# What is the Lebesgue density of $A$ and $B$ which answers a previous question?

Suppose we partition the reals into two sets $A$ and $B$ that are dense (with positive Lebesgue measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$, where Lebesgue measure $\lambda$ restricts outer measure $\lambda^{*}$ to sets measurable in the Caratheodory sense.

For $A$ and $B$ which answers this question, if $\mathbf{B}_{\varepsilon}(x)$ is a closed ball of radius $\varepsilon$ centered at $x\in\mathbb{R}$, what is: $$d_{\varepsilon}(x)=\frac{\lambda(A\cap \mathbf{B}_{\varepsilon}(x))}{2\varepsilon}$$ and $$d_{\varepsilon}^{\,*}(x)=\frac{\lambda(B\cap \mathbf{B}_{\varepsilon}(x))}{2\varepsilon}$$ in terms of $\varepsilon$ and $x$?

• I accepted your question so we can communicate here. Do you want to post this question also in MSE?

• @Paul Yes, please do so.

• Done: https://math.stackexchange.com/questions/4750841/what-is-the-lebesgue-density-sets-a-and-b

• You forgot to include a link to the first question (include the first one on math stack and the link to this question).

• Here's the link to the first question: https://math.stackexchange.com/q/4750001/125918

• You can also include this post.

• Done. But it it the link to the question which is currently closed.

• It will reopen eventually

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