Exercise 7.6.9

Given $D$ has measure zero, and since $D^{\alpha }\subseteq D$, we also know that $D^{\alpha }$ has measure zero. We can thus construct a countable open cover $\{H_1,\dots \}$ such that ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=1}^{\infty }|G_n|<\frac{𝜖}{4M}$.

Since $D^{\alpha }$ is closed, we can find a finite subcover $\{I_1,\dots ,I_P\}$. Finally, since this is a finite set, we can merge any overlapping intervals (which can only decrease the total length of the intervals), leaving us with the desired finite collection of disjoint open intervals.