This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.

I have a rough understanding of the answer; however, I'm unsure of the measures of $A$ and $B$.

Question: What is the measure of $A$ and $B$? If both measures are $1/2$, how do we change the answer so the measures are positive but unequal?

This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.

(In this case, I want to find $\lim_{t\to\infty} \lambda(A\cap [-t,t])/(2t)$ and $\lim_{t\to\infty} \lambda(B\cap [-t,t])/(2t)$, where $\lambda$ is the Lebesgue measure restricting the Lebesgue outer measure $\lambda^{*}$ on sets measurable in the Caratheodory sense.)

I have a rough understanding of the answer; however, I'm unsure of the measures of $A$ and $B$.

Question: What is the measure of $A$ and $B$? If both measures are $1/2$, how do we change the answer so the measures are positive but unequal?