This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.

0. During lecture I have sometimes skipped some arguments or not given as much detail as you may have wanted. If there was a result that in particular required of you some effort to complete in detail, please state it here and show me how you filled in the gaps left in lecture. Also, if there is a result for which you do not see how to fill in the details, let me know as well, as I may have overlooked something and it may be worth going back over it in class.

1. Give an example of a bounded set for which

does not exist.

2. Compute .

3. From the book, solve exercises 1.1.3, 1.1.5, 1.1.6, and 1.1.15.

[To get you started on 1.1.3: First verify in that assigns value 0 to any point. For this, use monotonicity and translation invariance, arguing first that for any . Then find that in terms of , and use this to find for any box with rational coordinates. Use this to compute for any box, and conclude by analyzing arbitrary elementary sets.

Note we essentially solved 1.1.15 in class, but under the assumption that 1.1.6 holds.]

4. From the book, solve Exercises 1.1.7-10. Make sure to explain in 1.1.9 why Tao’s definition of compact convex polytopes coincides with what should be our intuitive definition. Please also verify that convex polytopes are indeed convex.

(For a nice argument verifying that indeed , at least for even values of , see the paper “On the volumes of balls” by Blass and Schanuel, available here.)

5. From the book, solve exercise 1.1.11.

(If you are not comfortable with linear algebra beyond size , at least argue in the plane and in .)

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A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

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No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]

For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.

(I have added a missing prime in the hint on question 3.) Thanks to Tara for noticing it.