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 CLASS ON THE DUE DATE HAVE  GOOD ODDS  OF GETTING INTO THE GRADER'S HANDS AND BEING GRADED.      IN THE EVENT OF A SPECIAL PROBLEM (ILLNESS, ETC.) CONTACT ME BY E-MAIL. UNLESS I'VE MADE A SPECIAL ALLOWANCE FOR SOME REASON, PAPERS RECEIVED AFTER I'VE GONE HOME ON THE DUE DATE WILL NOT BE GRADED AND HENCE WON'T RECEIVE ANY CREDIT.
DON'T PUT ANY PAPERS IN MY MAILSLOT SINCE I CHECK IT ONLY SPORADICALLY.
                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 25

    1.   Read Chapters 1 and 2 in Bartle's book and browse through 3 and 4.

    2.   Hand in solutions for the following problems:
           (i) Chapter 2, p. 15-17: 2D-2H, 2I, 2J,2W . Hints: In 2I, DON'T try to construct
any non-measurable sets, instead just blithely assume that such sets exist and use one or more of them
to define a function with the indicated properties.  In 2W, use certain collections of real intervals.
         (ii) Show that every finitely additive function lambda from the ring of half-open  real intervals
into the real numbers is of the form lambda_F for some function F.  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.
        (iii)  Suppose (X,A,mu) is a finite measure space, i.e., mu(X) is finite.  Also suppose all of the singleton sets {x},x in X, belong to the sigma algebra A.  Let D be the set of points
x in X for which w(x) = mu({x}) >0. Show that D is necessarily countable.  Let lambda be the measure defined on A by lambda(B) = sum of the numbers w(x) over all points x in B. Show that nu(B)=
mu(B) - lambda(B) defines a measure on the sigma-algebra A for which nu({x}) =0 for every x in X.
True or False?  Exactly the same result is true for any sigma-finite measure space.  Why?

        3.  Speculate about a generalization of 2(ii) to set functions on the ring of half-open
rectangles in R^n for n>2.  Don't hand in your speculations.  Instead, if you're interested
in learning whether your speculations are correct, ask me in class or send me an e-mail.

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ASSIGNMENT #2.  DUE THURSDAY, FEBRUARY 1

       Do the following problems from Bartle's book:
            Chapter 3, pp. 24-26, #3I-3N, 3P, and 3U

        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,... 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 8

    1.  Do the following problems from Chapter 4, pp. 37-40:  #4I, 4J, 4K, and 4R-4U.
For 4T, use the Lebesgue Dominated Convergence Theorem (which we'll discuss on Tuesday)
to get an easy proof.  A proof without the LDCT is nasty.  As usual, in problems like 4U asking for a counter-example, use as your measure space either the natural numbers with the counting measure
or the real line with Lebesgue measure (i.e., the measure assigning the interval length to any bounded
interval).  I think you'll find these Chapter 4 problems MUCH easier than the recent Chapter 3
assignment.  You may want to spend a few minutes browsing through the non-assigned problems--
I thought they were either really easy or "unnecessarily convoluted" and "uninteresting".

     2.  Let X be a random variable on a probability space (S, script A, P).  As we discussed in class a few weeks ago, associated with X is a function F (called the cumulative probability distribution
function for X) defined by F(x) = P(X<= x) and then we can use F to define a measure lambda
on the Borel subsets of the real line by lambda (a,b] = F(b)-F(a).  A common notation for lambda
is dF, more precisely, under an integral sign, we write (dlambda)(x)=dF(x).  We say X has moments of orders 1, 2, ..., k if the integral over the sample space S of |X|^k with respect to P is finite. (Note that for j < k, the integral of |X|^j is then automatically finite since this function is bounded from above
by the maximum of 1 and |X|^k). Using the 5.1 definition of integrals of functions which may take on both positive and negative values, we then can integrate X^j over S with respect to S for each j between 1 and k. Go back to the definition of integrals of non-negative functions using simple functions to show that the integral of X^j with respect to P coincides with the integral over the real line of the monomial
x^j with respect to the measure dF.  Then go on to show that if Phi is some Borel measurable function from the real line to itself for which Phi composed with X is integrable over S with respect to P, then the value of the integral coincides with the integral over the real line of Phi(x) with respect to dF(x).

Remark.  The result of this problem is used "all the time" in probability courses.  Often it's
assumed that there is a function f(x) (called the probability density function of X) for which
dF(x) = f(x) dx.  What's going on here is the assumption that the measure lambda is absolutely
continuous with respect to Lebesgue measure on the real line in the sense of 4.9, 4.11, 5.2.  This will seem like a "reasonable assumption" after we prove the Radon-Nikodym theorem in Chapter 8.

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ASSIGNMENT #4   DUE THURSDAY, FEB. 15

        1.    Do the following problems from Chapter 5, pp. 48-51:
                    #5I, 5L, 5M, 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 hence not all that interesting.  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.

        2.    In class, we showed that, when a measure lambda is defined in terms of another measure mu and a measurable non-negative function f by equation 4.9 on page 34, then  the integral of another measurable function g with respect to lambda exists precisely when the integral of gf with respect to mu exists and the two integrals are then equal.  We call f the Radon-Nikodym derivative of lambda
with respect to mu and denote it by f = d(lambda)/d(mu); f is only well defined up to a set of mu-
measure zero.  Also mentioned in class was the Radon-Nikodym Theorem (to be proved in Chapter 8)
stating that, in the sigma-finite case, f exists if and only if every mu-null set is also a lambda-null set.
Use these two statements to show the following:
            (i) Show that f is >0  mu-a.e. if and only if lambda and mu have exactly the same null
sets and then 1/f = d(mu)/d(lambda), i.e., mu is defined in terms of lambda and 1/f using 4.9.
            (ii) 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), then fg = d(lambda)/d(nu).  For obvious reasons, this
is sometimes called the measure-theoretic chain rule.
[It's obvious from the linearity properties of integrals 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 change-of-variable theorems for integrals reduce to the existence of Radon-Nikodym derivatives.]

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ASSIGNMENT #5   DUE THURSDAY, FEB. 22

        1.  DO THE FOLLOWING PROBLEMS FROM CHAPTER 6, pp. 61-64:
              #6F, #6H, #6J, #6M, #6T, #6U

        2.  In class we showed that, on any measure space, when a function f is in L^r for all
r in an interval I about p, then the function N(r) = ||f||_r is continuous at p.  When f is in L^p for some finite p and also is in L^{infinity}, show that N(r) is continuous at infinity in the sense that N(r)
approaches N(infinity) as r goes to infinity.  [Those who took 4111 may recall that there was
a 4111 homework problem similar to this].  There is no loss of generality in
assuming that ||f||_infinity = 1 (why?).  Deduce from this that N(r)^r <= N(p)^p and conclude by
taking the rth root on both sides that  lim sup N(r) <=1.  Now show lim inf N(r) >= 1 by using
the fact that, for any epsilon > 0, there is a set E of positive measure on which the values of
f lie in the interval from 1 - epsilon to 1.
 

        3.  Do Problem #6L on p. 63.
                There is an easy way to do this problem (along with lots of not so easy ways).  We suppose
f is a function on the natural numbers and use the counting measure on the natural numbers.  It's enough to show that if f is in L^p and q> p, then ||f||_q <= ||f||_p with strict inequality when f(i) is non-zero for at least two indices.  The case when q = infinity is very easy (why?).  When q is finite, first note that it's enough to treat the case when ||f||_p = 1 (why?).  Then let a_i = |f(i)|^p so the sum of the a_i's is 1.  Put r =q/p so r>1 and observe that |f(i)|^q = (a_i)^r.  Now finish the proof.

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ASSIGNMENT #6  DUE THURSDAY,  MAR. 1

            DO THE FOLLOWING PROBLEMS FROM CHAPTER 7, pp. 77-78:
            #7A, #7B, #7C, #7D, #7I (Hint:  by the proof of the Riesz-Fischer Theorem, a subsequence of the given sequence will converge a.e. to f, so it's enough to show that when a function f is the limit
a.e. of a sequence of characteristic/indicator functions, then f is equal a.e. to a characteristic/indicator function), 7J, #7K, #7L

            There's no need to spend a lot of time agonizing over the technical proofs in Chapter 7.  Instead,
concentrate on puzzling over the diagrams at the end of the Chapter and the proof of Egoroff's Theorem.  The above exercises give a host of counter-examples (convergence in one mode but not another) and may help to clarify the diagrams.
            Convergence in measure is very popular among probability theorists but practically never used
outside of probability theory.  What's useful about convergence in measure is that it's equivalent to convergence in L^p when members of the sequence are dominated by a function in L^p and that convergence in measure implies that a subsequence converges a.e.  The only important thing about almost uniform convergence is Egoroff's Theorem:  on spaces of finite measure, almost uniform convergence is the same thing as a.e. convergence.  As we'll discuss in class, on sigma-finite spaces, comparison of convergence modes boils down to just three types:  uniform convergence, strong convergence = a.e. convergence, and weak convergence = measure convergence.  As the names
suggest, uniform implies strong and strong implies weak.  In most situations, uniform convergence
is "too strong" in the sense that it applies only in very special cases.  This leaves just strong and weak.
These names crop up in a number of results in probability such as the strong and weak laws of large
numbers.
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ASSIGNMENT #7 DUE THURSDAY, MAR. 8 [However, if you won't be on campus at the end of next week, or if you're very busy with other courses, or just tired out and needing a break, it will be O.K. to turn in the assignment on Monday, Mar. 19.    If none of the above alternatives apply to you,
I suggest doing the assignment and handing it in late next week in order to clear the decks for a week
of "measure free" vacation]

        DO THE FOLLOWING PROBLEMS FROM THE TEXTBOOK:

        CHAPTER 7, pp. 77-79, #7Q, #7S, #7T  (in 7T, take phi to be an easy function
such as phi(x)=x^2)

        CHAPTER 8, pp. 93-95, #8D, #8K, #8M, #8S (in all of these problems, note that Bartle
requires a charge to be finite--in 8S, recall that a Banach space is a complete normed real vector space,
so the problem comes down to showing that a Cauchy sequence of finite charges converges to a finite charge)

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ASSIGNMENT #8 DUE THURSDAY, MAR. 29
             The assignment this week is intended to serve as a review for the Mar. 29-Apr. 2 take-home exam of previously covered material.   I should have mentioned a long time ago that Bartle has a somewhat strange way of stating problems.  Frequently, he follows the pattern of other texts:  If
...., show that ....  But nearly as frequently, he omits the "show that" and just makes a statement:
If ...., then ....Whenever you encounter such a statement in one of his problems, read it as "show that".

        DO THE FOLLOWING PROBLEMS FROM THE TEXTBOOK:

        1.  Easy Review Problems:   4Q, 5A, 5K, 6Q,6R

        2.  LDCT for Measure Convergence.  I skipped the proof of this theorem in class on the grounds
that the proof I know is similar to the proof of Egoroff's Theorem in that it involves not well motivated fiddling with lim sup and lim inf constructions.  Lan Xu was kind enough to point out to me a clever proof given in last year's 4121 class by Prof. Ilya Krishtal which follows the suggestion of Bartle in 7N and 70.  Complete the following outline of Prof. Krishtal's proof:
             (a)  First suppose we have a sequence f_n, n > 0, of non-negative measurable functions which converges in measure to a function f.   Let L be the lim inf of the integrals of the members of the sequence.  Pick a subsequence for which L is the limit of the integrals of members of the subsequence.
Now use Theorem 7.6 and Corollary 7.7 to pick a subsequence of the subsequence which converges a.e.   Apply the ordinary Fatou Lemma to this sub-subsequence to deduce that the integral
of  f  is less than or equal to L.  (it's only this last statement where you
need to complete the outline).
             (b)  Use the result in (a) to turn the proof of the LDCT for a.e. convergence into a proof of the
LDCT for measure convergence.

        3.  Problem 8V.  What Bartle is getting at in his initial remark is the theorem (proved by a variation of the Gram-Schmidt process) that every Hilbert space H (=complete normed vector space where the norm arises from a real or complex inner product <.,.> on H) admits orthonormal bases, i.e. collections of mutually perpendicular unit vectors e_i, i in some index set I, for which every vector v in H is expressed by the unconditionally converging infinite series of terms <v,e_i> e_i.   Use the existence of orthonormal bases to show that H is its own dual space in the sense that alpha is in H* if and only if there is a vector w in H such that alpha(v) = <v,w>  for every v in H.  Why does this immediately prove the Riesz Representation Theorem for the case p = 2?  Now go on to complete the outline of Bartle's proof in 8V of the Radon-Nikodym Theorem.

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ASSIGNMENT #9  DUE THURSDAY, APRIL 12

DO THE FOLLOWING PROBLEMS FROM CHAPTER 9, pp. 108-112:
        #9D, #9F, #9M, #9N (see the Note below), #9O, #9P (observe to yourself the comment in #9Q),
#9R , #9T

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.
Is the statement in #9N correct for the case when mu(X) is finite?
           If you're rushed for time, I strongly suggest looking first at #9T.  The other problems are
mostly routine ("boring"?) but #9T is interesting.

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ASSIGNMENT #10 DUE THURSDAY, APRIL 19

DO THE FOLLOWING PROBLEMS:
        1.  Modify slightly the argument given in class for Euclidean norms to show that
if ||.|| is any norm on R^n which is smooth in the sense that N(x) =||x|| is  C^{infinity} on R^n\{0}, then for each closed ball K of radius r >0 (relative to || . ||) about a point p
and each delta > 0, there is a C^{infinity} function H which is 1 on K, zero off the closed ball L
about p with radius r+delta, and has values in (0,1) at points in the interior of L which are not in K.
Note:  In the language used in class, H is a bump function for K and L.  As we vary the choice of norm, we get a huge collection of compact convex sets K and "magnifications" L.  Smoothness of
||.|| means that spheres relative to ||.|| are smooth surfaces: no corners, locally are graphs of C^{infinity} functions.  From the 4111 final, compact convex sets which are symmetric about 0
and have a smooth boundary surface are balls of radius 1 relative to a smoooth norm..

       2.  Let K be a convex polygon in R^2 with p a point in the interior of K.   For r>1, the r-magnification of K relative to p consists of all points of the form y=p+r(x-p) for x in K.
For each delta >0, construct a bump function for K and its (1+delta)-magnification relative to p.
Note:  Let <.,.> be the Euclidean inner product (usual dot product) on R^2. If K is an N-sided polygon, then there are lines L_1, L_2,....L_n  not passing through
p and points q_i on L_i for which the Euclidean distance d_i from p to q_i is the minimum distance from p to L_i. Letting u_i be the unit vector in the direction q_i -p,
 K ={x in R^2: <x-p,u_i>  <= d_i for i=1,...,N}
    ={points in R^2 on the same side of each L_i as p since <x-p,u_i> = d_i is the equation of L_i}.
Conversely, if we take any N positive numbers d_i and  unit vectors u_i, the set K defined above
is convex and contains p as an interior point but K is a polygon if and only if K is compact and K
is N-sided if and only if K intersects each L_i in a line segment of positive length.  You needn't bother
to prove these statements--just accept them as the easiest way to describe polygons in any plane.
When we move from R^2 to R^n for n>2, the L_i's become hyperplanes and we get a convenient
description of N-sided n-dimensional polyhedra.  It should be obvious that your description
of bump functions for plane polygons extends immediately to a description of bump functions
for n-dimensional polyhedra.

        3.  Next Tuesday we'll show that Lebesgue measure lambda_n is relatively Phi-invariant
for every invertible affine map Phi(x) = a +L(x).  Here a is any fixed point in R^n and L is any invertible linear map.   We'll show that lambda_n(Phi(A)) =|det L| lambda_n(A) for every Borel set A.
Suppose G is a subgroup of the group of all invertible affine maps, U is an open subset of R^n
mapped into itself by each member of G, and w is a continuous function from U to the positive
real numbers for which w(Phi(x)) = |det (linear part of Phi)| w(x) for each Phi in G and all x in U
Let mu be the measure on the Borel subsets of U for which mu is absolutely continuous with
respect to the restriction of lamba_n to Borel subsets of U with 1/w the Radon-Nikodym derivative of mu with respect to lambda_n.  Show that mu is Phi-invariant for all Phi in G.

        4.  (Special cases of 3).  (i)  Construct a regular measure on R\{0} which is invariant under each
multiplicative dilation map M_c(x)=cx, c and x non-zero.
            (ii)  View the non-zero complex numbers as U= R^2 \{(0,0)}.  Then complex multiplication
of z = x+iy by c=a+ib corresponds to applying to (x,y) the real linear map L on R^2 whose matrix has first row [a  -b] and second row [b  a]; hence |c|^2 = a^2 +b^2 is the determinant of L.  Use this to construct a regular measure on U which is invariant under the group of multiplicative "dilations" by non-zero complex numbers.
             (iii)  The group GL(n, R) of invertible n x n real matrices can be identified with the open
subset U of R^{n^2} =={all n x n matrices} defined as all x for which det x isn't zero.  Each x in
U can be regarded as an n-tuple of the n columns of x or as an n-tuple of the n rows of x.  Use this to show that, for each g in GL(n,R), both the linear map L_g(x) =gx and the linear map R_g(x) =xg, x in U, have determinant (det g)^n.  Then go on to construct a regular measure mu on U which is both L_g  invariant and R_g  invariant for each g in GL(n,R).
 

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ASSIGNMENT #11 DUE THURSDAY, APRIL 28

         1.  Let U = {(x,y) in R^2: y is not zero}.  Each (x,y) in U can be identified with
the affine map Phi= Phi_(x,y) on R defined by Phi(t) = x +yt.  This turns U into a group G, the
group multiplication operation being composition of the corresponding affine maps on R. First compute explicitly the left and right group translation operators L_g(h) =gh and R_g(h) =hg for each g in G and discover that they can be described by certain affine maps on U.  Then construct regular measures mu and nu on U for which mu is L_g invariant for all g and nu is R_g invariant for all g.
Note:   It should be clear how one can use 4(iii) in Assigment #10 to generalize  to a description of left and right invariant measures on the n + n^2 dimensional group of all invertible affine maps on R^n.  Don't bother to write  down this generalization.  In general, a left (respectively, right) Haar measure on a topological group is a measure mu (respectively, nu) which is defined and is finite on compact sets  and which is invariant under all left (respectively, right) translation operators.  With the mild assumption that there are neigborhoods of the identity whose closure is compact and that
the group is a countable union of compact sets, one can show that left and right sigma-finite
Haar measures exist and that each is unique up to a positive scalar multiple.  The above problems
illustrate how to construct Haar measures for groups which can be parametrized by open sets
in R^n for some n.

          2.   With X the set of positive real numbers and Y the unit sphere S^(n-1) about the origin in R^n,  the Cartesian product X x Y can be identified with  U = R^n\{0}, i.e. for any q in U,
q=rp with r the Euclidean distance from 0 to q and p a point on S^(n-1). In Calculus II and III, it's customary to exploit this by saying that polar coordinates (r,theta) express the area element in R^2 by rdrd(theta) while spherical coordinates (r,phi,theta) express the volume element in R^3 by
r^2 sin(phi) dr d(phi) d(theta).   Both of these statements are special cases of a general statement that lamba_n may be viewed as the product of a measure mu on X  with dmu/d(lamba_1) (r) = r^{n-1} and a measure sigma_{n-1} on S^{n-1} called Lebesgue spherical measure.
        (i)  Prove this statement in the following way.  S^(n-1) is the union of 2n hemispheres each of which is the graph of a smooth function from the unit ball in R^(n-1) into R^n.  Alternatively, one can regard S^(n-1) as a finite union of sets each of which admits an analog of spherical coordinates.  This means that there are lots of ways to describe S^(n-1) as the union of the ranges of  finitely
many maps Phi(t)==Phi(t_1,t_2,...,t_(n-1)from open sets in R^(n-1) into S^(n-1) for which the maps Psi(r,t) =rPhi(t) are continuously differentiable with non-vanishing Jacobians det(d Psi).  It's best to AVOID choosing specific Phi's.  Instead, show that for EVERY choice of Phi,
dPsi = r dPhi + Phi dr with (dPhi_t)(v) perpendicular to Phi_t for every t in the domain of Phi
and every vector v in R^(n-1).  Deduce from this that the matrix of (d Psi)_(r,t)  relative to
the standard basis in R x R^(n-1) and an orthonormal basis of R^n whose first member is Phi(t)
is block diagonal and |dPsi_(r,t)| = r^(n-1) |det (dPhi_t)|.  By the change
of variable theorem, this means that (lambda_n) composed with Psi is the product of measures r^(n-1) dr on R and mu_Phi on R^(n-1)where |det (dPhi_t)| is the Radon Nikodym derivative of mu_Phi with respect to Lebesgue measure on R^(n-1).  Finally, define  sigma_{n-1} be the measure on S^{n-1} which, for each choice of Phi, corresponds to mu_Phi via Phi on the range of Phi.   Why does this argument make sigma_{n-1} well defined?
       (ii)  Accepting the above decomposition of lambda_n into the product of mu and sigma_(n-1),
why does it follow that sigma_(n-1) is a measure invariant under the group of orthogonal linear
operators (each of which takes S^(n-1) onto itself)?

3.  Let f be a real-valued function on a closed, bounded rectangle R in R^n.  In class, we showed
that, if f is bounded and the subset N of R on which f is not continuous is a Lebesgue null set,
then f is Riemann integrable.  We also know that Riemann integrals are never defined for unbounded functions.  Complete the characterization of Riemann integrals by showing that, if  f is bounded and
N is NOT a Lebesgue null set, then f is NOT Riemann integrable on R.  Do this by justifying the following assertions:
        (i)  N is the union of the sequence of sets N_k, k>= 1,  consisting of points p
in R for which every open ball about p contains points x' and x'' which lie in R and for which
f(x') - f(x'') > 1/k. (Just use the definition of continuity).
        (ii)  We can fix a k for which N_k is not a Lebesgue null set.
        (iii)  For every partition of R into subrectangles R_i, we can pick selection rules * and **
picking out certain points (x_i)^* and (x_i)^** in R_i with f(x_i)^* - f(x_i)^** equal to zero
if the intersection of the interion of R_i with N_k is empty and otherwise >1/k.
         (iv)  Let S(f,*) and S(f,**) be the Riemann sums of f for the given partition and selection
rules defined in (iii).  Then S(f,*) -S(f,**)  > 1/k(outer Lebesgue measure of N_k) = fixed positive constant. (here you want to use subadditivity of outer Lebesgue measure
and the fact that boundaries of rectangles have zero Lebesgue measure)
          (v)  Since (iv) holds for every partition of R, f isn't Riemann integrable on R.
Note:  The subadditivity property of Lebesgue outer measure is all we need.  This allows
us to avoid making any measurability assumptions about f and any fussiness over whether N
is a Lebesgue measurable set.

__________________________________________________________________________________________________________________________

STUDY GUIDE FOR THE FINAL EXAM (closed book exam on Tuesday May 8, 6:00 p.m.-8:00 p.m.)

1.  Outline the course in a few pages.  You shouldn't need to include definitions in your outline or easy lemmas, just the statements of the major theorems.  Then study your outline until you've memorized
it; as long as your outline is reasonably concise, memorization won't take long.

2.  Go over the outline again, this time pausing to ask yourself what the key ideas are in the proofs
of the hard theorems (LMCT,Riesz-Fischer,  Egoroff, Hahn-Jordan, Radon-Nikodym/Lebesgue Decomposition, Riesz Representation,  Caratheodory, Fubini) and how the entire proof goes with the straightforward theorems (Continuity property of measures, Fatou's Lemma and LDCT assuming the LMCT,  Young, Hoelder, and Minkowski inequalities ,  applications of Caratheodory theorem to characterization of measurable sets for what we called continuous regular Borel measures on R^n, density of smooth functions in L^p for such measures when p is finite,  invariance properties of Lebesgue measure, lemma on existence of non-measurable sets).  The first time you do this, you'll probably want to have scratch paper at hand to jot things down, going back to your notes or the text if you get stuck.  The second or third time, you shouldn't have to write anything down.  If your description of key ideas in a hard theorem takes more than a third of a page to write out, you've gotten into too much detail.

3.  Rapidly go over the EASY problems in each chapter of Bartle, concentrating on the easy problems appearing in the above homework assignments.  Don't fuss about hard problems and don't waste any time studying solutions to the tough take-home exam problems.

4.  The Final Exam will consist of asking you to state a few results (hence #1),  very briefly
discuss the key ideas in one of the hard theorems and give the proof of one of the straightforward
theorems (hence #2), then pick some easy problems from a list to work out (hence #3).