This is a follow up to [this question](https://math.stackexchange.com/q/4747105/1176963), 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](https://en.m.wikipedia.org/wiki/Carath%C3%A9odory%27s_criterion).)


I have a rough understanding of the [answer](https://math.stackexchange.com/a/4747177/1176963); 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](https://math.stackexchange.com/a/4747177/1176963) so the measures are positive but unequal?



What is the measure of two sets which partition the reals into subsets of positive measure?
We can make those limits be any two positive numbers that add to $1$, or we can make them nonconvergent. The reason is that in any interval, we can construct $A$ and $B$ on that interval by using the iterated fat-Cantor set construction (as described [here](https://twitter.com/JDHamkins/status/1683929594175401984)), and we can make $A$ have arbitrarily small measure on that one interval, which is what bof refers to in his comment.


By repeating those small-measure sets on successive intervals, or by using large-measure sets as desired on successive intervals, we can control the limit value of your asyptotic measure so as to realize any desired value or a nonconvergent value.


