- Assignment 1 (Due Aug 30). The statements of the following
problems can be found here.
- To turn in: Billingsley, exercises 2.5(b), 2.9, 2.11.
- Suggested: Billingsley, exercises 2.3, 2.4, 2.7, 2.8, 2.10.

- Assignment 2 (Due Sep 6).
- To turn in: Billingsley, exercises 2.13, 10.2.
- Suggested: Billingsley, exercises 2.17, 10.1, 10.3.

- Assignment 3 (Due Sep 13).
- To turn in: Billingsley, exercises 3.11, 3.12(b–c).
- Suggested: Billingsley, exercises 3.13.

- Assignment 4 (Due Sep 20).
- To turn in: Billingsley, exercises 2.19, 12.9 (hints)

- Assignment 5 (Due Sep 27).
- To turn in: Billingsley, exercises 13.1, 13.8.
- Hint for 13.8: Show that {x: f(x) < t} is open for all real t. If your solution is more than just a few lines, you're working too hard. BTW, where Billingsley says "for each ε" he means "for each ε > 0."

- Suggested: Billingsley, exercises 12.1, 12.11, 13.6.
- Hints for 12.1: For the linear (i.e., 1-dimensional)
case, consider the function f(x) = μ((0, x]) and
use Cauchy's equation from Appendix A20 of
Billingsley.
The k-dimensional case can be done using induction on k. To go from k to k+1 dimensions, let B be a fixed linear Borel set, and consider μ(A x B), as A varies over the k-dimensional Borel sets. You can show that this defines a translation invariant measure on the k-dimensional Borel sets, and by the induction hypothesis, this measure must be a constant multiple of k-dimensional Lebesgue measure, with the constant α(B) possibly depending on B.

Next show that as B varies over the linear Borel sets, α(B) defines a translation invariant measure, which implies that α(B) is a constant multiple of Lebesgue measure on the real line. For this note that α(B) = μ(A x B) where A is a k-dimensional Borel set with Lebesgue measure 1 (you should be able to write down at least one such set).

Now put these two pieces together to show that the μ-measure of a k+1 dimensional rectangle is a constant multiple of its volume, and invoke the uniqueness theorem for σ-finite measures.

- Note: In exercise 12.11, the claim that C supports the corresponding measure is false without the additional hypothesis that G is continuous, so you must assume continuity for (only) this part of the problem; try to find a counterexample in the discontinuous case.

- Hints for 12.1: For the linear (i.e., 1-dimensional)
case, consider the function f(x) = μ((0, x]) and
use Cauchy's equation from Appendix A20 of
Billingsley.

- To turn in: Billingsley, exercises 13.1, 13.8.
- Assignment 6 (Due Oct 4)
- To turn in: Billingsley, exercises 13.9, 13.10.
- Suggested: Billingsley, exercises 13.11, 13.14,
14.1, 14.3, 14.4.
- In problem 14.3, φ represents the quantile
function of F. You should probably use the notation
from class for this instead, i.e, F
^{−1}instead of φ.

- In problem 14.3, φ represents the quantile
function of F. You should probably use the notation
from class for this instead, i.e, F

- Assignment 7 (Due by noon on Oct 19).
- To turn in: Billingsley, exercises 16.4, 16.8.
- Suggested: Billingsley, exercises 16.2.

- Assignment 8 (Due by noon on Oct 26).
- To turn in: Billingsley, exercises 17.6, 17.8.
- Hint for 17.8: Use symmetry and Billingsley's equation (17.8) for the first part.

- Suggested: Billingsley, exercises 17.5, 17.7.

- To turn in: Billingsley, exercises 17.6, 17.8.
- Assignment 9 (Due Nov 1)
- To turn in: Billingsley, exercises 32.6, 32.9.
- In part (a) of exercise 32.6, because dν/dρ is uniquely defined only up to ρ-a.e. equality, the statement dν/dρ = (dν/dμ) × (dμ/dρ) should be interpreted to mean that the right-hand side is a "version" of dν/dρ. The easiest and most clearly correct way to show this is to show that the right-hand side satisfies the operational definition of dν/dρ, i.e., that it is indeed a density for ν with respect to ρ. In fact, this must hold regardless of the versions of dν/dμ and dμ/dρ used in constructing the right-hand side, though this does not really affect the proof in any way. Similar comments apply to parts (b) and (c).
- In problem 32.9, assume only that ν and
ν
^{0}are σ-finite. There is no need to assume that μ and ν are finite. Note, however, that if ν is finite then so is ν^{0}, whereas σ-finiteness of ν does not imply σ-finiteness of ν^{0}. Thus we must also assume that ν^{0}is σ-finite in order to generalize the result to σ-finite ν.

- Suggested: Billingsley, exercises 32.10.
- In problem 32.10, you need only assume that ν and
ν
_{n}are σ-finite. There is no need to assume that the measures involved are finite.

- In problem 32.10, you need only assume that ν and
ν

- To turn in: Billingsley, exercises 32.6, 32.9.
- Assignment 10 (Due Nov 16)
- To turn in: Billingsley, exercises 18.6, 18.13, 32.5.
- Problem 18.6 is straightforward once you prove that "the set on the right is measurable" (meaning that it is contained in the product σ-field). I guess the easiest way to do this is to first show that the functions g(ω, y) = y and h(ω, y) = f(ω) are measurable from the product space into the reals.
- In problem 32.5, it is very easy to see that the
"σ-field of vertical strips" is a
sub-σ-field of the "planar" (i.e.,
two-dimensional) Borel sets. It is then immediate
that the restriction of planar Lebesgue measure to
this σ-field is indeed a measure. You can just
say this without further elaboration. It is also easy
to show that ν, as defined, is a measure on the
σ-field of vertical strips, so please do not
write more than a few lines about this either. (Of
course you should ask me about any of this that is not
completely clear to you).
Having said all that, once you write down the measures assigned by μ and ν to a typical set A × R in the σ-field of vertical strips, it will be easy to see that ν ≪ μ. To finish the problem, it is helpful to know that in order for a real-valued function f on the plane to be measurable with respect to the σ-field of vertical strips, the value of f(x,y) must depend only on x, or more precisely, f must have the form f(x,y) = g(x), where g is Borel measurable. Of course if you use this fact, you should prove it.

- Suggested: Billingsley, exercises 18.1, 18.2, 18.3,
18.4, 18.8, 18.10, 18.14, and
this
problem.
- Problem 18.2 is fairly complicated. Billingsley's hint lays out the basic approach. Near the end you need to recognize that the σ-field generated by a countable partition of a space consists of the class of all unions of sets in the partition (together with the empty set), which is easy to show.
- For problem 18.3, Billingsley's hint together with problem 18.1 makes the problem easy once you recall that the nonmeasurable set contructed in class is not only not a Borel set, it is not Lebesgue measurable either.

- To turn in: Billingsley, exercises 18.6, 18.13, 32.5.
- Assignment 11 (Due Nov 29): Turn in problems 4 and 5 from this sheet.