Wisniewool

2022-07-03

All examples of a dense and co-dense set I have seen are either of full Lebesgue measure or of measure zero. For instance, in restriction to the unit interval $\mathbb{I}=\left[0\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1px}{0ex}},\phantom{\rule{thinmathspace}{0ex}}1\right]$, we could have respectively $\mathbb{I}\cap \mathbb{Q}$ or $\mathbb{I}\setminus \mathbb{Q}$. What I am looking for is a dense and co-dense subset $A\subset \mathbb{I}$ such that
$\mathrm{m}\left(A\right)=\mathrm{m}\left(\mathbb{I}\setminus A\right)=\frac{1}{2}.$
I have attempted this task sequentially by, ever more finely, nibbling holes out of subintervals of $\mathbb{I}$ and partially back-filling the previously created holes. It's easy to approach half measure at each step, but I can't see how to to get convergence.

conveneau71

Expert

You can take $A=\left[0,\frac{1}{2}\right]\cup \left(\left[\frac{1}{2},1\right]\cap \mathbb{Q}\right)$

Ciara Mcdaniel

Expert

Let $C$ be a fat Cantor set with measure 1/2. Set $A=C\cup \mathbb{Q}\cap \left[0,1\right]$ and you're done.