1.6

$\dagger$ Note that it was simpler for me to first consider problem 1.7.

Since the Lebesgue measure is translation-invariant, it suffices to only consider balls centered at the origin. Theorem 1.4 gives that $B_1{(0)} = \bigcup_{k=1}^{\infty} Q_k$, where $\left\lbrace Q_n \right\rbrace_{n \in \mathbb{N}}$ is a countable collection of pairwise almost-disjoint cubes. Thus, $$v_d = m(B_1{(0)}) = \sum_{k=1}^{\infty} m(Q_k)$$ By $\dagger$, any dilation of the unit ball in $\mathbb{R}^d$ is equivalent to letting $\delta = (r, r, \ldots, r)$. It follows from the main argument in 1.7 that:

$m(B_r{(0)}) = m(\delta B_1{(0)}) = r^d m(B_1{(0)}) = r^d v_d$, as desired.