What is the Lebesgue measure of a open interval intersected with a generalized Cantor set with positive Lebesgue measure? In Folland's book Real Analysis: Modern Techniques and Their Applications, p. 39 has an explanation of how to construct a generalized Cantor set with positive measure.
For reference, the construction of the generalized Cantor set involves starting with $K_0 = [0,1]$ removing the open interval of length $\alpha_1$ (for $\alpha_1 \in (0,1)$) centered at the midpoint, and at each step $j$, creating $K_j$ by removing the open middle $\alpha_j^{th}$ from each interval in $K_{j-1}$.
After we construct a generalized Cantor set $K$ with positive measure $\beta \in (0,1)$, I want to know how to calculate the measure of any open interval, call it $V$ intersected with $K$. I am guessing that it is just the length of the $V$ inside $[0,1]$ times $\beta$, and I would like to know how to rigorously show this, assuming that my guess is correct. Thanks.

The guess is not correct. Choose an odd $n=2m+1\in\Bbb Z^+$ large enough so that $\frac1n<\alpha_1$. Then
$$\beta=\sum_{k=0}^{n-1}m\left(K\cap\left(\frac{k}n,\frac{k+1}n\right)\right)\,,$$
so
$$m\left(K\cap\left(\frac{k}n,\frac{k+1}n\right)\right)>0$$
for some $k\in\{0,\ldots,n-1\}$, but
$$m\left(K\cap\left(\frac{m}n,\frac{m+1}n\right)\right)=m(\varnothing)=0,.$$
