Math 623-- Real Analysis I

Fall 2024

Prof. Andrea R. Nahmod

Office: LGRT 1588
Tel : (413) 545 6031
Email: mylastname at umass dot edu

Class Meeting: Tuesdays and Thursdays 1:00 pm - 2:15 pm in LGRT Room TBD
Office hours: By Appointment in Zoom; by appointment in person (usually on Wednesdays) and by Email.

Book :   Real Analysis - Measure Theory, Integration and Hilbert Spaces by Elias M. Stein and Rami Shakarchi
            Princeton Lectures in Analysis, Vol. III (2005)    Princeton University Press
Note: The very early editions of the book have an Erratum for Theorem 4.2 Chapter 1: which can be found here(click).
Note that this was corrected in the later editions.

Topics:   This is the first part of a 2-semester introduction to Real Analysis: Math 623 in the Fall, and in the Spring Math 624 which covers
part of Vol. IV of Stein&Shakarchi also. The prerequisites for this class is a working knowledge undergraduate Analysis
(as for example taught in classes like M523H and M524 at UMass Amherst).

In the Fall semester we will cover the following material from Stein-Shakarchi's Vol III:

1) Measure theory: Lebesgue measure and Integrable functions (Chapter 1)
2) Integration theory: Lebesgue integral, convergence theorems, Fubini theorem (Chapter 2)
3) Differentiation and Integration. The Hardy Littlewood maximal function, covering lemmas, Lebesgue differentiation, approximations to the identity. Functions of bounded variation and absolutely continuous functions (Chapter 3)
4) We will start with Hilbert Spaces (Chapter 4+part of Chapter 5) which we'll reprise in M624

NOTE: Abstract measure theory (starting in Chapter 6) will be developed in in full in M624 together with other topics from Stein Shakarchi Volume IV.

In this semester we will cover most - though not all sections above and some additional relevant topics.

Announcements will be posted here.

There will be two exams:   Exam 1: Will be in class


                                             Exam 2: Will be a take home exam.

Grading Policy: Homework + Class Participation(50%) -- Exams (25% each).

If you feel sick please DO NOT attend class and contact me ASAP so I can help
you keep up. You'll not lose your class participation credit.

Homeworks will be posted here in a cummulative fashion and with specific due dates (no late homeworks unless mutually agreed upon due to extenuating circumstances.)

If you fall ill please contact me ASAP about alternative arrangements to turn in your homework

HOMEWORK SETS (click)

Some useful notes:

Ordering&Zorn Notes

Egorov's Theorem

Lusin's Theorem

On Fubini and Product Sets ( end of Section 3 Chapter 2 of Stein&Shakarchi).

End of Section 1 Chapter 3

Good Kernels and Approx. to the Identity

Lemma 3.2 Chapter 3 Section 3.1 (Stein-Shakarchi Vol III)

Part 1(ii) of Theorem 3.4 Chapter 3 Section 3.1 (Stein-Shakarchi Vol III)

Mandatory Information:

Accommodation Statement
The University of Massachusetts Amherst is committed to providing an equal educational opportunity for all students. If you have a documented physical, psychological, or learning disability on file with Disability Services (DS), you may be eligible for reasonable academic accommodations to help you succeed in this course. If you have a documented disability that requires an accommodation, please notify me within the first two weeks of the semester so that we may make appropriate arrangements. For further information, please visit Disability Services (https://www.umass.edu/disability/)

Academic Honesty Statement
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