I have a set defined by: $D_0 = [0, 1]$.
$D_1$ is obtained from D0 by removing an open interval of length $1/4$ from the middle, so $D_1 = [0, 3/8] \cup [5/8, 1]$.
$D_2$ is obtained from $D_1$ by removing an open interval of length $1/16$ from the middle of each of the $2$ intervals, so $$D_2 = \left[0, \frac{5}{32}\right] \cup \left[\frac{7}{32}, \frac 3 8\right] \cup \left[\frac 5 8, \frac{25}{32}\right] \cup \left[\frac{27}{32}, 1\right]$$
$D_n$ is obtained from $D_{n−1}$ by removing an open interval of length $1/4^n$ from the middle of each of the $2^{n−1}$ intervals. Set $$D = \bigcap_{n=0}^\infty D_n$$
Is $D$ countable? Is $D$ Borel? And is $D$ measurable? And justify your answers.
I think D is uncountable as $D$ becomes the union of closed intervals that is in total length $1/2$. And each interval would have a bijection with the real numbers $[0,1]$ which is uncountable. I think it is Borel as it can be constructed from countable unions and countable intersections of closed sets. I also think it's measurable as I think the Lebesgue measure is $1/2$ but I'm really not sure.
Any help is appreciated, thanks in advance.