| Problem Set |
Chapter |
Problems |
|---|---|---|
| 23 | ∗34 | For this problem, we will write A ~ B whenever d(A, B) (as defined in L.7 and L.5) is zero. Show that ~ is an equivalence relation on the power set of Rp. That is, show that, for all subsets A, B and C of Rp: (i) A ~ A, (ii) A ~ B implies B ~ A, and (iii) A ~ B and B ~ C imply A ~ C. |
| ∗35 | Suppose that (Bn) is a sequence of finitely μ-measurable sets, and that d(Bn, A) → 0. Show that A is finitely μ-measurable. | |
| ∗36 | Let K ⊂ Rp be compact. Show that if μ*(K) = 0, then the upper (Jordan) content of K is 0 as well. | |
| ∗37 | Recall that, for an object F (like a fractal) that displays self-similarity (in the sense that N copies of F may be put together to form a larger version of F, one whose lengths are magnified M times from the original F) we take the similarity dimension of F to be the number d which satisfies the equation M d = N. Find the similarity dimension of the Cantor middle-thirds set. | |
| n | manditory problem |
| ∗n | manditory problem, not from text |
| (n) | helper problem |
| [n] | ungraded problem |
| [∗n] | ungraded problem, not from text |
| {n} | optional problem |