INTRODUCTION TO MEASURE AND INTEGRATION

Prerequisites & course information

Requisites: MATH 522 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Spring 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in measure theory (e.g., measures and measurability, Lebesgue integral, absolute continuity, product measures, Lp spaces, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., that a measure is absolutely continuous with respect to another one, that a sequence of functions converge almost everywhere or in measure, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of measure theory (e.g., Fatou's Lemma, Egoroff's theorem, Hahn decomposition theorem, monotone convergence theorem, dominated convergence theorem, Holder's inequality, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, etc.). Audience: Both Grad & Undergrad 5. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 6. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in measure theory (e.g., measures and measurability, Lebesgue integral, absolute continuity, product measures, Lp spaces, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., that a measure is absolutely continuous with respect to another one, that a sequence of functions converge almost everywhere or in measure, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of measure theory (e.g., Fatou's Lemma, Egoroff's theorem, Hahn decomposition theorem, monotone convergence theorem, dominated convergence theorem, Holder's inequality, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, etc.). Audience: Both Grad & Undergrad 5. Convey arguments in oral and written forms using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 6. Identify applications of course content in current areas of research. Audience: Graduate