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 $x$?
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 $x$?
Bharathk98
91
Answer
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Paul F
200
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According to the same user on Reddit (https://www.reddit.com/r/mathematics/comments/15lzpev/what_is_the_asymptotic_density_and_lebesgue/jvlsx4w/?utm_source=share&utm_medium=ios_app&utm_name=ioscss&utm_content=1&utm_term=1&context=3), they realized the answer is wrong and this answer (https://math.stackexchange.com/a/4747177/125918) needs to be adjusted.
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Paul I found the true answer to the first question here: https://math.stackexchange.com/a/4754256/125918 ; however, I need to answer it with this question.
The answer is accepted.
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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).
What is the link. Please provide it here.
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