# Find $A\subseteq [0,1]$ such that $\lim\limits_{\varepsilon\to0}\frac{m(A\cap[0,\varepsilon])}{\varepsilon}=\frac{1}{2}$

Find a measurable set $$A\subseteq [0,1]$$ such that $$\lim\limits_{\varepsilon\to0}\frac{m(A\cap[0,\varepsilon])}{\varepsilon}=\frac{1}{2}$$

I'm also interested in a set $$B\subseteq [0,1]$$ such that $$\forall\alpha\in(0,1)\quad \frac{m(B\cap[0,\alpha])}{\alpha}=\frac{1}{2}$$

## My Attempt

I thought to try constructing:

1. Fat Cantor set

This does not answer the second question, and I'm not sure if $$0$$ is a density point for fat cantor set, or not.

1. Splitting $$[0,1]$$ by the binary expansion of every element.

Maybe something like:

$$\forall x\in[0,1]$$ define $$i(x)$$ to the the first index in the binary expansion of $$x$$ in which $$1$$ appear.

Then, $$x\in A\iff \sum\limits_{j=i(x)}^{2i(x)} x_j = 1 \mod 2$$

Intuitively, It seems that $$m(A)=m([0,1]\setminus A)$$, but I'm not sure how to prove it formally.

Thanks in advance for any help.

A hint for the first part: let $$\{x_k\}\downarrow 0$$ such that $$x_0=1$$ and $$x_k>x_{k+1}$$ for all $$k\in \Bbb N_{\geqslant 0}$$, also choose some $$r\in [0,1]$$. Now set $$m_k:=r\ell _k$$ for $$\ell _k:=x_k-x_{k+1}$$, also we set $$E_k:=[x_{k+1},x_{k+1}+m_k)$$. Then $$|E_k|=m_k$$, and setting $$E:=\bigcup_{k\geqslant 0}E_k$$ we found that $$|E\cap [0,x_k)|/x_k=r$$ for all $$k$$. Pick $$r=1/2$$ in your case and try to see how to choose $$(x_k)$$ to make the density of $$E$$ at zero equal to $$1/2$$.

A hint for the second part: use the Lebesgue differentiation theorem to show that it cannot be possible.

• Thanks for the answer! I didn't understand the sentence "try to see how to choose $(x_k)$ to make the density of $E$ at zero equal to $1/2$". I thought you proved that $$|E\cap [0,x_k)|/x_k=r\text{ for all k}$$ regardless of the choice of $(x_k)$ as long as $\{x_k\}\downarrow 0$ Feb 9, 2020 at 11:58
• @TheHolyJoker because you need to show that the limit exists. Just a sequence is not enough to show that the limit exists. You need to see what values have $|E\cap [0,t)|/t$ for any chosen $t$ in between of the sequence to figure out how this sequence must be to ensure that the limit exists Feb 9, 2020 at 12:00