This set is due Feb. 29 at the beginning of lecture. Let me know if more time is needed or anything like that. Problem 4 was incorrect as stated; I have fixed it now. Thanks to Tara Sheehan for bringing the problem to my attention.
1. We have used repeatedly that if is Riemann integrable, then it is bounded. We briefly discussed this in class, but I did not give a proof. Please prove it.
2. Recall that Cantor’s middle third set is defined as where and, for each , is the result of removing from each of the closed intervals making up their (open) middle third, so
etc. Show that is a Jordan measurable set of measure 0.
3. Solve exercise 1.1.12 from the book. To be specific, if is Jordan null and , you need to prove that is also measurable, and its measure is 0. (In particular, every subset of the Cantor set is Jordan measurable.)
4. Modify the construction of the Cantor set as follows: Let be a constant, . Define as , where and, for each , is the result of removing from each of the closed intervals making up their (open) middle th portion, so each of these intervals is replaced with where and (In particular, .)
Also, for , define so that at stage we remove from each interval its middle portion of length (and if the interval has length less than , then we remove it).
Check that all are Jordan measurable of measure 0, and find (with proof) all the values of for which is Jordan measurable. (If proceeding in this generality is not possible, at least prove that is not Jordan measurable.)
On the other hand, all the sets are Lebesgue measurable, as we will see in lecture.
5. Suppose that are continuous functions, and that uniformly. Suppose that is continuous. Prove that
6. Suppose that is continuous, and that for all Prove that is the constant zero function. This is a theorem of Hausdorff.
What if we are only given that the integrals vanish for integers ?
The quantities are called moments. They are particularly useful in probability theory and in physics. (For example, in classical mechanics, if for all , one thinks of as a mass density, and of the zero-th moment as its associated mass; the quantity is the center of gravity, and is the moment of inertia about 0.)
7. Hausdorff’s moment problem asks whether the moments of a continuous uniquely determine . If is defined on a bounded interval, the answer is yes, by the result from the previous problem. But this is no longer the case if we consider functions defined on unbounded intervals (where, of course, the integrals are now interpreted as improper integrals). For a specific example, define by
Show that for all
Hint: Let . Show that
Conclude that is real. Consider its imaginary part and, by an appropriate change of variables, deduce that all the moments
8. A trigonometric series is a series , where the series is understood as
Such a series is a Fourier series iff there is a function periodic of period and integrable on the interval such that for all integers . Not all trigonometric series are Fourier series; for example, is not a Fourier series. (You do not need to prove this.)
Suppose that is continuous and periodic of period . Show that as . This is a particular case of the Riemann-Lebesgue theorem.
(Cantor used this result in 1870 to prove that if for all , then for all . Set theory was born of his research in generalizations of this result.)
Extra Credit Problem.
Suppose that is a trigonometric series, and that the are bounded. The result of formally integrating twice is the Riemann function of , defined by
where the means that is excluded. Show that is continuous (though in general, it is no longer periodic). One would expect that exists and equals , but this is in general not true. However, one can prove a variant of this:
Define the second symmetric derivative of a function by
Prove that if exists, then so does and . Give an example where exists but does not.
Prove Riemann’s first lemma: If is a trigonometric series with bounded coefficients (that converges at the point ) then exists and equals .
For much more on this topic, see for example
- Alexander Kechris, Set theory and uniqueness for trigonometric series, 1997,
available at Kechris’s webpage.