# What is the measure of two sets which partition the reals into subsets of positive measure?

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?

• How could both measures be $1/2$? Doesn't $\mathbb R$ have measure $\infty$? What measure are you using?
– bof
Aug 4 at 0:43
• Given a number $\varepsilon\gt0$ we can constract an $F_\sigma$ set $A\subset\mathbb R$ so that (i) for every nonempty open set $U\subseteq\mathbb R$, both $U\cap A$ and $U\setminus A$ have positive Lebesgue measure;, and (ii) the set $A$ has Lebesgue measure less than $\varepsilon$.
– bof
Aug 4 at 0:48
• @bof I made edits. Aug 4 at 0:53

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), and we can make $$A$$ have arbitrarily small measure on that one interval, which is what bof refers to in his comment.