I have a three part question i'm trying to work through.
First part is in title,
(a) Suppose that there exists a measurable set E s.t. $m(E)=1$, and from this show then there then must be a measurable $F \subset E$ s.t. $m(F)=1/2$. [Hint: consider the function $f(x) = m(E \cap (-\infty,x])$
For this part, I want to invoke the intermediate value theorem to show that there must be an x for which this function has measure $1/2$, although i'm not sure if I'm allowed to do that.
(b) There is a closed set $F$, consisting entirely of irrationals, such that $F \subset E$ and $m(F)=1/2$
For this part, my first instinct would be to use the same $F$ that I found in part (a) and then remove all the rational numbers from it, this set will still have a measure of one half because I only removed a countable subset from the original set. As far as gaurenteeing that this set is closed, I haven't the foggiest idea.
(c) There is a compact set $F$ with empty interior such that $F \subset E$ and $m(F)=1/2$
I think this part will become more clear once I have a better understanding of part (b), although gaurenteeing that the compact set $F$ that i'm looking for has an empty interior is a bit intimidating. Definitely going to need you're guys's help on this one!!