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 .)

6. From the book, solve exercise 1.1.13.

7. From the book, solve exercise 1.1.17.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Wednesday, January 25th, 2012 at 1:08 pm and is filed under 515: Analysis II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

The paper MR1029909 (91b:03090). Mekler, Alan H.; Shelah, Saharon. The consistency strength of "every stationary set reflects". Israel J. Math. 67 (1989), no. 3, 353–366, that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular. They refer to MR […]

Start with Conway's base 13 function $c $ (whose range on any interval is all of $\mathbb R $), which is everywhere discontinuous, and note that if $f $ only takes values $0$ and $1$, then $c+f $ is again everywhere discontinuous (since its range on any interval is unbounded). Now note that there are $2^\mathfrak c $ such functions $f $: the characteris […]

Yes, there are such sets. To describe an example, let's start with simpler tasks. If we just want $P\ne\emptyset$ with $P^1=\emptyset$, take $P$ to be a singleton. If we want $P^1\ne\emptyset$ and $P^2=\emptyset$, take $P$ to be a strictly increasing sequence together with its limit $a$. Then $P^1=\{a\}$. If we want $P^2\ne\emptyset$ and $P^3=\emptyset$ […]

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]

No, not even $\mathsf{DC}$ suffices for this. Here, $\mathsf{DC}$ is the axiom of dependent choice, which is strictly stronger than countable choice. For instance, it is a theorem of $\mathsf{ZF}$ that for any set $X$, the set $\mathcal{WO}(X)$ of subsets of $X$ that are well-orderable has size strictly larger than the size of $X$. This is a result of Tarski […]

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