1. Math
  2. Advanced Math
  3. 1 exercise 329 of pugh this exercise outlines a proof...

Question: 1 exercise 329 of pugh this exercise outlines a proof...

Question details

(1) Exercise 3.29 of Pugh. This exercise outlines a proof that a closed interval [a, b with a 〈 b does not have measure zero. For part(iv) of item (b), use the fact that la,b] is a connected metric space (Corollary 2.48) along with the fact that you can reorder the intervals I, however you want. For part (v), note that an open interval of a, b might be a half-open interval of R, or even a closed interval (for example, [a, c) where a 〈 c 〈 b, or [a,b] itself). Theorem 2.50 might be useful for this part. You may assume without proof that a connected open subset of [a, b is an open interval; this follows from the corresponding result for R, which is a corollary of Exercise 2.31 (a) Finally, if youre not able to show 〈 in part (v), just show-and move on: this will suffice for part (vi).

*2. Prove that the interval [a, b is not a zero set. (a) Explain why the following observation is not a solution to the problem Every open interval that contains [a,b] has length 〉 b-a. (b) Instead, suppose there is a bad covering of [a, b by open intervals {I, whose total length is 〈 b-a, and justify the following steps. (i) It is enough to deal with finite bad coverings (ii) Let B {1.. . . , In} be a bad covering such that n is minimal among all bad coverings (iii) Show that no bad covering has n-1 so we have n 〉 2 (iv) Show that it is no loss of generality to assume a E 11 and 11 「メ1 (v) Show that 1-11 U 12 is an open interval and |1|く|11| + 21 (vi) Show that B- {I,I3, , Im) is a bad covering of [a, b with fewer intervals, a contradiction to minimality of n

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution