Hey all, can I have a quick check on this?
Problem: Construct a totally disconnected compact set such that where denotes the Lebesgue measure.
Solution Sketch: Let be an enumeration of the rationals. To each rational associate an open interval . Take and . Clearly is compact, while
(aside: it's clear I can get this as close to 1 as I like if I choose my intervals differently)
so . is totally disconnected because it contains no rationals, while connected subsets of with more than 1 point contain rationals (intermediate value property of connected subsets of R).
Thanks for your help. I'm studying this stuff on my own, so while I feel okay about a lot of problems I have no one to check my work.
Since that is okay, there is another part of this question, actually that I wouldn't mind getting verification on:
Problem, Part 2: Let denote the characteristic function of and let be lower semicontinuous. Show that, actually, so that cannot be approximated from below as in the Vitali-Carathedory Theorem.
Solution Sketch: It suffices to show is empty. Suppose otherwise, i.e. there is an . By lower semicontinuity is open; thus there is an such that . Because the rationals are dense, there exists satisfying . Thus, ; this contradicts the fact that because by our construction of .