Sorry that the question title is unclear, I didn't know how to ask it.
Take set $A \subseteq [0,1]$, measurable. Does there exist a sequence $x_1,x_2,\dots$ such that $\forall x_i$,
\begin{align*} \displaystyle\frac{\mu\left([x_i,x_{i+1}]\cap A \right)}{\mu\left([x_i,x_{i+1}]\right)} \in \{0,1\} \end{align*}
where $\mu$ is the Lebesgue measure?
No, this is not always possible, and in fact it can fail in an extreme way: we can have Lebesgue measurable (even Borel!) $A$ such that both $A$ and the complement of $A$ have positive-measure intersection with every nontrivial interval. For such an $A$ and any points $x_1< x_2$, we will never have $${\mu([x_1, x_2]\cap A)\over \mu([x_2,x_1])}\in\{0, 1\}.$$ See Construct a Borel set on R such that it intersect every open interval with non-zero non-"full" measure, and also Construction of a Borel set with positive but not full measure in each interval.
