Hey all, can I have a quick check on this?

Problem:Construct a totally disconnected compact set $\displaystyle K \subset R$ such that $\displaystyle m(K) > 0$ where $\displaystyle m$ denotes the Lebesgue measure.

Solution Sketch:Let $\displaystyle r_1, r_2, ...$ be an enumeration of the rationals. To each rational associate an open interval $\displaystyle U_i = (r_i - 2^{-i - 2}, r_i + 2^{-i - 2})$. Take $\displaystyle V = \bigcup_{i = 1} ^ \infty U_i$ and $\displaystyle K = [0, 1] \cap V^c$. Clearly $\displaystyle K$ is compact, while

$\displaystyle \displaystyle

m(K) = m([0, 1]) - m([0, 1] \cap V) \ge 1 - m(V)

$

$\displaystyle \displaystyle

\ge 1 - \sum_{i = 1} ^ \infty m(U_i) = 1 - \sum_{i = 1} ^ \infty 2^{-i - 1} = 1/2

$

(aside: it's clear I can get this as close to 1 as I like if I choose my intervals differently)

so $\displaystyle m(K) > 0$. $\displaystyle K$ is totally disconnected because it contains no rationals, while connected subsets of $\displaystyle R$ with more than 1 point contain rationals (intermediate value property of connected subsets of R).