{"id":295,"date":"2023-06-13T13:24:49","date_gmt":"2023-06-13T20:24:49","guid":{"rendered":"https:\/\/live-usc-dornsife.pantheonsite.io\/larry-goldstein\/?page_id=295"},"modified":"2023-10-27T14:16:18","modified_gmt":"2023-10-27T21:16:18","slug":"math-525a-real-analysis","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/","title":{"rendered":"Math 525a &#8211; Real Analysis"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <p><big><strong>Topics:<\/strong>\u00a0Measure and integration over abstract measure spaces, Radon-Nikodym theorem, Fubini&#8217;s theorem, convergence theorems, differentiation.<\/p>\n<p><strong>Instructor:<\/strong>\u00a0<a href=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/\">Larry Goldstein<\/a>, KAP 406D, x02405<br \/>\n<\/big><big>Lecture\u00a039735R<\/big><big>, KAP 140<\/big><big><br \/>\n<\/big><big>Office hours: M 12:50-1:50, W 11-12<\/big><br \/>\n<big><br \/>\n<strong>Grader:<\/strong>\u00a0Fei Wang, wang828 at usc dot edu<br \/>\n<strong>Office hours:<\/strong>\u00a0M 5-7 and F 11-12 in the Math Center<\/big><\/p>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <p>&nbsp;<\/p>\n<h2 style=\"text-align: left;\"><big>Materials<\/big><\/h2>\n<p><big><\/big><big><strong>Required Text:<\/strong>\u00a0Real Analysis, Modern Techniques and Their Applications, Second Edition<br \/>\nGerald Folland, Wiley Interscience.\u00a0<a href=\"http:\/\/www.math.washington.edu\/~folland\/Homepage\/reals.pdf\">Errata<\/a><br \/>\n<\/big><br \/>\n<big><strong>Supplemental References:\u00a0<\/strong><br \/>\n<\/big><\/p>\n<ul>\n<li><big>\u00a0Real and Complex Analysis, Walter Rudin\u00a0 \u00a0 \u00a0<\/big><\/li>\n<li><big>\u00a0Real Analysis, Royden<\/big><\/li>\n<li><big>\u00a0Measure Theory, Halmos<\/big><\/li>\n<\/ul>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <h2 style=\"text-align: left;\"><big>Assignments<\/big><\/h2>\n<p><big><\/big><big>Assignments are not finalized until their due dates appear in\u00a0bold.<\/big><\/p>\n<p>Chapter 1:<\/p>\n<p>1, 3, 4, 5 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<big>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Due 9\/8<\/big><br \/>\n<big>6, 8, 9, 12, <a href=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/wp-content\/uploads\/sites\/221\/2023\/10\/14.5.pdf\">14.5<\/a>, 15\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Due\u00a09\/17<br \/>\n17, 18, 19, 20, 22a, 23 \u00a0 \u00a0 \u00a0 \u00a0 Due\u00a09\/28<br \/>\n27, 29, 30, 31, 32\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Due\u00a010\/9<\/big><\/p>\n<p><big>Chapter 2: <\/big><\/p>\n<p><big>3,4,5,6,7,8,9,11\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Due 10\/20<\/big><br \/>\n<big>\u00a013,14,16,17,20,21,23 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Due\u00a010\/30<br \/>\n<\/big>\u00a0<big>25,28,32,34,36,40,44<\/big> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<big>Due\u00a011\/10\u00a0<\/big><br \/>\n<big>46,48,49,50,51\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Due\u00a011\/17<\/big><\/p>\n<p><big>Chapter 3: \u00a0<\/big><\/p>\n<p><big>1,2,3,4,7,<\/big><big>8,10,13,16,17* \u00a0 \u00a0 \u00a0 <\/big><big>\u00a0 \u00a0Due 12\/1<\/big><big><br \/>\n22,23,25,26<\/big>\u00a0\u00a0<big><br \/>\n28,30,31,37,40,41,42<\/big>\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<big>\u00a0 Due\u00a012\/9<\/big><\/p>\n<p>*) You may take f to be real valued, to avoid the use of the complex Lebesgue-Radon-Nikodym<\/p>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <h2><big>Exams and Grading Policy<\/big><\/h2>\n<p>Grading Policy<\/p>\n<ul>\n<li>25% Homework: If you are a Math student and have access to the graduate mailroom, please drop off your homework in Fei Wang&#8217;s mailbox, 4th floor KAP. Otherwise, submit at the front desk of the Math Center, KAP 263, or just slide them under the door of KAP 248B. No electronic submissions. Points on homework turned in past the due date are reduced by 20% for each week late.<\/li>\n<li>40% Two Midterms<\/li>\n<\/ul>\n<p>Midterm 1: Monday Sept 28th, coverage 1.1-1.4<br \/>\n200 possible points: High score 200, 3rd quartile 155, median 123, 1st quartile 83. Average 123<\/p>\n<p>Midterm 2: Monday November 2nd, coverage 1.5, 2.1-2.4<br \/>\n140 possible points: High score 136, 3rd quartile 123, median 67, 1st quartile 36. Average 74<\/p>\n<ul>\n<li>35% Final Exam, Wednesday, December 9th, 8-10AM, will be comprehensive.<\/li>\n<\/ul>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <div align=\"center\">\n<h2 style=\"text-align: left;\" align=\"center\"><small><small><strong><big>Links<\/big><\/strong><\/small><\/small><strong><big>\u00a0<\/big><\/strong><\/h2>\n<\/div>\n<p><a href=\"http:\/\/links.jstor.org\/sici?sici=0002-9890%28197903%2986%3A3%3C151%3ATBP%3E2.0.CO%3B2-D\"><big>The Banach Tarski Paradox<\/big><\/a><br \/>\n<big>Karl Stromberg<i><br \/>\nThe American Mathematical Monthly<\/i>, Vol. 86, No. 3. (Mar., 1979), pp. 151-161.<\/big><\/p>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n\n\n\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <h5>Important Dates and Information<\/h5>\n<ul>\n<li>September 11<sup>th<\/sup>, last day to register and add, or to drop without mark of W<\/li>\n<li>November 13<sup>th<\/sup>, last day to drop a class with mark of W.<\/li>\n<li>December 4<sup>th<\/sup>, classes end.<\/li>\n<\/ul>\n<p><big><a href=\"https:\/\/classes.usc.edu\/term-20153\/calendar\/\">Full registration calendar<\/a><br \/>\n<\/big><a href=\"https:\/\/dornsife.usc.edu\/ase\/statement-on-academic-conduct-and-support-systems\/\">Statement of Academic Conduct<\/a><\/p>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":370,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-295","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Math 525a - Real Analysis - Larry Goldstein<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Math 525a - Real Analysis - Larry Goldstein\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/\" \/>\n<meta property=\"og:site_name\" content=\"Larry Goldstein\" \/>\n<meta property=\"article:modified_time\" content=\"2023-10-27T21:16:18+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/\",\"url\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/\",\"name\":\"Math 525a - Real Analysis - Larry Goldstein\",\"isPartOf\":{\"@id\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/#website\"},\"datePublished\":\"2023-06-13T20:24:49+00:00\",\"dateModified\":\"2023-10-27T21:16:18+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-525a-real-analysis\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Math 525a &#8211; 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