Mon. and Wed. 2:30-3:45pm; CU-Bldg, 641
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Due |
Problems |
Notes |
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Wed. Dec. 13th |
Oral presentations |
Jeff 11.5 and 11.6 Don 11.7 |
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Mon. Dec. 11th, 5pm |
WRITTEN PROJECTS DUE Oral presentations. |
Mon 11.3 and 11.4 Nathan 10.1, 10.2, and 10.8 |
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Wed. Dec. 6th |
Oral presentations |
Dmitry 12.1 and 12.2 Jason 11.1 and 11.2 |
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Mon. Dec. 4th |
Homework sheet |
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Wed. Nov 15th |
EXAM 2 |
Covers material through the end of Chapter 4 |
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Wed. Nov. 8 |
Ch 4: Extra Problem 5 through Extra Problem 7 |
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Wed. Nov. 1 HW 6 |
Ch 3: 29 Ch 4: 1a through 7b (including Extra Problem 4) |
Note that 1b is extra credit. |
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Wed. Oct. 18 |
Ch 3: 18, 20, 22a,b; 25 |
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EXAM 1 Mon. Oct. 9th |
Material through Section 3.4 |
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Monday Oct. 2 |
Problems from Ch 3: 3, 4, 8, 10-12 |
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Mon. Sept. 25 |
Problems 44a, b, c, d from Ch 2, extra problems 2 and 3, Ch3: 1 and 2. |
Test 1 will cover material through S3.4. |
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Wed. Sept. 13 |
Problems from Chapter 2 listed below, 11a through 31. |
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Wed. Aug. 30 HW 1 |
All problems on assignment sheet from Chapter 2 up to and including 9a,b. Also, extra problem 1. |
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CHAPTER 2:
3, 4b, 4c, 4d, 5b, 5d, 5e
6 should be (inf E <= sup E)
7, 9a,b; (can use results of problem 13),
extra problem 1: Write down the corresponding conditions (i) and (ii) given in problem 13, for the lim inf of x_n.
11a,b,c,d; 15,
24 (don't have to prove it), 25, 26, 27, 31
extra problem: For all a, b in R, the set of real numbers, show that ||a|-|b||<= |a-b|
44a,b,c,d;
extra problem 2: Give an example of a sequence of continuous functions which converges pointwise to a discontinuous function.
extra problem 3: Let f(x) be defined as follows: If x is irrational, then f(x) = 0. If x is rational, then if x=m/n in irreducible form then f(x) = 1/n, where n is always taken to be positive, and f(0)=1. Show that this function is continuous at every irrational point and discontinuous at every rational point.
CHAPTER 3:
1, 2 (hint: use proposition 1.2, p. 17),
3, 4 For problems 1-3 assume that countable additivity implies finite additivity. Problem 4 is not trivial.
8
10, 11, 12
18 (your counterexample can assume each value more than once, but it must be measurable), 20 (use the identities to show that the sum and product of 2 simple functions are simple: must show that the resulting functions are *single* valued),
22a,b; 25
29
CHAPTER 4:
1a, extra credit: 1b (function f(x) is from 1a). Note that even if f_n are Riemann integrable which monotonically increase to a function f, the limit and integration may not be interchangeable.
extra problem 4: True or False: The characteristic function of the Cantor set is Riemann integrable. Justify (prove).
3 (hint: consider A_n = {x: f(x) > 1/n},
4 modified (Show the equivalence of the two definitions of the Lebesgue integral - assume simple functions can only take on finite values and that for each definition the measure of the space where the simple/bounded functions are not zero is finite),
5, 7a, 7b
extra problem 5: Suppose that f_n is a sequence of measurable non-negative functions on a measurable set E which converges (point-wise) almost everywhere to f. Prove that, if int_E f_n dx <= M < infinity, for all n, then f is integrable and int_E f dx <= M.
10a,
extra
problem 6a: Prove that f(x) = 1/(1+x^2) is Lebesgue
integrable on the real line.
extra problem 6b: Prove that f(x) = |x|^{-1/2}(1+x^2)^{-1}
is Lebesgue integrable on
the real line.
12, 15a, 15b, 15c; 16
extra problem 7: Show the equivalence of the definitions of convergence in measure given in class and given in the text.
Homework
11 (pdf file)