Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure Theory textbook. The free online/pdf version is linked at the end of this post.
i). Give an example of a compact set $K \subset \mathbb{R}$ of positive measure such that $m(K \cap I) < |I|$ for every interval $I$ of positive length.
I am given a hint to first construct an open dense subset of $[0,1]$ of measure strictly less than 1.
Here is my approach:
Since $K$ is compact, then it is contained in a countable union of open sets $\bigcup_ {n=1}^{\infty}B_n$. We want this set to be a dense subset of $[0,1]$ with measure strictly less than 1. So consider the set $E= \mathbb{Q}\cap[0,1]$. Then we have that $E=\mathbb{Q}\cap[0,1]=\bigcup_{k=1}^{\infty}[q_k,q_k]$ for some enumeration of the rationals. This implies that $\sum_{k=1}^{\infty}0=0$. Since $K\subset E$, then $m(K\cap I) \leq m(E\cap I) < |I|$.
ii). Give an example of a measurable set $E\subset \mathbb{R}$ such that $0<m(E\cap I)<|I|$ for every interval $I$ of positive length.
For this part, I am given the hint to consider the compliment of the set $K$ above. If so, then won't $K$ be a countable intersection of closed sets?
As always, any help is greatly appreciated, thanks in advance.
Link to the free textbook: http://terrytao.files.wordpress.com/2011/01/measure-book1.pdf
