**RUTGERS UNIVERSITY -- NEW BRUNSWICK
**

**DEPARTMENT OF MATHEMATICS **

**MATHEMATICS 501 -- FALL 1999 -- ARCHIVE PAGE**

*"Omnia disce,
videbis postea nihil esse superfluum. Coarctata scientia jucunda non est.''*

*Hugh of St.
Victor (c. 1078 or 1096?--1141)*

- Textbook(s):
- Required: Richard L. Wheeden and Antoni
Zygmund,
*Measure and Integral: An Introduction to Real Analysis*, ISBN 0-8247-6499-4. - Course Notice
- Inst.: Bertram Walsh, Hill 728, (732) [44]5-3733
- Course Materials:
- 9/3/99 and 9/7/99 Class meeting notes [* first 20 pages]
- (Old) metric-space notes from Math 503 Fall 1988
- All the Filter Stuff * in one place, along with proofs of logical equivalence for the various forms of the choice axiom. Read what you need. Last revised 0930 EDT 9/16/99.
- General Topology 1* including material on theorems of Arzela-Ascoli & Baire. Last revised 0900 EDT 9/25/99.
- 9/23/99 Necessary and Sufficient Conditions for Riemann[-Stieltjes] Integrability [* 9/23/99]
- 9/10/99 ff. BV & Riemann-Stieltjes integration [* New version 9/26/99--please see note]
- 9/28/99 Stone-Weierstrass Theorem [* 9/28/99]
- General Topology 2* including many things bearing the name G. Cantor. Last revised 1330 EDT 10/03/99.
- Introductory material about sigma-algebras, measures etc. can be found at the beginning of Wheeden & Zygmund's Ch. 10, p. 161 ff.
- Lebesgue Measure 1 [* now complete (8 pages). If you have pp. 1-6, copy only pp. 7-8 (though there was a small typo on p. 2). Covers 9/30/99 to 10/05. Last revised 1020 EDT 10/06/99.]
- More Riemann-Stieltjes and Lebesgue-Stieltjes integration [* unedited--the page references go with Royden's RV textbook. I will leave this posted as is, but it will soon be rewritten and reposted to conform to W & Z's Ch. 11 and connect to the R-S integration material above. 9/30/99]
- (Old) Stone-Weierstrass notes from Math 502 Spring 1997 including, e.g., the Bernstein approach, the Bohman-Korovkin theorem, and miscellaneous pieces of analysis [not required]
- Lebesgue Measure 2 [* puts the 10/19/99 dominated convergence mess into a form a human might possibly understand, and gives a record of the other proceedings of that date.]
- We will omit §4 of Wheeden &
Zygmund's Ch. 5 until we have the Fubini-Tonelli theorems (Ch. 6) available.
The material is easier to understand in a "two-dimensional" context,
and one can make
*a posteriori*identifications with Riemann-Stieltjes integrals at one's leisure. - Lebesgue Measure 3 [* The theorems of Tonelli and Fubini in R^(n+m); a little bit different from the textbook, and one might find the counterexample amusing.]
- Abstract Measures 1 [* Definitions, the Hahn-Jordan decomposition, the lattice of finite measures on a measurable space and the L^p spaces. There are a little more than 20 pp. here: the first 8 contain corrections of a minor typo in the previous posting with no other changes, but the remaining material is new (11/08/99).]
- Abstract Measures 2 [* More of the same (pp. 21-45), through the atomic/continuous decomposition, the absolutely continuous/singular decomposition, and the Radon-Nikodym theorem {only finite measures so far} and some Hilbert space material. Typos in the original posting have been repaired (11/27/99).]
- Abstract Measures 3 [* Finishes the Hilbert space material (pp. 45-51) (11/30/99).]
- Abstract Measures 4 [* Products of finitely many measure spaces (pp. 52-57) (12/06/99). These notes will be concluded with infinite products of probability spaces and will remain posted well into the Spring semester.]

- Examination Announcements and Materials:
- Midterm Exam 10/28/99. Closed book. Questions will be drawn from among
the following Twenty Questions.*
Please note the typo in question 4: The function
*"g"*of two variables was called*"f"*in a couple of places in the first posting of these questions. This is repaired in the version that is now (1855 EDT 10/14/99) posted. Apologies to all; thanks to N. Fefferman for finding this. Also question 14: for an l. s. c. function to take a finite minimum, one must require that it be finite*somewhere.*(Repaired 1815 EDT 10/19/99.) The functions in question 20 should be assumed measurable. - The final examination is scheduled for 1-4 PM EST on Wednesday, December 15th (same location, Hill 423). We can probably overstay our welcome in the room if necessary, but not by a large amount of time. Questions will be drawn from among the usual twenty questions: here they are. Twenty Questions. If you have not looked at this recently, please check the repairs in qq. 13 and 20. (12/14/99)
- Here, now that the dust has settled, are solutions for the twenty questions given for the final. Contributors include Aaron, Aobing, Eva, Jonathan, Kia, Laura, Matt, Michael, Pieter, Saša, Stephen, and your humble servant, who takes responsibility for any errors in the file.

The asterisk(s) * in the preceding lists indicate files in small type whose legibility on some terminals may be marginal. You're almost surely using an Acrobat reader to read these files: clicking on a magnification icon and selecting "fit visible" may improve things. You may find that the best solution for the legibility problem is to produce hard copies (or to obtain them from the instructor if you are unable to produce them yourself).

- Wheeden & Zygmund Ch. 1: 3, 9, 12, 13, 14 (in R^n), 17, 18
(use the structure theorem
**(1.10);**do not imitate the proof of the Tietze theorem for normal or metric spaces). Due--oh, say 9/21 or so. - Examples 2, including W & Z problems from Ch. 2. Due about 10/8.
- Baire Category Theorem applications for your amusement. No due date.
- Wheeden & Zygmund Ch. 3: 9, 12 (generalize to R^n x R^m),
13, 14. These are not due, ever; read for
the midterm instead! You might also try to do 27 by a category-theorem
argument without an explicit construction; you will need a norm on the
continuous functions of bounded variation such that this (normed vector)
space is complete in the metric (try |
*f(a)*| +*V[f;a,b]*). - Wheeden & Zygmund Ch. 4: 3, 11, 12, 18, 20. The last of these
can be extended to the case in which
*E*is any compact subset of R^n, since we have the Tietze extension theorem in that setting. Again, these are not due, ever; but I thought it would be good to pick out some interesting exercises from this chapter and its predecessor. - Wheeden & Zygmund Ch. 5: 2, 3 (what's the problem?), 6, 9, 10, 11, 13, 20, 21. I will pick out some of these to hand in after the mid-term.

Note: This version has been corrected from the previously-posted one. Modifications start at about p. 22, so you do not have to re-print anything but what Acrobat will think are pp. 22--33 (the last few pages are marked "31-cont." for the sake of consistency later on).

last revised 1554 EST 8/29/2000