: x the work you submit is entirely your own. x you do not communicate with other students on the topic of this assessment for the whole time the assessment is live. x you do not obtain advice or contribution from any third party, including proof- readers, friends, or family members. x For advice on academic integrity, you can see the University Library’s Academic Integrity Toolkit. You should note that: 1. Failure to adhere to these requirements will be considered a breach of the Academic Misconduct regulations (available here), where the offences of cheating, plagiarism, collusion, copying, and commissioning are particularly relevant. 2. Your exam answers will be run through Turnitin, and the usual similarity reports will be available to markers. Please read and note this statement of originality: By submitting this work I certify that: 1. it is my own unaided work; 2. the use of material from other sources has been properly and fully acknowledged in the text; 3. neither this piece of work nor any part of it has been submitted in connection with another assessment; 4. I have read the University’s definition of plagiarism, guidance on good academic practice and the guidelines set out above; and 5. I will comply with the requirements these place on me. I acknowledge the University may use appropriate software to detect similarities with other third-party material, in order to ensure the integrity of the assessment. I understand that if I do not comply with these requirements the University will take action against me, which if proven and following the proper process may result in failure of the year or part and/or my removal from membership of the University. With best wishes and good luck for your take-home-exams over the coming period. Please read the instructions below before you start the exam. April/May 2020 MA3MTI 2019/0 A 800 UNIVERSITY OF READING MEASURE THEORY AND INTEGRATION (MA3MTI) Two hours Answer ALL questions in section A and at least TWO questions from section B. (If more than two questions from section B are attempted then marks from the BEST two section B questions will be used. If the exam mark calculated in this way is less than 40%, then marks from the other section B questions which have been attempted will be added to the exam mark until 40% is reached). vc1 Page 2 SECTION A 1. (a) Prove that the real intervals (1,1) and (0,1) have the same cardinality. [5 marks] (b) Prove that the real intervals (0, 1) and (1,1) have the same cardinality. [5 marks] (c) Consider the set S of all functions f : N! R. Prove that S is uncountable. [10 marks] 2. (a) Let (X,F) be a measurable space, and let f : X ! R be a function. Define what it means for f to be a measurable function. [2 marks] (b) Suppose that f : X ! R is measurable, and let a, b 2 R be such that a < b. Prove from the definition that {x 2 X : a < f(x) < b} 2 F . [12 marks] (c) Suppose that f : X ! R is measurable, and let U ⇢ (0, 1) be an open set. Prove that {x 2 X : f(x) 2 U} 2 F . Hint: Every open set U ⇢ (0, 1) is a countable union of open intervals. [6 marks] MA3MTI 2019/0 A 800 vc1 Page 3 3. (a) Let (X,F , µ) be a measure space. State the dominated convergence theorem. [5 marks] (b) Calculate the limit lim n!1 Z 1 1 n sin(x/n) x3 dx. Hint: Recall that | sin x| x for all x 0. [15 marks] MA3MTI 2019/0 A 800 vc1 Turn over Page 4 SECTION B 4. Let X be a set. (a) Define what a -algebra F on X is. [5 marks] (b) Let (X,F) be a measurable space, and let (An)1n=1 be a sequence of measurable sets. Prove that A = {x 2 X : x belongs to exactly one set from the sequence} is measurable. [15 marks] 5. Let f : [1,1)! R be Lebesgue-integrable, f 2 L([1,1)). Prove that F (t) = Z 1 1 e xt 2 f(x) dx, t 2 R, defines a continuous function F : R! R. [20 marks] 6. Evaluate the limit lim n!1 Z 1 1 1 + (cos x)n x2 dx. Justify your answer. [20 marks] MA3MTI 2019/0 A 800 vc1 Page 5 7. Let (X,F , µ) be a measure space, and let f : X ! [0,1] be measurable and such that R X f dµ <1. (a) Let fn = n log ⇣ 1 + fn ⌘ . Prove that fn ! f almost everywhere. [8 marks] (b) Prove that lim n!1 Z X n log ✓ 1 + f n ◆ dµ = Z X f dµ. [12 marks] Here log denotes the natural logarithm. Hint: Recall that log(1 + x) x for x 0. [End of Question Paper] MA3MTI 2019/0 A 800 vc1