Exercise 33 (Real Analysis - Folland):
"There exists a Borel set $A \subset [0,1]$ such that $0 < m(A \cap I) < m(I)$ for every subinterval $I$ of $[0,1]$. (Hint: Every subinterval of [0,1] contains Cantor-type sets of positive measure)."
Here is an answer from Rudin ("Walter Rudin (1983) Well-Distributed Measurable Sets, The American Mathematical Monthly, 90:1, 41-42"):
"Let $I=[0,1]$, and let CTDP mean compact totally disconnected subset of $I$, having positive measure. Let $\langle I_n\rangle$ be an enumeration of all segments in $I$ whose endpoints are rational.
Construct sequences $\langle A_n\rangle,\langle B_n\rangle$ of CTDP's as follows: Start with disjoint CTDP's $A_1$ and $B_1$ in $I_1$. Once $A_1,B_1,…,A_{n−1},B_{n−1}$ are chosen, their union Cn is CTD, hence $I_n \setminus C_n$ contains a nonempty segment $J$ and $J$ contains a pair $A_n,B_n$ of disjoint CTDP's. Continue in this way, and put
$$A=\bigcup_{n=1}^{\infty}A_n.$$
If $V \subset I$ is open and nonempty, then $I_n \subset V$ for some $n$, hence $A_n \subset V$ and $B_n \subset V$. Thus
$$0 < m(A_n) \le m(A \cap V) < m(A \cap V)+m(B_n) \le m(V);$$the last inequality holds because A and Bn are disjoint. Done. "
I have read Rudin's solution for this exercise but I don't know how to construct two sequences $\langle A_n\rangle,\langle B_n\rangle$ as well as understand this answer. Moreover, I have been trying to construct a Borel set satisfies exercise's request. In detail, I begin to construct a Cantor type set with positive measure. I choose a sequence $\{a_n\}_{n=1}^{\infty}$ such that $$a_n = \dfrac{(1-2^{-3})2^{n-1}}{3^{n}},$$ for all $n \ge 1$.
I remove an open interval $\Delta_{11}$ with the same midpoint as $[0,1]$ and length equal to $a_1$. Then, from each of the remainder, I continue to remove an open interval with midpoint same as the remainder and length equal to $\dfrac{a_2}{2}$, namely $\Delta_{21}$ and $\Delta_{22}$. Obviously, $m(\Delta_{21})=m(\Delta_{22})=\dfrac{a_2}{2}$. Continue with this process. Set$$A=[0,1]\setminus \Big(\displaystyle\bigcup_{n=1}^{\infty}\displaystyle\bigcup_{i=1}^{2^{n-1}}\Delta_{ni} \Big).$$So, $A$ is a Borel set and $m(A)=1-\displaystyle\sum_{n=1}^{\infty}a_n=\dfrac{1}{2^3}$.
However, I don't know how to continue from that point. Please give me some hints regard this exercise.
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