According to Lebesgue's Density Theorem:
Let $\mu$ be the Lebesgue outer measure, and let $A\subseteq\mathbb{R}$ be a Lebesgue measurable set. Then the limit: $$\lim_{h\to0}\frac{\mu\left(A\cap\left(x-h,x+h\right)\right)}{2h}$$ exists and equals to either 0 or 1, for almost every $x\in\mathbb{R}$.
Is there a known example of a set $A\subseteq\mathbb{R}$ for which the claim above doesn't hold?
If not, Is there a known proof of existance of such a set?
(A set like this, if exists, is clearly non-measurable by the theorem above).
Best Answer
The answer is no.
Following the notation in the question, I will use $\mu$ to denote the Lebesgue outer measure; I will use $|\cdot|$ to denote the Lebesgue measure.
We will use the following standard result:
Sketch of proof: for each $n$, let $A_n$ denote an open superset of $A$ with $\mu(A_n) \leq \mu(A) + \frac1n$. Let $\hat{A} = \cap_n A_n$. This shows that $\hat{A}$ is measurable and $\mu(\hat{A}) = \mu(A)$. Given $M$ measurable we note that $\mu(A) = \mu(A\cap M) + \mu(A\cap M^c)$. Combining $\mu(\hat{A}\cap M) + \mu(\hat{A}\cap M^c) = \mu(A\cap M) + \mu(A\cap M^c)$ and the inequalities $\mu(A\cap M) \leq \mu(\hat{A}\cap M)$ and $\mu(A\cap M^c) \leq \mu(\hat{A}\cap M^c)$ we get the result. QED
Given a set $A$ and a point $x$, denote by $$ \overline{\delta}(A,x) = \limsup_{h \searrow 0} \frac{\mu(A\cap(x-h,x+h))}{2h} $$ and $$ \underline{\delta}(A,x) = \liminf_{h\searrow 0} \frac{\mu(A \cap (x-h,x+h))}{2h} $$
By our standard result we have that $$\overline{\delta}(A,x) = \overline{\delta}(\hat{A},x), \qquad \underline{\delta}(A,x) = \underline{\delta}(\hat{A},x)$$
By Lebesgue differentiation theorem, we have for a.e. $x$ that $\overline{\delta}(\hat{A},x) = \underline{\delta}(\hat{A},x) \in \{0,1\}$. Hence the same holds for $A$.
So how exactly is the case different between measurable and non-measurable sets? For measurable sets $M,N$ we have $\mu(M\cap N) + \mu(M^c\cap N) = \mu(N)$. For a potentially non-measurable $A$ we only have $\mu(A \cap N) + \mu(A^c \cap N) \geq \mu(N)$. That is, the problem with non-measurability is pronounced when you take the complement.
Translated to the current set up, what we have is that for a measurable set, we have
$$ \overline{\delta}(M,x) = 1 - \underline{\delta}(M^c,x) $$
but for potentially non-measurable sets we only have
$$ 1 \leq \overline{\delta}(A,x) + \underline{\delta}(A^c,x) \leq 2 $$
In fact, one can prove the following (before you ask: the underlines are correct, we are using that $\overline{\delta} \geq \underline{\delta}$ trivially):
In fact, if I am not making a mistake, the arguments in the above-linked paper prove the following result, which seems to be the closest positive answer to the question you posed.