MATH 4121 HOMEWORK ASSIGNMENTS, SPRING '08
INSTRUCTOR'S POLICY: TRY HARD TO HAND IN
ASSIGNMENTS IN CLASS ON THE INDICATED DUE DATE AND, IF YOU'RE UNABLE TO ATTEND,
GIVE YOUR COMPLETED ASSIGNMENT TO A FRIEND TO HAND IN. PAPERS SLIPPED
UNDER MY OFFICE DOOR WITHIN AN HOUR AFTER
IT'S PERFECTLY O.K. TO ASK FOR A HINT DURING OFFICE HOURS. I'LL USUALLY
BE QUITE "GENEROUS" WITH HINTS, MAY COME VERY CLOSE TO COMPLETING THE
PROBLEM. REMEMBER THAT IT'S ALSO PERFECTLY O.K. TO DISCUSS PROBLEMS WITH
OTHERS AS LONG AS YOU INDIVIDUALLY WRITE OUT YOUR OWN SOLUTIONS AND PENCIL IN A
NOTE IDENTIFYING THE OTHERS ON THE TOP OF YOUR PAPER (SEE ACADEMIC INTEGRITY
SECTION). THERE'S NO DEDUCTION AT ALL FOR SUCH IDENTIFYING NOTES
BUT THERE MIGHT BE A SUBSTANTIAL DEDUCTION WHEN IT'S CLEAR THERE'S BEEN
COLLABORATION BUT NO NOTES APPEAR.
ASSIGNMENT #1. DUE THURSDAY, JANUARY 24
1. Read Chapters 1 and 2 in Bartle's book and start reading Chapter 9.
2. Hand in
solutions for the following problems:
(i) Chapter 2, p. 15-17: 2D-2H, 2J, 2K,2V,2W . Hint:
In 2W, use certain collections of real intervals.
(ii) For F a monotone increasing function from R,the
set of real numbers, into R, define lambda_F(a,b] =F(b) – F(a) when a< b and lambda_F (A) as the sum of the numbers lambda_F(I_k),
k=1,…,N, when A is the disjoint union of half open intervals I_k, k=1,…,N. Show that every finitely additive
function lambda from the ring R^(1) consisting of finite unions of
half-open real intervals into the
non-negative real numbers is of the form lambda_F
for some monotone increasing function F. Here finite additivity
of lambda means that lambda(union of A
and B) =lambda (A)+lambda(B) when A and B are disjoint
sets in R^(1). Note that lambda doesn’t uniquely determine
F; instead, lambda_F doesn’t change
when we replace F by F+d for any real number d.
We’ll show in class that lambda_F has a
unique extension to a measure on the Borel subsets of
the real line if and only if F is continuous from the right.
Hint: Pick some real number c and define F(x) =lambda((c,x])
if x>c, F(x)=0 if x=c, and F(x)=-lambda((x,c] if
x<c. Then check that
lambda((a,b]) = F(b) -F(a) for all a<b.
In class, we’ll generalize 2(ii) to finitely additive set functions on the ring R^(n) of finite unions of half-open rectangles in R^n for n>1 using nth order difference operators. You may want to speculate on how this will come about from a generalized version of the hint but don’t bother to hand in your speculations. After we prove the Caratheodory Extension Theorem, this will give us an explicit description of all measures on the sigma-algebra consisting of the Borel subsets of R^n which are finite on compact sets.
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ASSIGNMENT #2. DUE THURSDAY, January 31
Do the
following problems from Bartle's book:
Chapter 3, pp. 24-26, #3I-3N, 3P, and 3U
Hint: If you’ve never heard of the
(standard) Cantor set, it is a closed subset Cof the
interval [0,1]with two equivalent definitions The constructive definition
starts by removing the middle open third (1/3,2/3) from [0,1] to yield the two
disjoint closed intervals [0,1/3] and [2/3,1] from which we delete the open
middle thirds (1/9,2/9) and (7/9,8/9) to yield 4 closed intervals each of
length 1/9. We then continuing deleting middle thirds; after k steps, we’re
left with 2^k closed intervals each of length 1/3^k. This process gives
us a monotone decreasing sequence of closed sets C_k
with Lebesgue measure (2/3)^k and the limit set C
has Lebesgue measure 0 since (2/3)^k goes to
zero as k goes to infinity.
The non-constructive definition of C is
all numbers in [0,1] having a trinary expansion as a
sum of terms b_j(1/3)^j, j=1,2,…, where each b_j is either 0 or 2. One has to check that the open
thirds deleted in the constructive definition consist of the numbers in [0,1] with
trinary (or base 3) expansions having some of the
base 3 “digits” equal to 1. There is a little fussiness here
about those rational numbers having two trinary
expansions. Thus 1/3 has both the trinary
expansion (.1000…)_3 using 1 as well as 0 as digits and the expansion
(.0222….)_3 using only 0’s and 2’s as digits. The point
of the non-constructive definition is that with all b_j’s
being 0 or 2 , we can put a_j=b_j/2 to get a typical
binary expansion ( .a_1a_2….)_2 of some number in [0,1]. This gives
a 1-1 correspondence between C and [0,1] and is the simplest way to see that C
is uncountable. C is the easiest example of an uncountable subset with Lebesgue measure 0. What problem 3U is asking
for is a variation on the construction of the standard Cantor set so that the
limit set has positive measure but, like C, doesn’t contain any non-empty
open intervals. For example, you might want to try removing the
middle one fourth or middle one fifth at each step or, if you wish, vary
the size of the fraction being removed as the steps procede.
Be sure to carefully explain why your limit set has positive measure and
doesn’t contain any non-empty open intervals.
When you write up these and future problems, please DON'T use Bartle's notation for sigma algebras since it's very confusing for anyone else to read. As in
class, use a script version of letters
like A, B, C, S, T,... for sigma algebras,
saving X,Y,Z,... for spaces of points in general measure theory and for random variables in probability theory.
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ASSIGNMENT #3 DUE Thursday, February 6
DO THE FOLLOWING PROBLEMS FROM CHAPTER 9,
pp. 108-112:
#9D, #9F-#9K
(total of 7 problems). However, ignore the last statement in 9J since we
haven’t yet discussed integrals.
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ASSIGNMENT #4 DUE Thursday, February 13
DO THE FOLLOWING PROBLEMS FROM CHAPTER 9, pp.
108-112:
#9M, #9N (see the Note below), #9O, #9P
(observe to yourself the comment in #9Q),
#9R . ALSO do the Extra Problem (actually,
3 problems rolled into 1) described after the Note.
Note: Problem #9N is NOT correct as
stated. Show that it's not correct for the case X = real line
with mu =Lebesgue measure and with the algebra script A being the
algebra generated by finite unions of half-open bounded intervals (a,b]. Take the set B to be the union of subintervals of
(n,n+1] having length 1/2^|n| as n ranges
over all integers. Check that mu*(B) is finite
but mu'(B) is infinite. Then show that we can turn #9N into a correct
statement if we assume that the algebra script A exhausts our space X in the
sense defined in class for semi-rings and that the measure mu
is sigma-finite. [For the real line example, this boils down to
proceeding as we did in class with the algebra above replaced by the bigger
algebra consisting of sets whose intersection with (n,n+1] is a finite
union of half-open intervals (a,b] for each integer
n].
Extra Problem: For n>1 and F a function
from (bold face R)^n to the real numbers, do the following:
Show
that the set function delta^n F on the semi-ring script H^(n) consisting of products R of
half-open intervals I_j =(a_j,b_j),
j=1,2,…,n, is additive. We call each such R a half-open rectangle
with extreme vertices a=(a_1,a_2,…,a_n) and b
=(b_1, b_2,…,b_n). R has a total of 2^n
vertices c=(c_1,c_2,…c_n) with each c_j being either a_j or b_j and we define epsilon(c) to be +1 if the number of
indices for which c_j=b_j
is even and epsilon(c)= -1 otherwise. By definition, (delta^n
F) is the sum of the terms epsilon(c) F(c) over the vertices of R.
You’ll find it easiest to first show additivity
when each side (a_j, b_j)
is partitioned into subintervals—perhaps you’ll want to warm up
with the case n=2 to see how all unwanted terms cancel out. Then for an
arbitrary break-up of R into finitely many sub-rectangles, you can take a
partition of the sides of R for which each of the sub-rectangles are
partitioned into smaller rectangles and deduce the desired additivity
on H^(n).
(ii) Show that the compactness argument done in class (and in the textbook) for
n=1 generalizes to showing that the set function delta^n
F is a measure on the exhaustive semi-ring H^n
if F is “monotone increasing” in the sense that (delta^n F) (R) is non-negative for every R and continuous
“to the right and up” in the sense that, for each b as above, F(b)
is equal to the limit of F(x) as x approaches b over points x=(x_1,x_2,…x_n) for which each x_j is
greater than or equal to b_j and the inequality is
strict for at least one j.
Generalize
the extra problem in Assignment #1 to show that every measure on the algebra of
n-dimensional Borel sets which is finite on bounded
sets is the unique extension of delta^n F for some F
satisfying the conditions in (ii).
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ASSIGNMENT #5 DUE
THURSDAY, MAR. 6
1. Do the following problems from Chapter 5, pp. 48-51:
# 5L, 5M, 50, 5P, 5Q, 5R, 5T
I think you'll find most of these problems fairly straightforward albeit a bit
on the "dry"
side, i.e. not all that
surprising and not all that hard. On 5L, just use triangle ineguality arguments, not bothering to try to make a
dominated convergence argument. On 5P, you'll likely want to use the reverse triangle
inequality: | |x| - |y| | <= |x-y|. On 5R, mimic the proof
of Corollary 5.9.
As mentioned in class, when lambda
and mu are sigma-finite measures defined on the same
sigma algebra script S of subsets of some set X, then the Radon-Nikodym Theorem (Chapter 8—we’ll prove it in a
few weeks time) says that lambda is absolutely continuous with respect to mu (commonly denoted lambda < < mu)
in the sense that mu(N) = 0 implies lambda(N) = 0 if
and only if there is a non-negative measureable
function f for which lambda = lambda_f is given as in
equation 4.9 on page 34. We then call f the Radon-Nikodym
derivative of lambda with respect to mu and denote it
by f = d(lambda)/d(mu).
Since we showed in class that lambda_f=lambda_g if and only in f = g a.e.,
Radon-Nikodym derivatives are only well defined up to
a set of measure zero.
(a) Suppose f=d(lambda)/d(mu). First show that the integral with respect to lambda of
any non-negative simple function phi is equal to
the integral of f phi with respect to mu,
then go on to show using the Monotone Convergence theorem that the
integral with respect to lambda of any non-negative measureable
function g is equal to the integral of fg with
respect to mu, and finally deduce the same result for
measureable functions integrable
with respect to lambda.
(b)
For f as in (a), show that f is >0 mu-a.e.
if and only if lambda and mu have exactly the same
null sets and then deduce either directly or from the Radon-Nikodym Theorem that 1/f = d(mu)/d(lambda).
(c)
If we have three measures lambda, mu, nu on the same sigma-algebra of subsets of some X
and f= d(lambda)/d(mu), g=
d(mu)/d(nu), show that fg = d(lambda)/d(nu) a.e. For obvious reasons, this is sometimes called the
measure-theoretic chain rule.
[Clearly, (b) and (c) are reminiscent
of properties of ordinary derivatives. Two trivial additional familiar
properties are that the Radon-Nikodym
derivative
of a sum of two measures
with respect to a third measure is the sum of the Radon-Nikodym derivatives
of the summands (when they exist) and that the Radon-Nikodym
derivative of c (lambda) with respect to mu is c d(lambda)/d(mu) for any
c>0. Later we'll discuss cases where Radon-Nikodym
derivatives "reduce" to "ordinary" derivatives. We'll
also show how the “hard” change-of-variable theorem for integrals
(as opposed to the “soft” change of variable result discussed in
class) reduces to the existence of Radon-Nikodym
derivatives.]
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ASSIGNMENT #6 DUE Thursday, MARCH 20
DO THE
FOLLOWING EXERCISES FROM CHAPTER 6:
#6D,6F,6G,6H,6J,6K,6M
ALSO DO THE FOLLOWING PROBLEM COMBINING THE RESULTS FROM SOME OF THE ABOVE
PROBLEMS:
For I any interval in [1,infinity], there is a Lebesgue
measurable function f from R (real line) to R such that, relative to Lebesgue measure lambda_1 on R, f is in L_p
if and only if p belongs to I. Note: I can be open, closed, open on one side and closed on the other, bounded or not,
contain the “point” infinity or not, or I may be a degenerate
interval consisting of a single point. Using the fact that f is in L_p if and only if g=|f|^p is in
L_1, and then, for q>p, f is in L_q if and only if
g is in L_q/p, for intervals I other than {infinity},
there’s no loss of generality in assuming that I has left endpoint
1. In this way, a small modification in the 6H example with handle all
half-open intervals [a,b) and applying powers to the
6M example will handle all degenerate intervals other than
{infinity}.
ASSIGNMENT #7 DUE TUESDAY, APRIL 1
DO THE FOLLOWING EXERCISES FROM CHAPTER 6:
#6O,6P,6Q,
6R,6S (all of these are pretty straightforward)
ALSO DO THE FOLLOWING:
(1)
Show that #6O fails
for {p,q}={1,infinity} in the sense that, for f in L^p X,mu), we usually can’t
find g in L^q (X,mu) with qth norm 1 for which the integral of fg
is equal to the pth norm of f. What are the conditions on f for which such a g
does exist? When these conditions aren’t
met, show that we can pick a sequence of functions g_k
in L^{q} each of which has norm 1 for which the
integrals of fg_k are >0 and increase
monotonically to the pth norm of f. Assume that (X,mu) is sigma-finite to avoid “pathologies”.
(2)
When 1<=p <q <= infinity and f is in both L^p
(X,mu) and L^q (X,mu), show that the
function phi(r) = rth norm of f is continuous on the
interval (p,q).
For this, try to reduce the problem to consideration of a pair of
monotone sequences of real numbers.
ASSIGNMENT #8 DUE
TUESDAY, APRIL 8
DO THE FOLLOWING EXERCISES FROM
CHAPTERS 7 AND 10:
Chapter 7:
#7Q, 7V.
You should also quickly read over the other
Chapter 7 Exercises, the hard one being 7N.
The host of examples of functions with this property and not that one
are mostly examples we’ve previously studied. I don’t know why Bartle
dwells in 7R-7V on compositions of a continuous function phi with various types
of functions f; going through one such problem establishes the pattern.
Chapter 10: # 10.H, 10.K, 10.L, 10.O, 10.S
For 10.H, you can show F measurable
directly by using results mentioned in class, then deduce the statement about
the sets gamma(E).
Note that 10.J, 10.M, 10.N, 10.P,10.Q are
essentially trivial— just special cases of the Fubini-Tonelli
Theorem.