Math 623-- Real Analysis I

                                                           Fall 2017

                                                                    Prof. Andrea R. Nahmod

Office: LGRT 1338
Tel :    (413) 545 6031
Email:  mylastname at math dot umass dot edu

Class Meeting:   Tuesdays and  Thursdays  11:30 am - 12:45 pm  in LGRT 1322.
Office hoursWednesdays 1:00pm-2:30pm. Also: By appointment and virtually via 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. Some early editions of the book have an Erratum for Theorem 4.2 Chapter 1: which can be found here(click).

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 M524H 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 and Fubini theorem (Chapter 2)
3) Differentiation and Integration. Functions of bounded variation (Chapter 3)
4) Abstract measure theory (first part of Chapter 6)

In this semester we will cover most - though not all - of chapters 1, 2, 3(part),  6 (part) and some additional relevant topics.


Midterm Exam:  This will be an evening 2 hours exam --in class-- on a date TBD

Final Exam:  This will be a Take Home Exam given after the last class and due Friday 12/15/2017 no later than 11AM.

Grading Policy:  Homework + Class Participation(40 %) -- Midterm (30 %) -- Final (30 %)


Ordering&Zorn Notes

Egorov's Theorem

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

Good Kernels in Fourier Series (periodic functions) (from Stein-Shakarchi's Fourier Analysis Volume I)

Good Kernels and Convergence in L^1

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)

HOMEWORKS :  Homework will be posted here in a cummulative fashion and with specific due dates. No late homeworks will be accepted
-except for extraordinary circumstances (please talk to me
before the due date in those cases).