What examples are known of a dense and co-dense set of half measure?

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 $\Bbb I=[0\,\pmb,\,1]$, we could have respectively $\Bbb I\cap\Bbb Q$ or $\Bbb I\setminus\Bbb Q$. What I am looking for is a dense and co-dense subset $A\subset\Bbb I$ such that $$\operatorname{m}(A)=\operatorname{m}(\Bbb I\setminus A)=\tfrac12.$$ I have attempted this task sequentially by, ever more finely, nibbling holes out of subintervals of $\Bbb I$ 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.

Solution 1:

You can take $A=\left[0,\frac12\right]\cup\left(\left[\frac12,1\right]\cap\Bbb Q\right)$.

Solution 2:

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